2025 AP Precalculus Free-Response Questions Study Notes

Administrative and General Exam Instructions

• Exam Information: These are the AP® Precalculus Free-Response Questions for the 2025 exam by the College Board. AP Central is identified as the official online home for the AP Program at apcentral.collegeboard.org.
• Section II, Part A Directions:     * Time: 30 minutes for 2 free-response questions.     * Calculator Requirement: A graphing calculator is required. You may use a handheld or the application-provided calculator.     * Calculator Mode: Must be in radian mode.     * Supporting Work: All work must be shown in the free-response booklet using a pencil or a pen with black or dark blue ink. Answers without supporting work may not receive credit when such work is requested. Points are awarded based on correctness and completeness.     * Calculator Usage: Graphing calculators should be used for tasks including:         * Producing graphs and tables.         * Evaluating functions.         * Solving equations.         * Performing computations.     * Rounding and Precision:         * Avoid rounding intermediate computations.         * Unless otherwise specified, decimal approximations must be accurate to three places after the decimal point (e.g., $0.123$).         * It is suggested to store computed values (constants, functions, intermediate values) in the calculator to maintain maximum precision.     * Domain Assumption: Unless specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f(x)$ is a real number.     * Timing Warnings: The clock turns red when 5 minutes remain. Proctors will not provide other time updates or warnings.
• Section II, Part B Directions:     * Time: 30 minutes for 2 free-response questions.     * Calculator Usage: No calculator is allowed for this part of the exam.     * Domain and Trig Assumptions: Angle measures for trigonometric functions are assumed to be in radians. Solutions to equations must be real numbers. Exact values must be determined for expressions that can be calculated without a calculator (e.g., $\cos\left(\frac{2π}{9}\right)$, $\log_{2}(8)$, or $\sin^{-1}(1)$).     * Algebraic Rules: Terms should be combined using algebraic methods and rules for exponents and logarithms. For example, expressions like $2x + 3x$, $5^{2} \cdot 5^{3}$, and $\ln(3) + \ln(5)$ must be rewritten in equivalent forms ($5x$, $5^{5}$, and $\ln(15)$).

Question 1: Analysis of Functions f and g

• Function f Details:     * Function $f$ is defined for all real numbers and is strictly decreasing.     * Selected Values of f(x):         * When $x = -2$, $f(x) = 14$.         * When $x = -1$, $f(x) = 7$.         * When $x = 0$, $f(x) = 3.5$.         * When $x = 1$, $f(x) = 1.75$.         * When $x = 2$, $f(x) = 0.875$.
• Function g Details:     * $g(x) = -0.167x^3 + x^2 + 1.834x - 2$.
• Part A Tasks:     * i. Composite Function: Define $h(x) = (g \circ f)(x) = g(f(x))$. Find the value of $h(1)$ as a decimal approximation or indicate if it is undefined. Show supporting work.     * ii. Inverse Function: Find the value of $f^{-1}(3.5)$ or indicate if it is undefined.
• Part B Tasks:     * i. Roots: Find all values of $x$ (as decimal approximations) for which $g(x) = 0$. If no such values exist, indicate so.     * ii. End Behavior: Describe the end behavior of function $g$ as $x$ increases without bound ($x \to \infty$) using limit notation: $\lim_{x \to \infty} g(x)$.
• Part C Modeling:     * i. Model Type: Identify which function type best models $f$: linear, quadratic, exponential, or logarithmic.     * ii. Reasoning: Justify the choice based on the relationship between the change in output values and the change in input values. Reference specific values from the table (e.g., constant ratio or constant second difference).

Question 2: Quadratic Modeling of Song Plays

• Context: A musician releases a song on a streaming service (an online source for playing music). After several months (defined as $t = 0$), the musician uses an app to track the total number of plays in thousands since the song's release.
• Data Table:     * At $t = 0$ months: Total plays = $25$ thousand.     * At $t = 2$ months: Total plays = $30$ thousand.     * At $t = 4$ months: Total plays = $34$ thousand.
• Mathematical Model: The total number of plays is modeled by the quadratic function $D(t) = at^2 + bt + c$.
• Part A Tasks:     * i. System of Equations: Write three equations using the data points $(0, 25)$, $(2, 30)$, and $(4, 34)$ to solve for constants $a$, $b$, and $c$.     * ii. Constants: Find the values for $a$, $b$, and $c$ as decimal approximations.
• Part B Tasks:     * i. Average Rate of Change (AROC): Calculate the AROC of total plays (thousands per month) from $t = 0$ to $t = 4$. Show the computation: $\text{AROC} = \frac{D(4) - D(0)}{4 - 0}$.     * ii. Linear Estimation: Use the AROC from B(i) to estimate the total plays at $t = 1.5$ months. Show supporting work.     * iii. Comparison: Let $A(t)$ represent the estimate using AROC. It is given that A(1.5) < D(1.5). Explain why, in general, A(t) < D(t) for all $t$ such that 0 < t < 4. The explanation must reference the graph of $D$ (the quadratic model) and its geometric relationship to $A(t)$ (the secant line).
• Part C Domain Restriction:     * The quadratic model $D$ has exactly one absolute extremum (minimum or maximum).     * Explain how this vertex (minimum or maximum) can be used to determine a boundary for the domain of $D$ based on the real-world context of song plays over time.

Question 3: Periodic Motion of a Guitar String (No Calculator)

• Context: Guitar string vibration is modeled by a periodic function.
• Motion of Point X:     * At $t = 0$ seconds, Point X is at its highest position: $2\,mm$ above resting position ($h(0) = 2$).     * Sequence of motion: Starts at highest (2) → passes resting (0) → lowest (-2) → passes resting (0) → returns to highest (2).     * Frequency: This full motion occurs $200$ times in $1$ second.
• Function Definition: The sinusoidal function $h(t)$ models the distance from resting position. Positive values = above resting; Negative values = below resting.
• Part A Graph Analysis:     * The graph shows two full cycles with five labeled points: $F$, $G$, $J$, $K$, and $P$.     * Task: Determine possible coordinates $(t, h(t))$ for these five points based on the periodic motion described.
• Part B Function Form:     * Write the function in the form: $h(t) = a \sin(b(t + c)) + d$.     * Find the values for constants $a$, $b$, $c$, and $d$.
• Part C Interval Analysis:     * Let $t_1$ be the t-coordinate of $G$ and $t_2$ be the t-coordinate of $J$.     * i. Behavior on Interval (t1, t2): Identify if $h$ is positive/negative and increasing/decreasing.     * ii. Concavity: Describe the concavity of the graph on $(t_1, t_2)$ and determine if the rate of change of $h$ is increasing or decreasing.

Question 4: Algebraic and Trigonometric Manipulations (No Calculator)

• Part A Solving Equations:     * Given: $g(x) = 2 \log_{3}(x)$ and $h(x) = 4 \cos^2(x)$.     * i. Logarithmic Equation: Solve $g(x) = 4$ for $x$ in the domain of $g$.     * ii. Trigonometric Equation: Solve $h(x) = 3$ for $x$ in the interval $[0, \frac{2π}{9}]$.
• Part B Rewriting Expressions:     * i. Logarithm Combination: Given $j(x)$ involving multiple logs (e.g., $\log_{2}(x^2) + \log_{2}(x+3) - \log_{2}(x-1)$), rewrite as a single logarithm base $2$: $\log_{2}(\text{expression})$ without negative exponents.     * ii. Trigonometric Substitution: Given $k(x)$ involving functions like $\tan(x)$ and $\csc(x)$, rewrite the expression so that $\tan(x)$ appears exactly once and no other trigonometric functions are used.
• Part C Exponential Equation:     * Function: $m(x) = e^{2x} - e^x - 12$.     * Task: Find all input values in the domain of $m$ that yield an output of $0$ (solve $m(x) = 0$).