AP Precalculus study package

AP Precalculus is designed to be the equivalent of a one-semester, college-level precalculus course. The course deepens students' understanding of functions and their applications, preparing them for calculus and other advanced mathematics courses. A graphing calculator is permitted on the entire exam and is essential for several question types.

Exam Format

Section	Type	# of Questions	Time	Weight
Section I	Multiple-Choice Questions	40	120 minutes	62.5%
Section II	Free-Response Questions	4	60 minutes	37.5%

Total Exam Time: 180 minutes

Multiple-Choice Section

• 40 questions in 120 minutes (3 minutes per question)
• Graphing calculator permitted on ALL questions
• Both conceptual and computational questions
• Includes questions requiring calculator-based analysis (roots, intersections, regression)
• Covers all four units

Free-Response Section

• 4 questions in 60 minutes (15 minutes per question)
• Graphing calculator required for parts of most FRQs
• Questions typically have multiple parts (a, b, c, d)
• May include: function analysis, modeling with functions, solving equations, interpreting results in context
• Partial credit awarded — show all work

Course Units

Unit 1: Polynomial and Rational Functions (30–40% of exam)

Polynomial functions: end behavior, zeros, multiplicity, turning points, the Remainder and Factor Theorems. Rational functions: asymptotes (vertical, horizontal, slant/oblique), holes, domain restrictions, end behavior. Transformations of functions. Inverse functions.

Unit 2: Exponential and Logarithmic Functions (27–40% of exam)

Exponential functions: growth and decay, compound interest, the number e. Logarithmic functions: properties of logarithms, solving exponential and logarithmic equations. Logarithmic scales. Modeling with exponential and logarithmic functions. Inverse relationship between exponential and logarithmic functions.

Unit 3: Trigonometric and Polar Functions (30–35% of exam)

Trigonometric functions: sine, cosine, tangent and their graphs. Radian measure. Unit circle. Trigonometric identities: Pythagorean, double-angle, half-angle. Inverse trigonometric functions. Solving trigonometric equations. Polar coordinates and polar graphs. Parametric functions.

Unit 4: Functions Involving Parameters, Vectors, and Matrices (15–25% of exam)

Parameterized families of functions: how changing parameters affects graphs. Vectors: magnitude, direction, vector operations (addition, subtraction, scalar multiplication, dot product). Matrices: operations (addition, multiplication), determinants, inverse matrices, solving systems using matrices.

Study File Index

File	Description
00-Course-Overview.md	This file
01-Unit1-Notes-Polynomial-Rational.md	Comprehensive notes for Unit 1
02-Unit2-Notes-Exponential-Logarithmic.md	Comprehensive notes for Unit 2
03-Unit3-Notes-Trig-Polar.md	Comprehensive notes for Unit 3
04-Unit4-Notes-Parameters-Vectors-Matrices.md	Comprehensive notes for Unit 4
05-Unit1-Practice.md	Practice problems for Unit 1
06-Unit2-Practice.md	Practice problems for Unit 2
07-Unit3-Practice.md	Practice problems for Unit 3
08-Unit4-Practice.md	Practice problems for Unit 4
09-Full-Exam-Answers.md	Full practice exam with answer key
10-Summary.md	Condensed summary of all key concepts
11-Exam-Strategy.md	Tips and strategies for exam day
12-Audio-Script.md	Script for audio review recording

Calculator Policy

A graphing calculator is permitted and expected on the entire AP Precalculus exam. Recommended calculators include the TI-84 series. You should be comfortable with:

• Graphing functions and finding intersections and zeros
• Creating tables of values
• Using the numerical solver
• Performing regression analysis
• Working with matrices
• Converting between degrees and radians

Approximate Grading Scale

• 5: 60–70% of total points
• 4: 50–60%
• 3: 40–50%
• 2: 30–40%
• 1: Below 30%

---

Polynomial Functions

A polynomial function is a function of the form:

$$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$

where $n$ is a non-negative integer (the degree of the polynomial), $a_n \neq 0$ is the leading coefficient, and each $a_i$ is a real number (a coefficient). The constant term $a_0$ is the y-intercept of the graph.

Polynomials are classified by degree:

• Constant ($n = 0$): $f(x) = a_0$ — a horizontal line
• Linear ($n = 1$): $f(x) = a_1 x + a_0$ — a straight line
• Quadratic ($n = 2$): $f(x) = ax^2 + bx + c$ — a parabola
• Cubic ($n = 3$): $f(x) = ax^3 + bx^2 + cx + d$
• Quartic ($n = 4$)
• Quintic ($n = 5$)

  The domain of any polynomial function is all real numbers, $(-\infty, \infty)$. Polynomial graphs are continuous, smooth curves with no breaks, holes, or sharp corners.

End Behavior

The end behavior of a polynomial function describes what happens to $f(x)$ as $x$ approaches $\pm\infty$. End behavior is determined by the degree and the leading coefficient:

Even degree, positive leading coefficient: Both ends point upward: $f(x) \to +\infty$ as $x \to \pm\infty$. Example: $f(x) = x^2$ (U-shape)

Even degree, negative leading coefficient: Both ends point downward: $f(x) \to -\infty$ as $x \to \pm\infty$. Example: $f(x) = -x^2$ (upside-down U)

Odd degree, positive leading coefficient: Left end points down, right end points up: $f(x) \to -\infty$ as $x \to -\infty$ and $f(x) \to +\infty$ as $x \to +\infty$. Example: $f(x) = x^3$

Odd degree, negative leading coefficient: Left end points up, right end points down: $f(x) \to +\infty$ as $x \to -\infty$ and $f(x) \to -\infty$ as $x \to +\infty$. Example: $f(x) = -x^3$

Memory aid: Odd-degree polynomials have OPPOSITE end behaviors (one up, one down). Even-degree polynomials have SAME end behaviors (both up or both down).

Zeros and Their Multiplicity

A zero (or root or x-intercept) of a polynomial $f(x)$ is a value $c$ such that $f(c) = 0$. If $c$ is a zero, then $(x - c)$ is a factor of $f(x)$.

The multiplicity of a zero is the number of times the corresponding factor appears in the factorization. Multiplicity affects the graph's behavior at the x-intercept:

• Odd multiplicity (1, 3, 5, ...): The graph crosses the x-axis at the zero. If multiplicity is 1, the crossing is a smooth pass-through. If multiplicity is 3 or higher, the graph flattens near the zero before crossing.
• Even multiplicity (2, 4, 6, ...): The graph touches the x-axis and bounces back (turns around) at the zero. The graph does not cross the axis. Higher even multiplicities cause more flattening near the zero.

  Example: $f(x) = (x-2)^3(x+1)^2(x-5)$ has zeros at $x=2$ (multiplicity 3, crosses with flattening), $x=-1$ (multiplicity 2, bounces), and $x=5$ (multiplicity 1, crosses smoothly).

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every polynomial of degree $n$ with real or complex coefficients has exactly $n$ zeros (counting multiplicity) in the complex number system. This means:

• A quadratic ($n=2$) has exactly 2 zeros
• A cubic ($n=3$) has exactly 3 zeros
• A quartic ($n=4$) has exactly 4 zeros

  Some zeros may be complex (non-real), and complex zeros of polynomials with real coefficients always come in conjugate pairs: if $a + bi$ is a zero, then $a - bi$ is also a zero.

The Remainder and Factor Theorems

Remainder Theorem: If a polynomial $f(x)$ is divided by $(x - c)$, the remainder is $f(c)$. This means you can evaluate $f(c)$ by performing synthetic division or direct substitution.

Factor Theorem: $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$. In other words, $c$ is a zero of $f(x)$ if and only if $(x-c)$ is a factor.

These theorems are extremely useful for finding zeros and factoring polynomials.

Turning Points and Local Extrema

A turning point is a point where the graph of a polynomial changes from increasing to decreasing (local maximum) or from decreasing to increasing (local minimum). The graph of a polynomial of degree $n$ has at most $n - 1$ turning points.

• A linear function ($n=1$) has 0 turning points
• A quadratic ($n=2$) has at most 1 turning point (the vertex)
• A cubic ($n=3$) has at most 2 turning points
• A quartic ($n=4$) has at most 3 turning points

Transformations of Functions

Given a function $f(x)$, the following transformations apply:

• $f(x) + k$: Vertical shift UP by $k$ units
• $f(x) - k$: Vertical shift DOWN by $k$ units
• $f(x - h)$: Horizontal shift RIGHT by $h$ units
• $f(x + h)$: Horizontal shift LEFT by $h$ units
• $a \cdot f(x)$: Vertical stretch (if $|a| > 1$) or compression (if $0 < |a| < 1$); reflection across x-axis if $a < 0$
• $f(bx)$: Horizontal compression (if $|b| > 1$) or stretch (if $0 < |b| < 1$); reflection across y-axis if $b < 0$

  Key rule: Horizontal transformations work OPPOSITE to what you expect. $f(x - 3)$ shifts RIGHT (not left), and $f(2x)$ compresses horizontally (not stretches).

Inverse Functions

The inverse function of $f$, denoted $f^{-1}$, reverses the mapping of $f$. If $f(a) = b$, then $f^{-1}(b) = a$.

A function has an inverse that is also a function (the function is one-to-one or invertible) if and only if it passes the horizontal line test — no horizontal line intersects the graph more than once.

To find the inverse algebraically:

1. Replace $f(x)$ with $y$
2. Swap $x$ and $y$
3. Solve for $y$
4. Replace $y$ with $f^{-1}(x)$

   The domain of $f^{-1}$ equals the range of $f$, and the range of $f^{-1}$ equals the domain of $f$.

   Composition property: $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$ for all $x$ in the appropriate domains.

Rational Functions

A rational function is a function of the form:

$$f(x) = \frac{P(x)}{Q(x)}$$

where $P(x)$ and $Q(x)$ are polynomials and $Q(x) \neq 0$. The domain of a rational function is all real numbers except where the denominator equals zero.

Vertical Asymptotes

A vertical asymptote is a vertical line $x = a$ where the function values approach $\pm\infty$. Vertical asymptotes occur at values of $x$ that make the denominator zero, provided the factor does NOT cancel with a factor in the numerator.

If $Q(c) = 0$ and $P(c) \neq 0$, then $x = c$ is a vertical asymptote.

To determine whether the function approaches $+\infty$ or $-\infty$ on each side of the vertical asymptote, test values slightly to the left and right of $c$.

Horizontal Asymptotes

A horizontal asymptote describes the end behavior of a rational function — a horizontal line that the graph approaches as $x \to \pm\infty$.

Compare the degrees of the numerator and denominator:

• Degree of P < Degree of Q: The horizontal asymptote is $y = 0$ (the x-axis). The denominator grows faster, pulling the function toward zero. Example: $f(x) = \frac{1}{x}$
• Degree of P = Degree of Q: The horizontal asymptote is $y = \frac{a}{b}$, where $a$ is the leading coefficient of $P$ and $b$ is the leading coefficient of $Q$. Example: $f(x) = \frac{3x^2 + 1}{2x^2 - 5}$ has horizontal asymptote $y = \frac{3}{2}$
• Degree of P > Degree of Q: There is no horizontal asymptote. Instead, the function has a slant (oblique) asymptote or a polynomial asymptote found by polynomial long division.

Slant (Oblique) Asymptotes

When the degree of the numerator is exactly one more than the degree of the denominator, the rational function has a slant asymptote. Find it by performing polynomial long division and ignoring the remainder.

Example: $f(x) = \frac{x^2 + 3x + 1}{x - 2}$. Divide $x^2 + 3x + 1$ by $x - 2$ to get $x + 5 + \frac{11}{x-2}$. The slant asymptote is $y = x + 5$.

Holes (Removable Discontinuities)

A hole occurs when a factor in the numerator and denominator cancel. The hole is at the x-value that makes the canceled factor zero.

Example: $f(x) = \frac{(x-1)(x+2)}{(x-1)(x-3)}$ has a hole at $x = 1$ (the factor $(x-1)$ cancels) and a vertical asymptote at $x = 3$. To find the y-coordinate of the hole, substitute $x = 1$ into the simplified function: $\frac{1+2}{1-3} = -\frac{3}{2}$. The hole is at $(1, -3/2)$.

Graphing Rational Functions: Step-by-Step

1. Find the domain: Set denominator $\neq 0$
2. Find intercepts: y-intercept by evaluating $f(0)$; x-intercepts by setting $P(x) = 0$
3. Find vertical asymptotes: Set denominator $= 0$ (after canceling common factors)
4. Find horizontal/slant asymptotes: Compare degrees
5. Identify holes: Cancel common factors
6. Test intervals: Use test points in each interval determined by asymptotes and intercepts
7. Sketch the graph: Connect the pieces, approaching (but never touching) asymptotes

Key Formulas and Theorems for Unit 1

Concept	Formula/Rule
End behavior	Determined by degree + leading coefficient
Zeros	Solve $f(x) = 0$; each zero has multiplicity
Remainder Theorem	Remainder of $f(x) \div (x-c)$ equals $f(c)$
Factor Theorem	$(x-c)$ is a factor iff $f(c) = 0$
Max turning points	$n - 1$ for degree $n$
Vertical asymptote	$x = c$ where $Q(c) = 0$ and $P(c) \neq 0$
Horizontal asymptote	deg P < deg Q: $y=0$; deg P = deg Q: $y = a/b$
Slant asymptote	Polynomial division when deg P = deg Q + 1
Holes	Where common factors cancel

---

Exponential Functions

An exponential function has the form:

$$f(x) = a \cdot b^x$$

where $a$ is a nonzero constant (the initial value or y-intercept, since $f(0) = a \cdot b^0 = a$), and $b$ is a positive constant not equal to 1 (the base or growth/decay factor).

Growth vs. Decay

• Exponential growth: $b > 1$. The function increases as $x$ increases. Common examples include population growth, compound interest, and the spread of viruses.
• Exponential decay: $0 < b < 1$. The function decreases as $x$ increases. Common examples include radioactive decay, drug elimination from the body, and depreciation.

The Graph of Exponential Functions

The graph of $f(x) = a \cdot b^x$ has:

• A horizontal asymptote at $y = 0$ (the x-axis)
• A y-intercept at $(0, a)$
• The domain is $(-\infty, \infty)$
• The range is $(0, \infty)$ if $a > 0$, or $(-\infty, 0)$ if $a < 0$
• The function is always positive (if $a > 0$) or always negative (if $a < 0$)
• The function is strictly increasing (if $b > 1$) or strictly decreasing (if $0 < b < 1$)

  The graph passes through $(1, ab)$, since $f(1) = a \cdot b^1 = ab$. This point is useful for verifying your graph.

Transformations of Exponential Functions

Just like polynomial functions, exponential functions can be transformed:

• $f(x) = a \cdot b^x + k$: Vertical shift of $k$ units; horizontal asymptote becomes $y = k$
• $f(x) = a \cdot b^{x-h}$: Horizontal shift of $h$ units (remember: opposite direction)
• $f(x) = -a \cdot b^x$: Reflection across the x-axis

Compound Interest

Exponential functions model compound interest:

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

where:

• $A$ = final amount
• $P$ = principal (initial investment)
• $r$ = annual interest rate (as a decimal)
• $n$ = number of compounding periods per year
• $t$ = time in years

  Continuously compounded interest uses the formula:

  $$A = Pe^{rt}$$

The Number $e$

The number $e$ (Euler's number) is an irrational constant approximately equal to 2.71828. It is defined as the value that $(1 + 1/n)^n$ approaches as $n \to \infty$.

The function $f(x) = e^x$ is called the natural exponential function. It has the unique property that the slope of the tangent line at any point on the graph equals the y-coordinate of that point. This makes $e^x$ the most important function in calculus.

---

Logarithmic Functions

A logarithmic function is the inverse of an exponential function. The logarithm with base $b$ is defined as:

$$\log_b(x) = y \iff b^y = x$$

This means: "$\log_b(x)$ is the exponent to which $b$ must be raised to get $x$."

Properties of Logarithms

1. $\log_b(1) = 0$ because $b^0 = 1$ for any base $b$
2. $\log_b(b) = 1$ because $b^1 = b$
3. $\log_b(b^x) = x$ — the logarithm undoes the exponential
4. $b^{\log_b(x)} = x$ — the exponential undoes the logarithm

Product, Quotient, and Power Rules

These are the three essential properties for simplifying and expanding logarithmic expressions:

Product Rule: $\log_b(xy) = \log_b(x) + \log_b(y)$

Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$

Power Rule: $\log_b(x^n) = n \cdot \log_b(x)$

Important caution: There is NO formula for $\log_b(x + y)$. The sum inside a logarithm cannot be split. Similarly, $\log_b(x) + \log_b(y) \neq \log_b(x + y)$.

Common and Natural Logarithms

• Common logarithm: $\log(x) = \log_{10}(x)$. Base 10, used for pH, sound intensity (decibels), Richter scale.
• Natural logarithm: $\ln(x) = \log_e(x)$. Base $e$, the most important logarithm for calculus.

Change of Base Formula

To evaluate a logarithm with any base using a calculator (which typically only has $\log$ and $\ln$ buttons):

$$\log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log(x)}{\log(b)}$$

The Graph of Logarithmic Functions

The graph of $f(x) = \log_b(x)$ is the reflection of $f(x) = b^x$ across the line $y = x$ (since they are inverses).

• The domain is $(0, \infty)$ (only positive inputs)
• The range is $(-\infty, \infty)$ (all real outputs)
• There is a vertical asymptote at $x = 0$ (the y-axis)
• The x-intercept is at $(1, 0)$ since $\log_b(1) = 0$
• The graph passes through $(b, 1)$ since $\log_b(b) = 1$

  For $b > 1$: the function is increasing. For $0 < b < 1$: the function is decreasing.

Solving Exponential Equations

Strategy 1 — Same base: If both sides can be written as powers of the same base, set the exponents equal.

$2^{3x+1} = 2^{7} \implies 3x + 1 = 7 \implies x = 2$

Strategy 2 — Take logarithms: If the bases are different or cannot be made the same, take the natural log (or common log) of both sides and use the power rule.

$3^x = 7 \implies \ln(3^x) = \ln(7) \implies x \ln(3) = \ln(7) \implies x = \frac{\ln(7)}{\ln(3)} \approx 1.771$

Solving Logarithmic Equations

1. Isolate the logarithm on one side
2. Rewrite in exponential form
3. Solve for the variable
4. CHECK your answer(s) in the original equation — logarithmic equations can produce extraneous solutions (values that work algebraically but are not in the domain of the original logarithm)

   Example: $\ln(x+3) = 2 \implies e^2 = x + 3 \implies x = e^2 - 3 \approx 4.389$. Check: $\ln(4.389 + 3) = \ln(7.389) \approx 2$. ✓

Logarithmic Scales

Logarithmic scales are used when data spans many orders of magnitude:

• pH scale: $\text{pH} = -\log[H^+]$. Each unit decrease in pH represents a 10-fold increase in hydrogen ion concentration.
• Richter scale: Each unit increase represents a 10-fold increase in earthquake amplitude (approximately 31.6-fold increase in energy).
• Decibel scale: $\text{dB} = 10\log(I/I_0)$. Each 10 dB increase represents a 10-fold increase in sound intensity.

Exponential Modeling

When modeling real-world data with exponential functions, use your calculator's regression features:

• Exponential regression: Fits data to $y = a \cdot b^x$. Used when data shows constant percentage change.
• Key indicator of exponential data: The ratio of consecutive y-values is approximately constant. If $\frac{y_2}{y_1} \approx \frac{y_3}{y_2} \approx \frac{y_4}{y_3}$, the data is exponential.

Key Formulas for Unit 2

Concept	Formula
Exponential function	$f(x) = a \cdot b^x$
Compound interest	$A = P(1 + r/n)^{nt}$
Continuous interest	$A = Pe^{rt}$
Logarithm definition	$\log_b(x) = y \iff b^y = x$
Product rule	$\log_b(xy) = \log_b(x) + \log_b(y)$
Quotient rule	$\log_b(x/y) = \log_b(x) - \log_b(y)$
Power rule	$\log_b(x^n) = n \cdot \log_b(x)$
Change of base	$\log_b(x) = \ln(x)/\ln(b)$
Domain of $\log_b(x)$	$x > 0$

---

Radian Measure

Angles can be measured in degrees or radians. One full revolution equals 360° or $2\pi$ radians.

Degrees	Radians
0°	0
30°	$\pi/6$
45°	$\pi/4$
60°	$\pi/3$
90°	$\pi/2$
120°	$2\pi/3$
135°	$3\pi/4$
150°	$5\pi/6$
180°	$\pi$
270°	$3\pi/2$
360°	$2\pi$

Conversion formulas: $$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$ $$\text{degrees} = \text{radians} \times \frac{180}{\pi}$$

Arc length: The length of an arc subtended by angle $\theta$ (in radians) in a circle of radius $r$ is: $$s = r\theta$$

Sector area: The area of a sector with angle $\theta$ (in radians) in a circle of radius $r$ is: $$A = \frac{1}{2}r^2\theta$$

---

The Unit Circle

The unit circle is a circle centered at the origin with radius 1. Every point on the unit circle has coordinates $(\cos\theta, \sin\theta)$, where $\theta$ is the angle measured counterclockwise from the positive x-axis.

Key Coordinates on the Unit Circle

Angle	$\cos\theta$	$\sin\theta$	$\tan\theta$
0	1	0	0
$\pi/6$ (30°)	$\sqrt{3}/2$	$1/2$	$\sqrt{3}/3$
$\pi/4$ (45°)	$\sqrt{2}/2$	$\sqrt{2}/2$	1
$\pi/3$ (60°)	$1/2$	$\sqrt{3}/2$	$\sqrt{3}$
$\pi/2$ (90°)	0	1	undefined
$2\pi/3$ (120°)	$-1/2$	$\sqrt{3}/2$	$-\sqrt{3}$
$3\pi/4$ (135°)	$-\sqrt{2}/2$	$\sqrt{2}/2$	-1
$5\pi/6$ (150°)	$-\sqrt{3}/2$	$1/2$	$-\sqrt{3}/3$
$\pi$ (180°)	-1	0	0
$7\pi/6$ (210°)	$-\sqrt{3}/2$	$-1/2$	$\sqrt{3}/3$
$5\pi/4$ (225°)	$-\sqrt{2}/2$	$-\sqrt{2}/2$	1
$4\pi/3$ (240°)	$-1/2$	$-\sqrt{3}/2$	$\sqrt{3}$
$3\pi/2$ (270°)	0	-1	undefined

Memory strategy for the unit circle:

• In Quadrant I, all values are positive
• In Quadrant II, sine is positive ("All Students Take Calculus" → A(ll), S(ine), T(an), C(os))
• In Quadrant III, tangent is positive
• In Quadrant IV, cosine is positive
• Reflect Quadrant I values across axes to find other quadrants

Trigonometric Functions

Definitions

$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}$$ $$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}$$ $$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x} = \frac{\sin\theta}{\cos\theta}$$

The reciprocal functions: $$\csc\theta = \frac{1}{\sin\theta} \quad \sec\theta = \frac{1}{\cos\theta} \quad \cot\theta = \frac{\cos\theta}{\sin\theta}$$

Graphs of Trigonometric Functions

Sine function $f(x) = a\sin(bx + c) + d$:

• Amplitude = $|a|$
• Period = $\frac{2\pi}{|b|}$
• Phase shift = $-\frac{c}{b}$ (remember: horizontal shift is opposite sign)
• Vertical shift = $d$; midline is $y = d$
• Range: $[d - |a|, d + |a|]$
• Domain: $(-\infty, \infty)$

  Cosine function $f(x) = a\cos(bx + c) + d$:

• Same parameters as sine
• The cosine graph is the sine graph shifted $\pi/2$ to the left

  Tangent function $f(x) = a\tan(bx + c) + d$:

• There is NO amplitude for tangent
• Period = $\frac{\pi}{|b|}$ (half the sine/cosine period)
• Phase shift = $-\frac{c}{b}$
• Vertical asymptotes occur where $\cos(bx + c) = 0$
• Range: $(-\infty, \infty)$

Trigonometric Identities

Pythagorean Identities

$$\sin^2\theta + \cos^2\theta = 1$$ $$1 + \tan^2\theta = \sec^2\theta$$ $$1 + \cot^2\theta = \csc^2\theta$$

Double-Angle Identities

$$\sin(2\theta) = 2\sin\theta\cos\theta$$ $$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$ $$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$$

Note: There are THREE forms of the $\cos(2\theta)$ identity. Choose the one most convenient for your problem. If you know only $\cos\theta$, use $2\cos^2\theta - 1$. If you know only $\sin\theta$, use $1 - 2\sin^2\theta$.

Even/Odd Identities

$$\sin(-\theta) = -\sin\theta \quad (\text{sine is odd})$$ $$\cos(-\theta) = \cos\theta \quad (\text{cosine is even})$$ $$\tan(-\theta) = -\tan\theta \quad (\text{tangent is odd})$$

Sum and Difference Identities

$$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$$ $$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$$

---

Inverse Trigonometric Functions

Inverse trigonometric functions "undo" the trigonometric functions.

• $y = \arcsin(x) = \sin^{-1}(x)$: Domain $[-1, 1]$, Range $[-\pi/2, \pi/2]$
• $y = \arccos(x) = \cos^{-1}(x)$: Domain $[-1, 1]$, Range $[0, \pi]$
• $y = \arctan(x) = \tan^{-1}(x)$: Domain $(-\infty, \infty)$, Range $(-\pi/2, \pi/2)$

  Important: $\sin^{-1}(x)$ means the inverse sine, NOT $1/\sin(x)$ (which is $\csc(x)$).

Solving Trigonometric Equations

To solve trigonometric equations:

1. Isolate the trigonometric function
2. Use inverse trigonometric functions or algebraic methods
3. Find ALL solutions within the specified interval
4. For general solutions, add $2\pi n$ (for sine/cosine) or $\pi n$ (for tangent)

   Example: Solve $2\sin\theta - 1 = 0$ for $0 \leq \theta < 2\pi$. $\sin\theta = 1/2$ $\theta = \pi/6$ and $\theta = 5\pi/6$ (from the unit circle)

Polar Coordinates

A point in polar coordinates is written as $(r, \theta)$, where $r$ is the distance from the origin (pole) and $\theta$ is the angle from the positive x-axis (polar axis).

Conversion to rectangular: $$x = r\cos\theta \quad y = r\sin\theta$$

Conversion from rectangular: $$r = \sqrt{x^2 + y^2} \quad \theta = \arctan(y/x)$$

Polar Graphs

Key polar curves to know:

• Circle: $r = a$ (circle centered at origin with radius $|a|$)
• Cardioid: $r = a(1 + \cos\theta)$ — heart-shaped curve
• Rose curves: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$. If $n$ is odd, $n$ petals. If $n$ is even, $2n$ petals.
• Limaçon: $r = a + b\cos\theta$. If $a/b < 1$, has an inner loop. If $a/b = 1$, it's a cardioid. If $1 < a/b < 2$, it's dimpled. If $a/b \geq 2$, it's convex.

Parametric Functions

In parametric equations, both $x$ and $y$ are expressed as functions of a parameter $t$: $$x = f(t) \quad y = g(t)$$

To convert parametric to rectangular, eliminate $t$ by solving one equation for $t$ and substituting into the other.

To find the slope of a parametric curve: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

Key Formulas for Unit 3

Concept	Formula
Arc length	$s = r\theta$
Sector area	$A = \frac{1}{2}r^2\theta$
Pythagorean	$\sin^2\theta + \cos^2\theta = 1$
Sine graph	$a\sin(bx+c)+d$; Amp=$	a	$, Period=$2\pi/	b	$, Shift=$-c/b$
Double-angle sin	$\sin(2\theta) = 2\sin\theta\cos\theta$
Double-angle cos	$\cos(2\theta) = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
Polar to rectangular	$x = r\cos\theta, y = r\sin\theta$
Rectangular to polar	$r = \sqrt{x^2+y^2}, \theta = \arctan(y/x)$

---

Part A: Functions Involving Parameters

What Is a Parameter?

A parameter is a constant in a function whose value can vary to produce a family of related functions. Unlike a variable (which represents an input), a parameter controls the shape, position, or behavior of the graph itself.

For example, in the function family $f(x) = a(x - h)^2 + k$, the letters $a$, $h$, and $k$ are parameters. Changing any one of them produces a different parabola — but all of them share the same basic quadratic structure.

Linear Function Families

Consider the family $f(x) = mx + b$, where $m$ and $b$ are parameters.

Varying $m$ (the slope): When we fix $b$ and let $m$ change, every function in the family passes through the point $(0, b)$ — the y-intercept. Increasing $m$ makes the line steeper. When $m = 0$, we get a horizontal line $f(x) = b$. When $m$ is negative, the line slopes downward. All lines in this family share the common point $(0, b)$, which is called the locus of the family.

Varying $b$ (the y-intercept): When we fix $m$ and let $b$ change, every function has the same slope but a different y-intercept. The result is a collection of parallel lines. The lines are evenly spaced when $b$ increases by constant increments.

Quadratic Function Families

Consider $f(x) = a(x - h)^2 + k$.

Varying $a$: All parabolas in the family share the same vertex $(h, k)$. A larger $|a|$ produces a narrower parabola (steeper). A smaller $|a|$ produces a wider parabola. When $a$ changes sign, the parabola flips: $a > 0$ opens upward, $a < 0$ opens downward.

Varying $h$: All parabolas have the same shape (same $a$ and $k$) but are shifted horizontally. They all have the same "width" and the same y-coordinate of the vertex.

Varying $k$: All parabolas are shifted vertically. They share the same axis of symmetry $x = h$.

Radical Function Families

For $f(x) = a\sqrt{x - h} + k$:

• The domain is $[h, \infty)$
• The endpoint (starting point of the graph) is at $(h, k)$
• $a$ controls the steepness and direction
• All functions in the family with fixed $h$ and $k$ share the endpoint $(h, k)$

Rational Function Families

For $f(x) = \frac{a}{x - h} + k$:

• The vertical asymptote is always $x = h$
• The horizontal asymptote is always $y = k$
• $a$ controls how quickly the function approaches the asymptotes
• All functions in the family share the same asymptotes
• The center of each hyperbola is at $(h, k)$

Exponential Function Families

For $f(x) = a \cdot b^{x - h} + k$:

• The horizontal asymptote is $y = k$
• The y-value of the "starting point" (at $x = h$) is $a + k$
• $a$ controls the distance from the asymptote to the curve at $x = h$
• $b$ controls the growth/decay rate

Logarithmic Function Families

For $f(x) = a \log_b(x - h) + k$:

• The vertical asymptote is $x = h$
• The x-intercept occurs where $a \log_b(x - h) + k = 0$
• $a$ controls the steepness; negative $a$ reflects across the x-axis

Analyzing the Effect of Changing a Parameter

When asked how changing a specific parameter affects a function, think about:

1. Which feature stays the same? (intercepts, asymptotes, vertices, endpoints)
2. Which feature changes? (slope, steepness, position, direction of opening)
3. How does it change? (increases, decreases, shifts, reflects)

   Example: In the family $f(x) = \frac{3}{x - 2} + 1$, if we change 3 to 6, what happens? The asymptotes $x = 2$ and $y = 1$ remain the same, but the curve moves farther from the asymptotes in each direction. The function values are doubled at every point.

The Locus of a Function Family

The locus is the set of points common to ALL functions in the family. To find it, set the varying parameter and solve for the relationship that must hold regardless of the parameter's value.

Example: For $f(x) = m(x - 1) + 2$ where $m$ varies, find the locus. Since $f(1) = m(0) + 2 = 2$ regardless of $m$, the locus is the single point $(1, 2)$.

---

Part B: Vectors

What Is a Vector?

A vector is a quantity that has both magnitude (size) and direction. Vectors are used to represent displacement, velocity, force, and many other physical quantities.

A vector can be represented in several ways:

• Component form: $\langle a, b \rangle$ where $a$ is the horizontal component and $b$ is the vertical component
• Geometric form: an arrow drawn from an initial point to a terminal point
• Unit vector notation: $a\mathbf{i} + b\mathbf{j}$, where $\mathbf{i} = \langle 1, 0 \rangle$ and $\mathbf{j} = \langle 0, 1 \rangle$

Vector Magnitude

The magnitude (length) of vector $\mathbf{v} = \langle a, b \rangle$ is:

$$\|\mathbf{v}\| = \sqrt{a^2 + b^2}$$

This is the distance formula applied to the vector components.

Vector Direction

The direction angle $\theta$ of vector $\mathbf{v} = \langle a, b \rangle$ is the angle the vector makes with the positive x-axis:

$$\theta = \arctan\left(\frac{b}{a}\right)$$

Be careful with the quadrant — you may need to add $\pi$ (or 180°) if the vector is in Quadrant II or III.

Unit Vectors

A unit vector has magnitude 1. To find the unit vector in the direction of $\mathbf{v}$:

$$\hat{\mathbf{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \left\langle \frac{a}{\|\mathbf{v}\|}, \frac{b}{\|\mathbf{v}\|} \right\rangle$$

Standard unit vectors: $\mathbf{i} = \langle 1, 0 \rangle$ (points right) and $\mathbf{j} = \langle 0, 1 \rangle$ (points up).

Vector Operations

Vector addition: Add corresponding components.

$$\langle a_1, b_1 \rangle + \langle a_2, b_2 \rangle = \langle a_1 + a_2, b_1 + b_2 \rangle$$

Scalar multiplication: Multiply each component by the scalar.

$$c \cdot \langle a, b \rangle = \langle ca, cb \rangle$$

Vector subtraction: Subtract corresponding components.

$$\langle a_1, b_1 \rangle - \langle a_2, b_2 \rangle = \langle a_1 - a_2, b_1 - b_2 \rangle$$

The Dot Product

The dot product of $\mathbf{u} = \langle a_1, b_1 \rangle$ and $\mathbf{v} = \langle a_2, b_2 \rangle$ is:

$$\mathbf{u} \cdot \mathbf{v} = a_1 a_2 + b_1 b_2$$

The dot product is a scalar (a number), NOT a vector.

Geometric interpretation: $\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos\theta$, where $\theta$ is the angle between the vectors.

Using the Dot Product to Find Angles

$$\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|}$$

This lets you find the angle between any two vectors.

Orthogonal Vectors

Two vectors are orthogonal (perpendicular) if and only if their dot product equals zero:

$$\mathbf{u} \cdot \mathbf{v} = 0$$

This is an extremely useful condition for checking whether vectors are perpendicular.

Vector Decomposition (Resolution)

Any vector $\mathbf{v}$ can be decomposed into horizontal and vertical components:

$$\mathbf{v} = \|\mathbf{v}\| \cos\theta \, \mathbf{i} + \|\mathbf{v}\| \sin\theta \, \mathbf{j}$$

Example: A force of 50 N acts at 30° above the horizontal. The horizontal component is $50\cos(30°) = 50 \cdot \frac{\sqrt{3}}{2} \approx 43.3$ N. The vertical component is $50\sin(30°) = 50 \cdot \frac{1}{2} = 25$ N.

The Resultant Vector

When multiple vectors act on the same point, the resultant is the vector sum. To find the resultant, add all the vectors component-wise.

---

Part C: Matrices

What Is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with $m$ rows and $n$ columns is called an $m \times n$ matrix.

$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix} \quad \text{(a 2 × 3 matrix)}$$

A square matrix has the same number of rows and columns ($n \times n$).

Matrix Addition and Subtraction

Two matrices can be added or subtracted ONLY if they have the same dimensions. Add or subtract corresponding entries:

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix} + \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} a+e & b+f \\ c+g & d+h \end{bmatrix}$$

Scalar Multiplication

Multiply every entry of the matrix by the scalar:

$$k \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} ka & kb \\ kc & kd \end{bmatrix}$$

Matrix Multiplication

Matrix multiplication is the most important and most tested matrix operation. To multiply matrices $A$ and $B$ (written $AB$), the number of columns in $A$ must equal the number of rows in $B$.

For a $2 \times 2$ product:

$$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} e & f \\ g & h \end{bmatrix} = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}$$

Row-by-column rule: The entry in row $i$, column $j$ of the product is the dot product of row $i$ of $A$ with column $j$ of $B$.

Important: Matrix multiplication is NOT commutative. In general, $AB \neq BA$. Always multiply in the specified order.

The Identity Matrix

The identity matrix $I$ is the matrix equivalent of the number 1. Multiplying any matrix by $I$ leaves it unchanged.

$$I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \quad I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

The Determinant of a 2 × 2 Matrix

The determinant of $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is:

$$\det(A) = |A| = ad - bc$$

Properties of the determinant:

• If $\det(A) = 0$, the matrix is singular (not invertible)
• If $\det(A) \neq 0$, the matrix is invertible
• $\det(I) = 1$
• $\det(AB) = \det(A) \cdot \det(B)$
• Swapping two rows changes the sign of the determinant

The Inverse of a 2 × 2 Matrix

If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $\det(A) \neq 0$, then:

$$A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$

The inverse exists if and only if $\det(A) \neq 0$. To verify your inverse, check that $A \cdot A^{-1} = I$.

Using Matrices to Solve Systems of Equations

A system of linear equations can be written in matrix form:

$$A\mathbf{x} = \mathbf{b}$$

where $A$ is the coefficient matrix, $\mathbf{x}$ is the column vector of variables, and $\mathbf{b}$ is the column vector of constants.

To solve, multiply both sides by $A^{-1}$:

$$\mathbf{x} = A^{-1} \mathbf{b}$$

Example: Solve the system $$2x + 3y = 7$$ $$4x - y = 1$$

Write as: $\begin{bmatrix} 2 & 3 \\ 4 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 1 \end{bmatrix}$

$\det(A) = 2(-1) - 3(4) = -2 - 12 = -14$

$A^{-1} = \frac{1}{-14}\begin{bmatrix} -1 & -3 \\ -4 & 2 \end{bmatrix} = \begin{bmatrix} 1/14 & 3/14 \\ 2/7 & -1/7 \end{bmatrix}$

$\begin{bmatrix} x \\ y \end{bmatrix} = A^{-1} \begin{bmatrix} 7 \\ 1 \end{bmatrix} = \begin{bmatrix} 7/14 + 3/14 \\ 14/7 - 1/7 \end{bmatrix} = \begin{bmatrix} 10/14 \\ 13/7 \end{bmatrix} = \begin{bmatrix} 5/7 \\ 13/7 \end{bmatrix}$

Matrix Transformations

Matrices can represent geometric transformations in the plane:

• Reflection across x-axis: $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$
• Reflection across y-axis: $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$
• Reflection across y = x: $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
• Rotation by 90° counterclockwise: $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$
• Rotation by 180°: $\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}$
• Dilation (scaling): $\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$

  To apply a transformation, multiply the transformation matrix by the coordinate vector.

Key Formulas for Unit 4

Concept	Formula/Rule
Vector magnitude	$\	\langle a, b \rangle\	= \sqrt{a^2 + b^2}$
Unit vector	$\hat{\mathbf{v}} = \mathbf{v}/\	\mathbf{v}\	$
Dot product	$\mathbf{u} \cdot \mathbf{v} = a_1 a_2 + b_1 b_2$
Angle between vectors	$\cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\	\mathbf{u}\	\	\mathbf{v}\	}$
Orthogonal vectors	$\mathbf{u} \cdot \mathbf{v} = 0$
Determinant (2×2)	$\det(A) = ad - bc$
Matrix inverse (2×2)	$A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$
System solution	$\mathbf{x} = A^{-1}\mathbf{b}$
Vector decomposition	$\mathbf{v} = \	\mathbf{v}\	\cos\theta\,\mathbf{i} + \	\mathbf{v}\	\sin\theta\,\mathbf{j}$

1. What is the end behavior of $f(x) = -3x^4 + 2x^3 - 7x + 1$?

(A) $f(x) \to +\infty$ as $x \to +\infty$ and $f(x) \to +\infty$ as $x \to -\infty$
(B) $f(x) \to -\infty$ as $x \to +\infty$ and $f(x) \to -\infty$ as $x \to -\infty$
(C) $f(x) \to +\infty$ as $x \to +\infty$ and $f(x) \to -\infty$ as $x \to -\infty$
(D) $f(x) \to -\infty$ as $x \to +\infty$ and $f(x) \to +\infty$ as $x \to -\infty$

2. The polynomial $f(x) = (x+2)(x-1)^3(x+5)^2$ has zeros at $x = -2$, $x = 1$, and $x = -5$. At which of these zeros does the graph cross the x-axis?

(A) Only $x = 1$
(B) $x = -2$ and $x = 1$ only
(C) $x = 1$ and $x = -5$ only
(D) All three zeros

3. If $f(x) = x^3 - 4x^2 + x + 6$, what is $f(3)$?

(A) 0
(B) 3
(C) 6
(D) 12

4. What is the maximum number of turning points for a polynomial of degree 6?

(A) 3
(B) 4
(C) 5
(D) 6

5. Which of the following is a vertical asymptote of $f(x) = \frac{x^2 + 4x + 3}{x^2 - 9}$?

(A) $x = -3$
(B) $x = -1$
(C) $x = 3$
(D) Both $x = -3$ and $x = 3$

6. What is the horizontal asymptote of $f(x) = \frac{5x^3 - 2x + 1}{3x^3 + x^2 - 4}$?

(A) $y = 0$
(B) $y = \frac{5}{3}$
(C) $y = 1$
(D) There is no horizontal asymptote

7. If $f(x) = 2x^3 - 5x + 3$, then which statement is true about $f^{-1}$?

(A) $f^{-1}$ does not exist because $f$ is not one-to-one
(B) $f^{-1}$ exists and its domain is $(-\infty, \infty)$
(C) $f^{-1}$ exists and its domain is $(0, \infty)$
(D) $f^{-1}$ exists and its range is $(-\infty, \infty)$

8. The function $g(x)$ is $f(x) = x^2$ shifted 3 units left and 4 units down. Which equation represents $g(x)$?

(A) $g(x) = (x-3)^2 - 4$
(B) $g(x) = (x+3)^2 - 4$
(C) $g(x) = (x-3)^2 + 4$
(D) $g(x) = (x+3)^2 + 4$

9. What is the slant asymptote of $f(x) = \frac{2x^2 + 5x - 1}{x + 3}$?

(A) $y = 2x + 5$
(B) $y = 2x - 1$
(C) $y = x + 2$
(D) $y = 2x + 11$

10. The polynomial $P(x) = x^4 - 3x^3 + 5x^2 - 9x + 6$ has a zero at $x = 1$. What is the quotient when $P(x)$ is divided by $(x - 1)$?

(A) $x^3 - 2x^2 + 3x - 6$
(B) $x^3 - 3x^2 + 5x - 9$
(C) $x^3 + 2x^2 + 3x + 6$
(D) $x^3 - 2x^2 + 2x - 6$

---

Free-Response Questions

11. Consider the rational function $f(x) = \frac{(x-2)(x+1)}{(x+1)(x-4)}$.

(a) State the domain of $f$. (b) Identify all vertical asymptotes and holes. (c) Find the horizontal asymptote. (d) Describe the end behavior as $x \to \pm\infty$.

12. A polynomial $f(x)$ of degree 4 has the following properties:

• Leading coefficient is 2
• Zeros at $x = -1$ (multiplicity 1), $x = 3$ (multiplicity 2), and $x = 5$ (multiplicity 1)

  (a) Write $f(x)$ in factored form. (b) State the end behavior of $f$. (c) What is the maximum number of turning points? (d) At $x = 3$, does the graph cross or bounce off the x-axis? Explain.

  13. Let $f(x) = \frac{3x^2 - 12}{x^2 + x - 6}$.

  (a) Factor the numerator and denominator completely. (b) Identify any holes and vertical asymptotes. (c) Find the horizontal asymptote. (d) Find all x- and y-intercepts.

Answer Key

1. B — Even degree (4), negative leading coefficient: both ends point down.
2. B — $x = -2$ (multiplicity 1, odd → crosses) and $x = 1$ (multiplicity 3, odd → crosses). $x = -5$ (multiplicity 2, even → bounces).
3. A — $f(3) = 27 - 36 + 3 + 6 = 0$. Since $f(3) = 0$, $(x-3)$ is a factor.
4. C — A polynomial of degree $n$ has at most $n - 1$ turning points. For $n = 6$, at most 5.
5. C — Factor: $f(x) = \frac{(x+1)(x+3)}{(x-3)(x+3)}$. The $(x+3)$ cancels (hole at $x = -3$), leaving vertical asymptote at $x = 3$.
6. B — Same degree (3) in numerator and denominator: HA is ratio of leading coefficients, $5/3$.
7. A — $f(x) = 2x^3 - 5x + 3$ is a cubic with turning points, so it fails the horizontal line test. $f^{-1}$ does not exist as a function.
8. B — Shift left 3: $(x+3)^2$. Shift down 4: $-4$. Result: $(x+3)^2 - 4$.
9. B — Long division: $(2x^2 + 5x - 1) ÷ (x + 3) = 2x - 1$ with remainder 2. Slant asymptote: $y = 2x - 1$.
10. A — Synthetic division of $x^4 - 3x^3 + 5x^2 - 9x + 6$ by $(x-1)$ gives $x^3 - 2x^2 + 3x - 6$.

    FRQ 11: (a) Domain: $x \neq -1, x \neq 4$. (b) Hole at $x = -1$; vertical asymptote at $x = 4$. (c) HA: $y = 1$. (d) Same end behavior as $y = 1$.

    FRQ 12: (a) $f(x) = 2(x+1)(x-3)^2(x-5)$. (b) As $x \to +\infty$, $f(x) \to +\infty$; as $x \to -\infty$, $f(x) \to +\infty$ (even degree, positive leading coefficient). (c) At most 3. (d) Bounces — multiplicity 2 is even.

    FRQ 13: (a) $\frac{3(x-2)(x+2)}{(x+3)(x-2)}$ (b) Hole at $x = 2$; VA at $x = -3$. (c) $y = 3$. (d) x-intercept: $x = -2$; y-intercept: $f(0) = -12/-6 = 2$.

1. A population of bacteria doubles every 4 hours. If the initial population is 500, which function models the population $P(t)$ after $t$ hours?

(A) $P(t) = 500 \cdot 2^{4t}$
(B) $P(t) = 500 \cdot 4^{t/2}$
(C) $P(t) = 500 \cdot 2^{t/4}$
(D) $P(t) = 500 + 2^t$

2. What is the domain of $f(x) = \ln(x + 5)$?

(A) $x > -5$
(B) $x \geq -5$
(C) $x > 5$
(D) All real numbers

3. Simplify: $\log_3(27) + \log_3\left(\frac{1}{9}\right)$

(A) $0$
(B) $1$
(C) $3$
(D) $-1$

4. If $\log_4(x) = 3$, what is $x$?

(A) $12$
(B) $64$
(C) $81$
(D) $7$

5. Which expression is equivalent to $\ln\left(\frac{x^3}{e^2}\right)$?

(A) $3\ln(x) - 2$
(B) $3\ln(x) + 2$
(C) $\frac{3\ln(x)}{2}$
(D) $\ln(x^3) \cdot \ln(e^2)$

6. An investment of $2,000 earns 5% annual interest compounded quarterly. What is the value after 10 years?

(A) $2,000(1.05)^{10}$
(B) $2,000(1.0125)^{40}$
(C) $2,000(1.5)^{10}$
(D) $2,000(0.05)^{40}$

7. The half-life of a radioactive substance is 1600 years. If a sample initially contains 80 grams, how much remains after 4800 years?

(A) 20 grams
(B) 10 grams
(C) 40 grams
(D) 5 grams

8. What is the x-intercept of $f(x) = 3 \cdot e^{x-2} - 6$?

(A) $x = 0$
(B) $x = \ln(2) + 2$
(C) $x = 2$
(D) $x = \ln(3) - 2$

9. The pH of a solution is 4. What is the hydrogen ion concentration $[H^+]$?

(A) $10^{-4}$
(B) $4 \times 10^0$
(C) $10^4$
(D) $4$

10. Which of the following functions represents exponential decay?

(A) $f(x) = 3(1.5)^x$
(B) $f(x) = -2(0.8)^x$
(C) $f(x) = 0.5(2)^x$
(D) $f(x) = 4(1)^x$

---

Free-Response Questions

11. The population of a city is modeled by $P(t) = 120{,}000 \cdot e^{0.025t}$, where $t$ is the number of years since 2020.

(a) What was the population in 2020? (b) In what year will the population reach 200,000? (c) What is the annual growth rate as a percentage? (d) How long does it take for the population to double?

12. Solve each equation. Show your work and check for extraneous solutions.

(a) $4^{2x-1} = 32$ (b) $\log_2(x + 3) + \log_2(x - 1) = 4$ (c) $e^{3x} - 5e^x + 6 = 0$

13. A scientist measures the intensity of an earthquake and finds it is 1000 times more powerful than a reference earthquake.

(a) How many points higher is this earthquake on the Richter scale? (b) If the reference earthquake measures 3.0 on the Richter scale, what is the magnitude of the more powerful earthquake? (c) If a third earthquake measures 6.0 on the Richter scale, how many times more powerful is it than the reference earthquake?

---

Answer Key

1. C — Doubling time is 4 hours, so the growth factor per hour is $2^{1/4}$. After $t$ hours: $P(t) = 500 \cdot 2^{t/4}$.
2. A — The argument of $\ln$ must be positive: $x + 5 > 0 \implies x > -5$.
3. A — $\log_3(27) = \log_3(3^3) = 3$. $\log_3(1/9) = \log_3(3^{-2}) = -2$. Sum: $3 + (-2) = 1$. Wait — $3 + (-2) = 1$, not 0. The answer is (B). Correction: $\log_3(27) = 3$ and $\log_3(1/9) = -2$. So $3 + (-2) = 1$.
4. B — $\log_4(x) = 3 \implies 4^3 = x \implies x = 64$.
5. A — $\ln(x^3/e^2) = \ln(x^3) - \ln(e^2) = 3\ln(x) - 2$.
6. B — Quarterly compounding: $n = 4$, $r/n = 0.05/4 = 0.0125$, $nt = 4 \times 10 = 40$. Formula: $2000(1.0125)^{40}$.
7. B — After 4800 years = 3 half-lives. $80 \to 40 \to 20 \to 10$ grams.
8. B — $3e^{x-2} - 6 = 0 \implies e^{x-2} = 2 \implies x - 2 = \ln(2) \implies x = \ln(2) + 2$.
9. A — $\text{pH} = -\log[H^+] \implies 4 = -\log[H^+] \implies [H^+] = 10^{-4}$.
10. B — Base $0.8$ is between 0 and 1, so $0.8^x$ decays. The negative sign reflects but the function still decays in magnitude. Note: Both (B) is decay. Actually only (B) has $0 < b < 1$. (D) has $b = 1$, which is constant, not decay.

    FRQ 11: (a) $P(0) = 120,000$. (b) $120{,}000e^{0.025t} = 200{,}000 \implies e^{0.025t} = 5/3 \implies t = \ln(5/3)/0.025 \approx 20.4$ years, so approximately 2040. (c) 2.5% per year. (d) Double: $120{,}000e^{0.025t} = 240{,}000 \implies e^{0.025t} = 2 \implies t = \ln(2)/0.025 \approx 27.7$ years.

    FRQ 12: (a) $4^{2x-1} = 32$. Write as $2^{2(2x-1)} = 2^5$. So $4x - 2 = 5 \implies x = 7/4$. (b) $\log_2[(x+3)(x-1)] = 4 \implies (x+3)(x-1) = 16 \implies x^2 + 2x - 3 = 16 \implies x^2 + 2x - 19 = 0 \implies x = \frac{-2 \pm \sqrt{80}}{2} = -1 \pm 2\sqrt{5}$. Only $x = -1 + 2\sqrt{5} \approx 3.47$ is valid (both log arguments must be positive). (c) Let $u = e^x$: $u^3 - 5u + 6 = 0$. Note: $u^3 - 5u + 6 = (u - 2)(u^2 + 2u - 3) = (u-2)(u+3)(u-1) = 0$. So $u = 2, 1$ (reject $u = -3$ since $e^x > 0$). $e^x = 2 \implies x = \ln(2)$; $e^x = 1 \implies x = 0$.

    FRQ 13: (a) Each unit on the Richter scale = 10-fold increase. $1000 = 10^3$, so 3 points higher. (b) $3.0 + 3.0 = 6.0$. (c) $10^{6-3} = 10^3 = 1000$ times more powerful.

1. Convert $150°$ to radians.

(A) $\frac{5\pi}{6}$
(B) $\frac{2\pi}{3}$
(C) $\frac{3\pi}{4}$
(D) $\frac{7\pi}{6}$

2. What is $\sin\left(\frac{7\pi}{6}\right)$?

(A) $-\frac{1}{2}$
(B) $\frac{1}{2}$
(C) $-\frac{\sqrt{3}}{2}$
(D) $\frac{\sqrt{3}}{2}$

3. The function $f(x) = 4\sin\left(\frac{x}{3}\right) + 1$ has amplitude, period, and midline:

(A) Amplitude: 4, Period: $6\pi$, Midline: $y = 1$
(B) Amplitude: 4, Period: $\frac{2\pi}{3}$, Midline: $y = 1$
(C) Amplitude: 4, Period: $6\pi$, Midline: $y = 0$
(D) Amplitude: 2, Period: $6\pi$, Midline: $y = 1$

4. Which identity is equivalent to $\sin^2\theta$?

(A) $1 + \cos^2\theta$
(B) $1 - \cos^2\theta$
(C) $\frac{1}{1 + \tan^2\theta}$
(D) Both (B) and (C)

5. If $\cos\theta = \frac{3}{5}$ and $\theta$ is in Quadrant IV, what is $\sin\theta$?

(A) $\frac{4}{5}$
(B) $-\frac{4}{5}$
(C) $\frac{3}{4}$
(D) $-\frac{3}{4}$

6. What is $\cos(2\theta)$ if $\sin\theta = \frac{3}{5}$ and $\cos\theta = -\frac{4}{5}$?

(A) $\frac{7}{25}$
(B) $-\frac{7}{25}$
(C) $\frac{24}{25}$
(D) $-\frac{24}{25}$

7. The range of $f(x) = \arccos(x)$ is:

(A) $(-\infty, \infty)$
(B) $[0, \pi]$
(C) $[-1, 1]$
(D) $[-\pi/2, \pi/2]$

8. In polar coordinates, the point $(2, \pi/3)$ in rectangular coordinates is:

(A) $(1, \sqrt{3})$
(B) $(\sqrt{3}, 1)$
(C) $(-1, \sqrt{3})$
(D) $(2\sqrt{3}, 2)$

9. How many petals does the polar graph $r = 4\sin(3\theta)$ have?

(A) 3
(B) 6
(C) 4
(D) 9

10. Solve $\tan\theta = 1$ for $0 \leq \theta < 2\pi$.

(A) $\frac{\pi}{4}$ only
(B) $\frac{\pi}{4}$ and $\frac{5\pi}{4}$
(C) $\frac{\pi}{4}$ and $\frac{3\pi}{4}$
(D) $\frac{3\pi}{4}$ and $\frac{7\pi}{4}$

---

Free-Response Questions

11. Consider the function $f(x) = -2\cos(3x + \pi) + 5$.

(a) Find the amplitude, period, phase shift, and midline. (b) State the domain and range. (c) Determine the coordinates of a maximum point on the graph. (d) How many complete cycles occur in the interval $[0, 2\pi]$?

12. A Ferris wheel has a radius of 30 feet and its center is 35 feet above the ground. The wheel completes one revolution every 20 seconds. At time $t = 0$, a rider is at the lowest point.

(a) Write a sinusoidal function $h(t)$ that gives the rider's height above the ground at time $t$. (b) What is the rider's height after 7 seconds? (c) During the first revolution, when is the rider at the top of the wheel? (d) What is the first time the rider reaches a height of 50 feet?

13. The parametric equations $x(t) = 3\cos(t)$ and $y(t) = 2\sin(t)$ describe an ellipse for $0 \leq t \leq 2\pi$.

(a) Eliminate the parameter to find the Cartesian equation. (b) State the center, vertices, and co-vertices of the ellipse. (c) Find $\frac{dy}{dx}$ at $t = \pi/4$. (d) Convert the parametric form to polar form.

---

Answer Key

1. A — $150° \times \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6}$.
2. A — $7\pi/6$ is in Quadrant III where sine is negative. Reference angle $\pi/6$: $\sin = -1/2$.
3. A — Amplitude = $|4| = 4$. Period = $2\pi/|1/3| = 6\pi$. Midline = $y = 1$.
4. D — From $\sin^2\theta + \cos^2\theta = 1$: $\sin^2\theta = 1 - \cos^2\theta$. Also $\frac{1}{1+\tan^2\theta} = \frac{1}{\sec^2\theta} = \cos^2\theta$. Wait, that gives $\cos^2\theta$, not $\sin^2\theta$. Correction: $\frac{\tan^2\theta}{1+\tan^2\theta} = \sin^2\theta$. So the answer is (B) only.
5. B — Quadrant IV: sine is negative. $\sin^2\theta = 1 - (3/5)^2 = 1 - 9/25 = 16/25$, so $\sin\theta = -4/5$.
6. A — $\cos(2\theta) = \cos^2\theta - \sin^2\theta = 16/25 - 9/25 = 7/25$.
7. B — $\arccos(x)$ has range $[0, \pi]$.
8. A — $x = 2\cos(\pi/3) = 2 \cdot 1/2 = 1$; $y = 2\sin(\pi/3) = 2 \cdot \sqrt{3}/2 = \sqrt{3}$.
9. A — $r = a\sin(n\theta)$ with $n = 3$ (odd): 3 petals.
10. B — Tangent is positive in Quadrants I and III: $\theta = \pi/4$ and $\theta = 5\pi/4$.

    FRQ 11: (a) Amplitude: 2, Period: $2\pi/3$, Phase shift: $-\pi/3$ (shift left $\pi/3$), Midline: $y = 5$. (b) Domain: $(-\infty, \infty)$, Range: $[3, 7]$. (c) Maximum: When $-2\cos(3x+\pi) = 2$, i.e., $\cos(3x+\pi) = -1$. $3x + \pi = \pi \implies x = 0$. Point: $(0, 7)$. (d) Period $= 2\pi/3$. Number of cycles in $2\pi$: $2\pi \div (2\pi/3) = 3$.

    FRQ 12: (a) $h(t) = 35 - 30\cos\left(\frac{2\pi}{20}t\right) = 35 - 30\cos\left(\frac{\pi}{10}t\right)$. At $t = 0$: $h(0) = 35 - 30 = 5$ feet (lowest point). ✓ (b) $h(7) = 35 - 30\cos(7\pi/10)$. $\cos(7\pi/10) = \cos(126°) \approx -0.588$. $h(7) \approx 35 - 30(-0.588) = 35 + 17.6 = 52.6$ feet. (c) Top when $\cos(\pi t/10) = -1$, i.e., $\pi t/10 = \pi \implies t = 10$ seconds. (d) $35 - 30\cos(\pi t/10) = 50 \implies \cos(\pi t/10) = -1/2 \implies \pi t/10 = 2\pi/3 \implies t = 20/3 \approx 6.67$ seconds.

    FRQ 13: (a) $(x/3)^2 + (y/2)^2 = \cos^2 t + \sin^2 t = 1$, so $\frac{x^2}{9} + \frac{y^2}{4} = 1$. (b) Center: $(0,0)$. Vertices: $(\pm 3, 0)$. Co-vertices: $(0, \pm 2)$. (c) $\frac{dy}{dx} = \frac{2\cos t}{-3\sin t} = -\frac{2}{3}\cot t$. At $t = \pi/4$: $-\frac{2}{3} \cdot 1 = -\frac{2}{3}$. (d) $r\cos\theta = 3\cos t$ and $r\sin\theta = 2\sin t$. This does not simplify to a clean polar equation in terms of $\theta$ alone, so this is best left in parametric form.

1. In the function family $f(x) = a(x - 3)^2 + 5$, as the parameter $a$ varies over all real numbers, what point is common to every graph in the family?

(A) $(3, 5)$
(B) $(0, 5)$
(C) $(-3, 5)$
(D) $(3, 0)$

2. The function family $f(x) = m(x + 2) - 1$ produces a set of lines. As $m$ varies, what is the locus (common point)?

(A) $(-2, -1)$
(B) $(2, 1)$
(C) $(0, -1)$
(D) $(-2, 0)$

3. What is the magnitude of the vector $\mathbf{v} = \langle -3, 4 \rangle$?

(A) 1
(B) 5
(C) 7
(D) $\sqrt{7}$

4. If $\mathbf{u} = \langle 2, -1 \rangle$ and $\mathbf{v} = \langle -3, 4 \rangle$, what is $\mathbf{u} + \mathbf{v}$?

(A) $\langle -1, 3 \rangle$
(B) $\langle 5, -5 \rangle$
(C) $\langle -6, -4 \rangle$
(D) $\langle 1, -3 \rangle$

5. What is the dot product of $\mathbf{u} = \langle 3, 2 \rangle$ and $\mathbf{v} = \langle -1, 5 \rangle$?

(A) $-3 + 10 = 7$
(B) $-3 + 2 = -1$
(C) $3 - 10 = -7$
(D) $15 + 2 = 17$

6. Two vectors $\mathbf{a}$ and $\mathbf{b}$ are orthogonal. If $\mathbf{a} = \langle 4, -2 \rangle$, which could be $\mathbf{b}$?

(A) $\langle 2, 4 \rangle$
(B) $\langle 1, 2 \rangle$
(C) $\langle -4, 2 \rangle$
(D) $\langle 2, -4 \rangle$

7. What is the determinant of $A = \begin{bmatrix} 5 & 3 \\ -2 & 4 \end{bmatrix}$?

(A) $14$
(B) $26$
(C) $-14$
(D) $20$

8. Which matrix is the inverse of $B = \begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$?

(A) $\begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$
(B) $\begin{bmatrix} -3 & 1 \\ 5 & -2 \end{bmatrix}$
(C) $\begin{bmatrix} 2 & -1 \\ -5 & 3 \end{bmatrix}$
(D) $\begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$

9. For the family $f(x) = \frac{a}{x - 1} + 2$ where $a > 0$, which of the following is true for every function in the family?

(A) The domain is all real numbers
(B) The horizontal asymptote is $y = 0$
(C) The vertical asymptote is $x = 1$
(D) There is exactly one x-intercept

10. The transformation matrix $T = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$ represents:

(A) Reflection across the x-axis
(B) Reflection across the y-axis
(C) Rotation 90° counterclockwise
(D) Rotation 180°

---

Free-Response Questions

11. Consider the vectors $\mathbf{u} = \langle 6, 8 \rangle$ and $\mathbf{v} = \langle -3, 4 \rangle$.

(a) Find $\|\mathbf{u}\|$ and $\|\mathbf{v}\|$. (b) Compute the dot product $\mathbf{u} \cdot \mathbf{v}$. (c) Find the angle $\theta$ between $\mathbf{u}$ and $\mathbf{v}$. (d) Are $\mathbf{u}$ and $\mathbf{v}$ orthogonal? Explain. (e) Find the unit vector in the direction of $\mathbf{u}$.

12. Consider the function family $f(x) = a\sqrt{x + 4} - 3$.

(a) What is the domain of every function in this family? (b) What is the common point (endpoint) shared by all graphs in this family? (c) As $a$ increases from 1 to 5, how does the steepness of the graph change? (d) For $a = -2$, describe the graph's orientation.

13. Matrices $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & -1 \\ 1 & 2 \end{bmatrix}$.

(a) Compute $A + B$. (b) Compute $AB$. (c) Compute $BA$. (d) Show that $AB \neq BA$, confirming that matrix multiplication is not commutative. (e) Find $A^{-1}$.

---

Answer Key

1. A — When $x = 3$: $f(3) = a(0)^2 + 5 = 5$. Point $(3, 5)$ is on every graph.
2. A — When $x = -2$: $f(-2) = m(0) - 1 = -1$. The locus is $(-2, -1)$.
3. B — $\|\langle -3, 4 \rangle\| = \sqrt{9 + 16} = \sqrt{25} = 5$.
4. A — $\langle 2 + (-3), -1 + 4 \rangle = \langle -1, 3 \rangle$.
5. A — $3(-1) + 2(5) = -3 + 10 = 7$.
6. A — Dot product must equal zero. $\langle 4, -2 \rangle \cdot \langle 2, 4 \rangle = 8 - 8 = 0$. ✓
7. B — $\det(A) = 5(4) - 3(-2) = 20 + 6 = 26$.
8. A — $\det(B) = 2(3) - 1(5) = 1$. $B^{-1} = \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$.
9. C — For all functions in the family, the denominator is zero at $x = 1$, giving a vertical asymptote there.
10. C — $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 0 \end{bmatrix}$. This maps $(1,0) \to (0,1)$ and $(0,1) \to (-1,0)$, which is a 90° counterclockwise rotation.

    FRQ 11: (a) $\|\mathbf{u}\| = \sqrt{36+64} = 10$; $\|\mathbf{v}\| = \sqrt{9+16} = 5$. (b) $6(-3) + 8(4) = -18 + 32 = 14$. (c) $\cos\theta = 14/(10 \cdot 5) = 14/50 = 7/25 \implies \theta = \arccos(7/25) \approx 73.7°$. (d) No — the dot product is 14, not 0. (e) $\hat{\mathbf{u}} = \langle 6/10, 8/10 \rangle = \langle 0.6, 0.8 \rangle$.

    FRQ 12: (a) Domain: $[-4, \infty)$ (the radicand must be non-negative). (b) Endpoint: $f(-4) = a\sqrt{0} - 3 = -3$. Common point: $(-4, -3)$. (c) As $a$ increases, the graph becomes steeper — it rises (or falls) more rapidly as $x$ increases. (d) $a = -2$ reflects the graph across the x-axis. It starts at $(-4, -3)$ and decreases as $x$ increases (goes downward instead of upward).

    FRQ 13: (a) $A + B = \begin{bmatrix} 1 & 1 \\ 4 & 6 \end{bmatrix}$. (b) $AB = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}$. (c) $BA = \begin{bmatrix} -3 & -4 \\ 7 & 10 \end{bmatrix}$. (d) $AB \neq BA$ confirmed: $\begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \neq \begin{bmatrix} -3 & -4 \\ 7 & 10 \end{bmatrix}$. (e) $\det(A) = 4 - 6 = -2$. $A^{-1} = \frac{1}{-2}\begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 3/2 & -1/2 \end{bmatrix}$.

Polynomials

• Form: $f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_0$
• End behavior: Degree + leading coefficient → left end / right end
  • Even degree, $a > 0$: both up $\uparrow\uparrow$
  • Even degree, $a < 0$: both down $\downarrow\downarrow$
  • Odd degree, $a > 0$: down, up $\downarrow\uparrow$
  • Odd degree, $a < 0$: up, down $\uparrow\downarrow$
• Zeros (x-intercepts): Solve $f(x) = 0$
  • Odd multiplicity → crosses x-axis
  • Even multiplicity → bounces off x-axis
• Max turning points: $n - 1$
• Remainder Theorem: $f(x) \div (x-c)$ remainder $= f(c)$
• Factor Theorem: $(x-c)$ is a factor $\iff f(c) = 0$

Rational Functions

• Form: $f(x) = P(x)/Q(x)$
• Vertical asymptote: $x = c$ where $Q(c) = 0$ and $P(c) \neq 0$
• Hole: $x = c$ where $(x-c)$ cancels in both numerator and denominator
• Horizontal asymptote:
  • deg P < deg Q → $y = 0$
  • deg P = deg Q → $y = a/b$ (ratio of leading coefficients)
  • deg P > deg Q → no HA (use polynomial long division for slant asymptote)

Key Transformations

• $f(x) + k$ → shift up $k$
• $f(x - h)$ → shift right $h$ (opposite sign!)
• $af(x)$ → vertical stretch ($|a|>1$) or compression ($0<|a|<1$); flip if $a < 0$
• $f(bx)$ → horizontal compression ($|b|>1$) or stretch ($0<|b|<1$)

Inverse Functions

• Horizontal line test → function is one-to-one → inverse exists
• To find: swap $x$ and $y$, solve for $y$
• $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

Unit 2: Exponential and Logarithmic Functions (27–40%)

Exponentials

• Form: $f(x) = a \cdot b^x$
• $b > 1$ → growth; $0 < b < 1$ → decay
• y-intercept: $(0, a)$; horizontal asymptote: $y = 0$ (shifted: $y = k$)
• Compound interest: $A = P(1 + r/n)^{nt}$
• Continuous interest: $A = Pe^{rt}$
• Half-life: $t_{1/2} = \ln(2)/k$ for $A(t) = A_0 e^{-kt}$

Logarithms

• Definition: $\log_b(x) = y \iff b^y = x$
• Product rule: $\log(xy) = \log x + \log y$
• Quotient rule: $\log(x/y) = \log x - \log y$
• Power rule: $\log(x^n) = n\log x$
• Change of base: $\log_b(x) = \ln(x)/\ln(b)$
• Domain of $\log_b(x)$: $x > 0$
• y-intercept: $(1, 0)$; vertical asymptote: $x = 0$

Solving Equations

• Exponential with same base → set exponents equal
• Exponential with different bases → take $\ln$ of both sides
• Logarithmic → rewrite in exponential form; check for extraneous solutions

Logarithmic Scales

• pH $= -\log[H^+]$; Richter; Decibels

Unit 3: Trigonometric and Polar Functions (30–35%)

Radians

• $180° = \pi$ rad; arc length $s = r\theta$; sector area $A = \frac{1}{2}r^2\theta$

Unit Circle Key Values

Angle	$\sin$	$\cos$	$\tan$
$\pi/6$ (30°)	$1/2$	$\sqrt{3}/2$	$1/\sqrt{3}$
$\pi/4$ (45°)	$\sqrt{2}/2$	$\sqrt{2}/2$	$1$
$\pi/3$ (60°)	$\sqrt{3}/2$	$1/2$	$\sqrt{3}$

ASTC: All (QI), Sine (QII), Tangent (QIII), Cosine (QIV)

Trig Graphs: $f(x) = a\sin(bx+c)+d$

• Amplitude $= |a|$; Period $= 2\pi/|b|$ (tangent: $\pi/|b|$)
• Phase shift $= -c/b$; Midline $= y = d$
• Range $= [d - |a|, d + |a|]$

Key Identities

• $\sin^2\theta + \cos^2\theta = 1$
• $\sin(2\theta) = 2\sin\theta\cos\theta$
• $\cos(2\theta) = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta = \cos^2\theta - \sin^2\theta$

Inverse Trig

• $\arcsin$: domain $[-1,1]$, range $[-\pi/2, \pi/2]$
• $\arccos$: domain $[-1,1]$, range $[0, \pi]$
• $\arctan$: domain $(-\infty,\infty)$, range $(-\pi/2, \pi/2)$

Polar Coordinates

• $x = r\cos\theta$; $y = r\sin\theta$
• $r = \sqrt{x^2 + y^2}$; $\theta = \arctan(y/x)$
• Rose curves: $r = a\cos(n\theta)$ — odd $n$: $n$ petals; even $n$: $2n$ petals
• Limaçon: $r = a + b\cos\theta$

Parametric Equations

• $x = f(t)$, $y = g(t)$; slope $= (dy/dt)/(dx/dt)$

Unit 4: Parameters, Vectors, and Matrices (15–25%)

Parameters

• A parameter changes the graph's shape/position while keeping a common structure
• The locus is the point(s) common to ALL graphs in a family
• Key families: linear ($mx + b$), quadratic ($a(x-h)^2 + k$), radical ($a\sqrt{x-h} + k$), rational ($a/(x-h) + k$), exponential ($ab^{x-h} + k$)

Vectors

• Magnitude: $\|\langle a, b \rangle\| = \sqrt{a^2 + b^2}$
• Unit vector: $\hat{\mathbf{v}} = \mathbf{v}/\|\mathbf{v}\|$
• Dot product: $\mathbf{u} \cdot \mathbf{v} = a_1 a_2 + b_1 b_2$
• Angle: $\cos\theta = (\mathbf{u} \cdot \mathbf{v})/(\|\mathbf{u}\|\|\mathbf{v}\|)$
• Orthogonal: $\mathbf{u} \cdot \mathbf{v} = 0$
• Decomposition: $\mathbf{v} = \|\mathbf{v}\|\cos\theta\,\mathbf{i} + \|\mathbf{v}\|\sin\theta\,\mathbf{j}$

Matrices

• Add/subtract: Same dimensions only; add entry by entry
• Multiply: Columns of first = rows of second; row-by-column rule
• Determinant (2×2): $|A| = ad - bc$
• Inverse (2×2): $A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ (exists iff $|A| \neq 0$)
• Solve systems: $A\mathbf{x} = \mathbf{b} \implies \mathbf{x} = A^{-1}\mathbf{b}$
• Transformations: reflection ($y$-axis: $\begin{bmatrix}-1&0\\0&1\end{bmatrix}$), rotation 90° ($\begin{bmatrix}0&-1\\1&0\end{bmatrix}$)

Quick Reminders

• Calculator is allowed on the ENTIRE exam — use it strategically
• Show all work on FRQs — partial credit is generous
• Check extraneous solutions in logarithmic equations
• Horizontal transformations are OPPOSITE of what you expect
• Even multiplicity = bounce; odd multiplicity = cross

The AP Precalculus exam is 3 hours long:

• Section I: 40 multiple-choice questions in 120 minutes (50% weight by questions, 62.5% by scoring)
• Section II: 4 free-response questions in 60 minutes (37.5% weight)

  A graphing calculator is permitted and expected on the entire exam.

Before the Exam

Preparation Timeline

• 2+ months out: Work through all four units systematically. Master each unit before moving on.
• 1 month out: Complete full practice exams under timed conditions. Focus on weak areas.
• 1 week out: Review the summary sheet daily. Re-do missed problems from practice exams.
• Night before: Light review only. Sleep well. Pack your calculator, pencils, and eraser.

Calculator Setup

Ensure your calculator has these ready to use:

• Mode in radians (most trig problems use radians)
• "Diagnostic On" enabled for checking rREF results
• Window settings memorized: ZStandard, ZTrig
• Programs cleared (some calculators are checked)

What to Bring

• Government-issued ID
• Several sharpened #2 pencils
• Your approved graphing calculator with fresh batteries
• A backup calculator if possible
• A watch (no smartwatches allowed)

Section I: Multiple-Choice Strategy

Time Management

• 120 minutes ÷ 40 questions = 3 minutes per question
• If a question takes more than 2 minutes and you're stuck, mark it and move on
• Return to skipped questions in the last 15 minutes
• Never leave a question blank — there is no penalty for guessing

Question Types and Approach

Direct computation: Straightforward algebra or arithmetic. Use your calculator when possible to avoid arithmetic errors.

Conceptual: Tests understanding of definitions and properties. Read carefully — look for keywords like "always," "must," "which of the following."

Graph-based: May ask about features (asymptotes, intercepts, end behavior). Use your calculator to graph when uncertain.

Multi-step: Requires setting up an equation first, then solving. Identify what you're solving for before diving in.

Elimination Strategies

1. Eliminate answers that are clearly dimensionally wrong (e.g., time given in meters)
2. Eliminate answers with the wrong sign or wrong direction
3. For "which of the following" questions, test a single easy value (like $x = 0$ or $x = 1$) in each option
4. For end behavior questions, just check the leading term — no need to evaluate the whole polynomial

Calculator Strategies for MC

• Graph the function and use trace or table to check values
• Use the solver (or calc → zero) for finding roots
• Use intersect to find where two graphs cross
• For regression problems, enter data in lists and run the appropriate regression

Section II: Free-Response Strategy

Time Management

• 60 minutes ÷ 4 questions = 15 minutes per question
• Spend 2 minutes reading the full question and planning your approach
• Spend 12 minutes writing your solution
• Spend 1 minute checking your work

The Most Important Rule: SHOW YOUR WORK

The FRQ section awards partial credit generously. Even if you can't solve the whole problem, showing correct work for one part earns points.

Part-by-Part Approach

FRQs typically have parts (a), (b), (c), (d):

• Part (a) is usually the easiest — set up the expression, find the initial value, or do basic algebra
• Middle parts build on earlier parts. If you get stuck on (b), still attempt (c) and (d) — they may be independently solvable
• The last part often asks for interpretation in context. Even if your math is wrong in earlier parts, writing the correct interpretation based on your (incorrect) answer can still earn full credit for that sub-part

Writing Conventions

• Label every answer clearly: circle or box it
• Use proper mathematical notation ($\neq$, $\geq$, $\leq$, $\approx$)
• If you cross something out, make sure it's clearly crossed out — don't leave ambiguous scribbles
• Write legibly — graders can only give credit for what they can read

Common FRQ Topics

1. Function modeling: Given a real-world scenario, write the function, find specific values, interpret in context
2. Transformations: Given a base function and a transformation, write the new equation or describe the graph
3. Solving equations: Show the algebraic steps, not just the calculator answer
4. Analysis: Describe behavior — increasing/decreasing intervals, maximum/minimum values, asymptotic behavior
5. Inverse functions: Find the inverse, verify compositions, relate domains and ranges

When You're Stuck

• If you can't solve for an exact answer, set up the equation correctly — this earns partial credit
• If part (b) depends on part (a) and you couldn't solve (a), use a reasonable guess and proceed — you may earn full credit on (b) for correct method
• Draw a graph as a last resort — it may reveal the answer visually

Unit-Specific Strategies

Unit 1: Polynomials and Rationals

• For end behavior: look ONLY at the leading term. Ignore everything else.
• For rational functions: ALWAYS factor first. Common factors cancel → holes, not asymptotes.
• For inverse functions: check the horizontal line test before attempting algebra.

Unit 2: Exponentials and Logarithms

• For exponential equations: try matching bases first. If you can't, take the natural log.
• For logarithmic equations: ALWAYS check your answer in the original equation.
• For modeling: constant ratio of consecutive values → exponential model.
• Know the difference between $A = P(1+r/n)^{nt}$ and $A = Pe^{rt}$.

Unit 3: Trigonometric and Polar

• For sinusoidal graphs: identify $a$, $b$, $c$, $d$ from the equation. Period $= 2\pi/|b|$.
• For the unit circle: memorize the 16 key coordinates. They appear on every exam.
• For identities: you don't need to memorize all of them, but you MUST know the Pythagorean identity and double-angle formulas.
• For polar: be able to convert between polar and rectangular coordinates quickly.
• Make sure your calculator is in radian mode unless the problem specifies degrees.

Unit 4: Parameters, Vectors, and Matrices

• For parameter families: plug in specific values to find the locus.
• For vectors: draw a diagram. Decomposition into horizontal and vertical components is common.
• For matrices: the determinant and inverse formulas for 2×2 matrices appear frequently. Memorize them.
• For solving systems with matrices: set up $A\mathbf{x} = \mathbf{b}$, find $A^{-1}$, compute $A^{-1}\mathbf{b}$.

Mental Math and Common Mistakes

Top 10 Mistakes to Avoid

1. Forgetting that horizontal shifts are opposite direction ($f(x-3)$ shifts right)
2. Confusing $b^2$ with $(2b)^2$ or similar exponent errors
3. Forgetting to check extraneous solutions in logarithmic equations
4. Writing $\sin^{-1}(x)$ as $1/\sin(x)$ instead of the inverse sine
5. Wrong quadrant when finding trig values — always check ASTC
6. Mixing up the dot product formula (it's a scalar, not a vector)
7. Dividing by zero or forgetting domain restrictions
8. Confusing amplitude with period or midline
9. Incorrectly factoring — especially sign errors
10. Not using the calculator when you're allowed to

Speed Techniques

• Memorize common values: $\sqrt{2} \approx 1.414$, $\sqrt{3} \approx 1.732$, $\sqrt{5} \approx 2.236$
• Know that $\ln(2) \approx 0.693$, $\ln(10) \approx 2.303$
• Recognize $e \approx 2.718$ and $\pi \approx 3.14159$
• For the unit circle, remember: $1/2, \sqrt{2}/2, \sqrt{3}/2$ pattern across $\pi/6$, $\pi/4$, $\pi/3$

Final Checklist

• [ ] Calculator in radians
• [ ] Fresh batteries
• [ ] Review summary sheet one last time
• [ ] Know the 16 unit circle values
• [ ] Know the 2×2 inverse formula
• [ ] Know the dot product and magnitude formulas
• [ ] Remember: NO penalty for guessing on MC
• [ ] Remember: SHOW ALL WORK on FRQs
• [ ] Stay calm, stay focused, manage your time

This script is designed for a 25–30 minute audio recording. Read at a moderate, clear pace. Use pauses (marked with [PAUSE]) between sections to allow the listener to process information. Optionally, repeat key definitions.

---

Introduction (1 minute)

Welcome to the AP Precalculus audio review. This recording covers the key concepts, formulas, and strategies you need for exam day. We'll move through all four units, hitting the most frequently tested topics and common pitfalls.

Let's begin.

---

Unit 1: Polynomial and Rational Functions (6 minutes)

We start with polynomials. A polynomial has the form: a-sub-n times x-to-the-n, plus lower-degree terms, down to a constant. The degree is the highest power of x, and the leading coefficient is the number in front of that highest power.

End behavior depends on two things: the degree — even or odd — and the sign of the leading coefficient. Here's the quick rule. Even degree with positive leading coefficient: both ends point up, like a U. Even degree, negative: both ends point down, like an upside-down U. Odd degree, positive: the left end goes down and the right end goes up. Odd degree, negative: the left end goes up and the right end goes down. Think of it this way — odd-degree polynomials always go in opposite directions at the two ends. Even-degree polynomials go in the same direction.

[PAUSE]

Zeros are the x-intercepts, where the polynomial equals zero. Each zero has a multiplicity — how many times that factor appears. Odd multiplicity means the graph crosses the x-axis. Even multiplicity means it bounces off — it touches and turns around. So if you see a factor like x minus 2 to the third power, the graph crosses at x equals 2 but flattens out first. A factor like x plus 1 squared means the graph bounces at x equals negative 1.

The Remainder Theorem says: if you divide a polynomial by x minus c, the remainder is f of c. The Factor Theorem says: x minus c is a factor if and only if f of c equals zero. These are tested almost every year.

[PAUSE]

Now rational functions — these are fractions where both the top and bottom are polynomials. The domain excludes wherever the denominator equals zero.

Vertical asymptotes occur at x-values that make the denominator zero, but only if that factor doesn't cancel with the numerator. If a factor cancels, you get a hole — a removable discontinuity — not an asymptote.

For horizontal asymptotes, compare the degree of the numerator to the degree of the denominator. Numerator degree less than denominator degree: the horizontal asymptote is y equals zero. Same degree: the asymptote is the ratio of the leading coefficients. Numerator degree greater: no horizontal asymptote — use long division to find a slant asymptote.

One critical reminder: horizontal transformations work opposite to what you expect. The function f of the quantity x minus 3 shifts the graph 3 units to the RIGHT, not left. This trips up students on every exam.

---

Unit 2: Exponential and Logarithmic Functions (6 minutes)

Exponential functions have the form a times b-to-the-x. When b is greater than 1, it's exponential growth. When b is between 0 and 1, it's exponential decay. The y-intercept is always the point (0, a), and the horizontal asymptote is y equals zero (or y equals k if the function is shifted vertically).

The number e — approximately 2.718 — is the base of natural exponential functions. The formula for compound interest is A equals P times the quantity 1 plus r over n, raised to the power n-t, where P is principal, r is the rate, n is the number of compounding periods per year, and t is time. For continuously compounded interest, use A equals P times e-to-the-rt.

[PAUSE]

Logarithms are the inverse of exponential functions. Log-base-b of x equals y means b-to-the-y equals x. You need to know three logarithm properties cold. The product rule: log of x-y equals log x plus log y. The quotient rule: log of x over y equals log x minus log y. The power rule: log of x-to-the-n equals n times log x. The change of base formula lets you compute any logarithm using natural log: log-base-b of x equals ln x divided by ln b.

[PAUSE]

When solving exponential equations, first try to rewrite both sides with the same base. If that's not possible, take the natural log of both sides and use the power rule to bring the variable down. When solving logarithmic equations, isolate the logarithm, rewrite in exponential form, solve, and always — always — check your answer in the original equation. Logarithmic equations frequently produce extraneous solutions that are not in the domain.

The domain of a logarithm requires the argument to be positive. So log of x requires x greater than zero. The vertical asymptote is at x equals zero.

---

Unit 3: Trigonometric and Polar Functions (7 minutes)

Angles in precalculus are primarily measured in radians. Memorize the key conversions: 30 degrees is pi over 6, 45 degrees is pi over 4, 60 degrees is pi over 3, 90 degrees is pi over 2, and 180 degrees is pi.

The unit circle is essential. Every point on it has coordinates (cosine theta, sine theta). You must memorize the values at 0, pi over 6, pi over 4, pi over 3, pi over 2, pi, 3 pi over 2, and 2 pi. In Quadrant 1, all values are positive. In Quadrant 2, sine is positive. In Quadrant 3, tangent is positive. In Quadrant 4, cosine is positive. Remember ASTC: All Students Take Calculus.

[PAUSE]

The general sinusoidal function is f of x equals a times sine of the quantity b-x plus c, plus d. Here, a is the amplitude — the distance from the midline to the peak. The period is 2 pi divided by the absolute value of b. The phase shift is negative c over b. And the midline is y equals d.

For tangent, the period is pi over the absolute value of b — half the sine and cosine period.

[PAUSE]

You must know these identities: the Pythagorean identity: sine-squared theta plus cosine-squared theta equals 1. The double-angle formulas: sine of 2 theta equals 2 sine theta cosine theta, and cosine of 2 theta has three equivalent forms: cosine-squared theta minus sine-squared theta, or 2 cosine-squared theta minus 1, or 1 minus 2 sine-squared theta. Choose the form that's most convenient based on what information you have.

[PAUSE]

Polar coordinates use a radius r and an angle theta. Convert to rectangular using x equals r cosine theta and y equals r sine theta. Key polar curves to know: circles, cardioids, rose curves, and limaçons. For rose curves, if n is odd you get n petals; if n is even you get 2n petals.

Inverse trigonometric functions: arcsine has domain negative 1 to 1 and range negative pi over 2 to pi over 2. Arccosine has the same domain but range 0 to pi. Arctangent has domain all real numbers and range negative pi over 2 to pi over 2.

---

Unit 4: Parameters, Vectors, and Matrices (5 minutes)

Parameters are constants in a function that can vary to create a family of related functions. When a parameter changes, certain features of the graph stay the same while others change. The locus is the set of points common to every function in the family. To find the locus, plug in specific values to identify what doesn't change.

For example, in f of x equals m times the quantity x minus 2, plus 3, the variable parameter is m. No matter what m is, f of 2 always equals 3. So the locus is the single point (2, 3).

[PAUSE]

Vectors have magnitude and direction. The magnitude of vector v equals the components a and b is the square root of a-squared plus b-squared. The dot product of u and v equals a1-a2 plus b1-b2. The dot product is a scalar — a number, not a vector. Two vectors are orthogonal — perpendicular — if and only if their dot product is zero. The angle between two vectors is found using: cosine of theta equals the dot product divided by the product of the magnitudes.

[PAUSE]

For 2-by-2 matrices, the determinant is ad minus bc. If the determinant is zero, the matrix has no inverse — it's singular. The inverse formula for a 2-by-2 matrix is: one over ad minus bc, times the matrix with d, negative b, negative c, a. To solve a system of equations using matrices, write it as A-x equals b, then x equals A-inverse times b.

---

Final Tips (1 minute)

To wrap up, remember these key exam strategies. First, the calculator is allowed on the entire exam — use it. Second, there is no penalty for guessing on multiple-choice — never leave a question blank. Third, show all your work on free-response questions — partial credit is generous. Fourth, check for extraneous solutions in logarithmic equations. Fifth, make sure your calculator is in radian mode before the exam starts.

You've put in the work. Trust your preparation, manage your time carefully, and you'll do well. Good luck on the AP Precalculus exam.

---

End of Audio Script. Estimated total time: 26 minutes at moderate pace.