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AP Precalculus Free Response Questions

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  • Unit 1: Polynomial and Rational Functions (62)
  • Unit 2: Exponential and Logarithmic Functions (65)
  • Unit 3: Trigonometric and Polar Functions (49)
  • Unit 4: Functions Involving Parameters, Vectors, and Matrices (74)
Unit 1: Polynomial and Rational Functions

Analysis of a Rational Function with Factorable Denominator

A function is given by $$f(x)=\frac{x^2-5*x+6}{x^2-4}$$. Examine its domain and discontinuities.

Medium

Analysis of a Rational Function with Quadratic Components

Analyze the rational function $$f(x)= \frac{x^2 - 9}{x^2 - 4*x + 3}$$ and determine its key features

Medium

Analysis of Removable Discontinuities in an Experiment

In a chemical reaction process, the rate of reaction is modeled by $$R(x)=\frac{x^2-4}{x-2}$$ for $$

Easy

Analyzing a Rational Function with Asymptotes

Consider the rational function $$R(x)= \frac{(x-2)(x+3)}{(x-1)(x+4)}$$. Answer each part that follow

Medium

Analyzing Concavity in Polynomial Functions

A car’s displacement over time is modeled by the polynomial function $$f(x)= x^3 - 6*x^2 + 11*x - 6$

Medium

Analyzing End Behavior of Polynomial Functions

Consider the polynomial function $$P(x)= -2*x^4 + 3*x^3 - x + 5$$. Answer the following parts:

Easy

Average Rate of Change in Rational Functions

Let $$h(x)= \frac{3}{x-1}$$ represent the speed (in km/h) of a vehicle as a function of a variable x

Medium

Average Rate of Change of a Rational Function

For the rational function $$r(x)= \frac{4*x}{x+2}$$, answer the following:

Medium

Carrying Capacity in Population Models

A rational function $$P(t) = \frac{50*t}{t + 10}$$ is used to model a population approaching its car

Easy

Comparative Analysis of Polynomial and Rational Functions

A function is defined piecewise by $$ f(x)=\begin{cases} x^2-4 & \text{if } x\le2, \\ \frac{x^2-4}{x

Medium

Composite Function Analysis in Environmental Modeling

Environmental data shows the concentration (in mg/L) of a pollutant over time (in hours) as given in

Hard

Composite Function Transformations

Consider the polynomial function $$f(x)= x^2-4$$. A new function is defined by $$g(x)= \ln(|f(x)+5|)

Hard

Composite Functions and Inverses

Let $$f(x)= 3*(x-2)^2+1$$.

Medium

Constructing a Function Model from Experimental Data

An engineer collects data on the stress (in MPa) experienced by a material under various applied for

Medium

Designing a Rational Function to Meet Given Criteria

A mathematician wishes to construct a rational function R(x) that satisfies the following properties

Extreme

Determining Degree from Discrete Data

Below is a table representing the output values of a polynomial function for equally-spaced input va

Medium

Determining Domain and Range from Graphical Data

A function is represented by a graph with certain open and closed endpoints. A table of select input

Easy

Determining Polynomial Degree from Finite Differences

A function $$f(x)$$ is defined on equally spaced values of $$x$$, with the following data: | x | f(

Easy

Determining the Degree of a Polynomial via Differences

A function $$f(x)$$ is defined on equally spaced inputs and the following table gives selected value

Easy

Discontinuity Analysis in a Rational Function with High Degree

Consider the function $$f(x)=\frac{x^3-8}{x^2-4}$$. Answer the following:

Hard

Evaluating Limits and Discontinuities in a Rational Function

Consider the rational function $$f(x)=\frac{x^2-4}{x-2}$$, which is defined for all real $$x$$ excep

Medium

Evaluating Limits Involving Rational Expressions with Asymptotic Behavior

Consider the function $$f(x)=\frac{2*x^2-3*x-5}{x^2-1}$$. Answer the following:

Hard

Examining End Behavior of Polynomial Functions

Consider the polynomial function $$f(x)= -3*x^4 + 2*x^3 - x + 7$$. Answer the following parts.

Easy

Finding and Interpreting Inflection Points

Consider the polynomial function $$f(x)= x^3 - 6*x^2 + 9*x + 1$$. Answer the following parts.

Medium

Function Model Construction from Data Set

A data set shows how a quantity V changes over time t as follows: | Time (t) | Value (V) | |-------

Medium

Function Simplification and Graph Analysis

Consider the function $$h(x)= \frac{x^2 - 4}{x-2}$$. Answer the following parts.

Easy

Geometric Series Model in Area Calculations

An architect designs a sequence of rectangles such that each rectangle's area is 0.8 times the area

Easy

Graph Interpretation and Log Transformation

An experiment records the reaction time R (in seconds) of an enzyme as a power function of substrate

Medium

Inverse Analysis of a Modified Rational Function

Consider the function $$f(x)=\frac{x^2+1}{x-1}$$. Answer the following questions concerning its inve

Extreme

Inverse Analysis of a Quartic Polynomial Function

Consider the quartic function $$f(x)= (x-1)^4 + 2$$. Answer the following questions concerning its i

Hard

Inverse Analysis of a Transformed Quadratic Function

Consider the function $$f(x)= -3*(x-2)^2 + 7$$ with a domain restriction that ensures one-to-one beh

Medium

Inverse Analysis of an Even Function with Domain Restriction

Consider the function $$f(x)=x^2$$ defined on the restricted domain $$x \ge 0$$. Answer the followin

Easy

Inverse Function of a Rational Function with a Removable Discontinuity

Consider the function $$f(x)= \frac{x^2-4}{x-2}$$. Answer the following questions regarding its inve

Medium

Investigating End Behavior of a Polynomial Function

Consider the polynomial function $$f(x)= -4*x^4+ x^3+ 2*x^2-7*x+1$$.

Easy

Investigating Piecewise Behavior of a Function

A function is defined as follows: $$ f(x)=\begin{cases} \frac{x^2-9}{x-3} & x<3, \\ 2*x+1 & x\ge3

Medium

Logarithmic Equation Solving in a Financial Model

An investor compares two savings accounts. Account A grows continuously according to the model $$A(t

Medium

Modeling Inverse Variation with Rational Functions

An experiment shows that the intensity of a light source varies inversely with the square of the dis

Medium

Modeling Inverse Variation: A Rational Approach

A variable $$y$$ is inversely proportional to $$x$$. Data indicates that when $$x=4$$, $$y=2$$, and

Easy

Modeling with Inverse Variation: A Rational Function

A physics experiment models the intensity $$I$$ of light as inversely proportional to the square of

Easy

Office Space Cubic Function Optimization

An office building’s usable volume (in thousands of cubic feet) is modeled by the cubic function $$V

Hard

Parameter Identification in a Rational Function Model

A rational function modeling a certain phenomenon is given by $$r(x)= \frac{k*(x - 2)}{x+3}$$, where

Easy

Piecewise Function Analysis

Consider the piecewise function defined by $$ f(x) = \begin{cases} x^2 - 1, & x < 2 \\ 3*

Medium

Piecewise Function and Domain Restrictions

A temperature function is defined as $$ T(x)=\begin{cases} \frac{x^2-25}{x-5} & x<5, \\ 3*x-10 & x\g

Medium

Piecewise Function without a Calculator

Let the function $$f(x)=\begin{cases} x^2-1 & \text{for } x<2, \\ \frac{x^2-4}{x-2} & \text{for } x\

Medium

Polynomial Division in Limit Evaluation

Consider the rational function $$R(x) = \frac{2*x^3 + 3*x^2 - x + 4}{x - 2}$$.

Hard

Polynomial End Behavior and Zeros Analysis

A polynomial function is given by $$f(x)= 2*x^4 - 3*x^3 - 12*x^2$$. This function models a physical

Medium

Polynomial Long Division and Slant Asymptote

Perform polynomial long division on the function $$f(x)= \frac{3*x^3 - 2*x^2 + 4*x - 5}{x^2 - 1}$$,

Hard

Polynomial Long Division and Slant Asymptote

Consider the function $$P(x)= \frac{2*x^3 - 3*x^2 + x - 5}{x-2}$$. Answer the following parts.

Hard

Polynomial Transformation Challenge

Consider the function transformation given by $$g(x)= -2*(x+1)^3 + 3$$. Answer each part that follow

Easy

Predator-Prey Dynamics as a Rational Function

An ecologist models the ratio of predator to prey populations with the rational function $$P(x) = \f

Medium

Product Revenue Rational Model

A company’s product revenue (in thousands of dollars) is modeled by the rational function $$R(x)= \f

Medium

Rational Inequalities and Test Intervals

Solve the inequality $$\frac{x-3}{(x+2)(x-1)} < 0$$. Answer the following parts.

Medium

Regression Model Selection for Experimental Data

Experimental data was collected, and the following table represents the relationship between a contr

Extreme

Roller Coaster Curve Analysis

A roller coaster's vertical profile is modeled by the polynomial function $$f(x)= -0.05*x^3 + 1.2*x^

Medium

Solving a System of Equations: Polynomial vs. Rational

Consider the system of equations where $$f(x)= x^2 - 1$$ and $$g(x)= \frac{2*x}{x+2}$$. Answer the f

Hard

Temperature Rate of Change Analysis

In a manufacturing process, the temperature in a reactor is recorded over time. Using the table prov

Medium

Transformation and Reflection of a Parent Function

Given the parent function $$f(x)= x^2$$, consider the transformed function $$g(x)= -3*(x+2)^2 + 5$$.

Easy

Transformation in Composite Functions

Let the parent function be $$f(x)= x^2$$ and consider the composite transformation given by $$g(x)=

Easy

Trigonometric Function Analysis and Identity Verification

Consider the trigonometric function $$g(x)= 2*\tan(3*x-\frac{\pi}{4})$$, where $$x$$ is measured in

Medium

Zero Finding and Sign Charts

Consider the function $$p(x)= (x-2)(x+1)(x-5)$$.

Easy

Zeros and Complex Conjugates in Polynomial Functions

A polynomial function of degree 4 is known to have real zeros at $$x=1$$ and $$x=-2$$, and two non-r

Easy

Zeros and Factorization Analysis

A fourth-degree polynomial $$Q(x)$$ is known to have zeros at $$x=-3$$ (with multiplicity 2), $$x=1$

Medium
Unit 2: Exponential and Logarithmic Functions

Analyzing a Logarithmic Function from Data

A scientist proposes a logarithmic model for a process given by $$f(x)= \log_2(x) + 1$$. The observe

Medium

Analyzing Exponential Function Behavior

Consider the function \(f(x)=5\cdot e^{-0.3\cdot x}+2\). (a) Determine the horizontal asymptote of

Easy

Analyzing Exponential Function Behavior from a Graph

An exponential function is depicted in the graph provided. Analyze the key features of the function.

Easy

Analyzing Social Media Popularity with Logarithmic Growth

A social media analyst is studying the early-stage growth of a new account's followers. Initially, t

Extreme

Arithmetic Savings Plan

A person decides to save money every month, starting with an initial deposit of $$50$$ dollars, with

Easy

Arithmetic Sequence Analysis

Consider an arithmetic sequence with initial term $$a_0 = 5$$ and constant difference $$d$$. Given t

Easy

Arithmetic Sequence Derived from Logarithms

Consider the exponential function $$f(x) = 10 \cdot 2^x$$. A new dataset is formed by taking the com

Hard

Arithmetic Sequence in Savings

A student saves money every month and deposits a fixed additional amount each month, so that her sav

Easy

Bacterial Growth Model

In a laboratory experiment, a bacteria colony doubles every 3 hours. The initial count is $$500$$ ba

Medium

Comparing Exponential and Linear Growth in Business

A company is analyzing its revenue over several quarters. They suspect that part of the growth is li

Medium

Competing Exponential Cooling Models

Two models are proposed for the cooling of an object. Model A is $$T_A(t) = T_env + 30·e^(-0.5*t)$$

Hard

Composite Function Analysis: Identity and Inverses

Let $$f(x)= 2^x$$ and $$g(x)= \log_2(x)$$.

Medium

Composite Function Involving Exponential and Logarithmic Components

Consider the composite function defined by $$h(x) = \log_5(2\cdot 5^x + 3)$$. Answer the following p

Extreme

Composite Functions Involving Exponential and Logarithmic Functions

Let $$f(x) = e^x$$ and $$g(x) = \ln(x)$$. Explore the compositions of these functions and their rela

Easy

Composition and Transformation Functions

Let $$g(x)= \log_{5}(x)$$ and $$h(x)= 5^x - 4$$.

Hard

Composition of Exponential and Logarithmic Functions

Consider the functions $$f(x)= \log_5\left(\frac{x}{2}\right)$$ and $$g(x)= 10\cdot 5^x$$. Answer th

Medium

Compound Interest Model with Regular Deposits

An account offers an annual interest rate of 5% compounded once per year. In addition to an initial

Hard

Compound Interest vs. Simple Interest

A financial analyst is comparing two interest methods on an initial deposit of $$10000$$ dollars. On

Medium

Compound Interest with Periodic Deposits

An investor opens an account with an initial deposit of $$5000$$ dollars and adds an additional $$50

Medium

Data Modeling: Exponential vs. Linear Models

A scientist collected data on the growth of a substance over time. The table below shows the measure

Medium

Determining an Exponential Model from Data

An outbreak of a virus produced the following data: | Time (days) | Infected Count | |-------------

Medium

Domain Restrictions in Logarithmic Functions

Consider the logarithmic function $$f(x) = \log_4(x^2 - 9)$$.

Hard

Earthquake Intensity and Logarithmic Function

The Richter scale measures earthquake intensity using a logarithmic function. Suppose the energy rel

Easy

Earthquake Intensity on the Richter Scale

The Richter scale defines earthquake magnitude as \(M = \log_{10}(I/I_{0})\), where \(I/I_{0}\) is t

Medium

Earthquake Magnitude and Logarithms

The Richter scale is logarithmic and is used to measure earthquake intensity. The energy released, \

Hard

Estimating Rates of Change from Table Data

A cooling object has its temperature recorded at various time intervals as shown in the table below:

Hard

Exponential Decay and Half-Life

A radioactive substance decays according to an exponential decay function. The substance initially w

Medium

Exponential Equations via Logarithms

Solve the exponential equation $$3 * 2^(2*x) = 6^(x+1)$$.

Hard

Exponential Function Transformations

Consider an exponential function defined by f(x) = a·bˣ. A graph of this function is provided in the

Medium

Exponential Function with Compound Transformations and Its Inverse

Consider the function $$f(x)=2^(x-2)+3$$. Determine its invertibility, find its inverse function, an

Easy

Finding Terms in a Geometric Sequence

A geometric sequence is known to satisfy $$g_3=16$$ and $$g_7=256$$.

Easy

Finding the Inverse of an Exponential Function

Given the exponential function $$f(x)= 4\cdot e^{0.5*x} - 3,$$ find the inverse function $$f^{-1}(

Medium

Fitting a Logarithmic Model to Sales Data

A company observes that its sales revenue (in thousands of dollars) based on advertising spend (in t

Hard

Fractal Pattern Growth

A fractal pattern is generated such that after its initial creation, each iteration adds an area tha

Medium

General Exponential Equation Solving

Solve the equation $$2^{x}+2^{x+1}=48$$. (a) Factor the equation by rewriting \(2^{x+1}\) in terms

Hard

Geometric Sequence and Exponential Modeling

A geometric sequence can be viewed as an exponential function defined by a constant ratio. The table

Medium

Inverse and Domain of a Logarithmic Transformation

Given the function $$f(x) = \log_3(x - 2) + 4$$, answer the following parts.

Medium

Inverse Function of an Exponential Function

Consider the function $$f(x)= 3\cdot 2^x + 4$$.

Hard

Inverse Functions of Exponential and Log Functions

Let \(f(x)=4\cdot3^{x}\) and \(g(x)=\log_{3}(x/4)\). (a) Show that \(f(g(x))=x\) for all \(x\) in t

Easy

Inverse of an Exponential Function

Given the exponential function $$f(x) = 5 \cdot 2^x$$, determine its inverse.

Easy

Inverse Relationships in Exponential and Logarithmic Functions

Consider the functions \(f(x)=2^{(x-1)}+3\) and \(g(x)=\log_{2}(x-3)+1\). (a) Discuss under what co

Extreme

Logarithmic Analysis of Earthquake Intensity

The magnitude of an earthquake on the Richter scale is determined using a logarithmic function. Cons

Medium

Logarithmic Function and Its Inverse

Let $$f(x)=\log_5(2x+3)-1$$. Analyze the function's one-to-one property and determine its inverse, i

Easy

Logarithmic Inequalities

Solve the inequality $$\log_{2}(x-1) > 3$$.

Easy

Modeling Bacterial Growth with Exponential Functions

A research laboratory is tracking the growth of a bacterial culture. A graph showing experimental da

Medium

Parameter Sensitivity in Exponential Functions

Consider an exponential function of the form $$f(x) = a \cdot b^{c x}$$. Suppose two data points are

Hard

pH Measurement and Inversion

A researcher uses the function $$f(x)=-\log_{10}(x)+7$$ to measure the pH of a solution, where $$x$$

Easy

Piecewise Exponential and Logarithmic Function Discontinuities

Consider the function defined by $$ f(x)=\begin{cases} 2^x + 1, & x < 3,\\ 5, & x = 3,

Hard

Population Growth with an Immigration Factor

A city's population is modeled by an equation that combines exponential growth with a constant linea

Hard

Profit Growth with Combined Models

A company's profit is modeled by a function that combines an arithmetic increase with exponential gr

Hard

Radioactive Decay and Logarithmic Inversion

A radioactive substance decays such that its mass halves every 8 years. At time \(t=0\), the substan

Medium

Radioactive Decay Model

A radioactive substance decays according to the function $$f(t)= a \cdot e^{-kt}$$. In an experiment

Hard

Radioactive Decay Modeling

A radioactive substance decays with a half-life of $$5$$ years. A sample has an initial mass of $$80

Medium

Radioactive Decay Modeling

A radioactive substance decays according to the model N(t) = N₀ · e^(-k*t), where t is measured in y

Medium

Radioactive Decay Problem

A radioactive substance decays exponentially with a half-life of 5 years and an initial mass of $$20

Easy

Savings Account Growth: Arithmetic vs Geometric Sequences

An individual opens a savings account that incorporates both regular deposits and interest earnings.

Hard

Semi-Log Plot and Exponential Model

A researcher studies the concentration of a chemical over time using a semi-log plot, where the y-ax

Extreme

Shifted Exponential Function and Its Inverse

Consider the function $$f(x)=7-4*2^(x-3)$$. Determine its one-to-one nature, find its inverse functi

Hard

Solving Exponential Equations Using Logarithms

Solve the exponential equation $$5\cdot2^{(x-2)}=40$$. (a) Isolate the exponential term and solve f

Easy

Solving Logarithmic Equations and Checking Domain

An engineer is analyzing a system and obtains the following logarithmic equation: $$\log_3(x+2) + \

Hard

Transformations of Exponential Functions

Consider the base exponential function $$f(x)= 3 \cdot 2^x$$. A transformed function is defined by

Easy

Transformations of Exponential Functions

Consider the exponential function $$f(x) = 3 \cdot 2^x$$. This function is transformed to produce $$

Medium

Translated Exponential Function and Its Inverse

Consider the function $$f(x)=5*2^(x+3)-8$$. Analyze its properties by confirming its one-to-one natu

Easy

Validating the Negative Exponent Property

Demonstrate the negative exponent property using the expression $$b^{-3}$$.

Easy

Wildlife Population Decline

A wildlife population declines by 15% each year, forming a geometric sequence.

Easy
Unit 3: Trigonometric and Polar Functions

Analysis of Rose Curves

A polar curve is given by the equation $$r=4*\cos(3*θ)$$ which represents a rose curve. Analyze the

Medium

Analyzing Sinusoidal Function Rate of Change

A sound wave is modeled by the function $$f(t)=4*\sin(\frac{\pi}{2}*(t-1))+5$$, where t is measured

Hard

Analyzing the Tangent Function

Consider the tangent function $$T(x)=\tan(x)$$.

Easy

Combining Logarithmic and Trigonometric Equations

Consider a model where the amplitude of a cosine function is modulated by an exponential decay. The

Hard

Comparing Sinusoidal Functions

Consider the functions $$f(x)=\sin(x)$$ and $$g(x)=\cos\Bigl(x-\frac{\pi}{2}\Bigr)$$.

Easy

Composite Function Analysis with Polar and Trigonometric Elements

A radar system uses the polar function $$r(\theta)=5+2*\sin(\theta)$$ to model the distance to a tar

Medium

Concavity in the Sine Function

Consider the function $$h(x) = \sin(x)$$ defined on the interval $$[0, 2\pi]$$.

Medium

Converting and Graphing Polar Equations

Consider the polar equation $$r=2*\cos(\theta)$$.

Medium

Converting Complex Numbers to Polar Form

Convert the complex number $$3-3*\text{i}$$ to polar form and use this representation to compute the

Medium

Coterminal Angles and Unit Circle Analysis

Identify coterminal angles and determine the corresponding coordinates on the unit circle.

Easy

Damped Oscillations: Combining Sinusoidal Functions and Geometric Sequences

A mass-spring system oscillates with decreasing amplitude following a geometric sequence. Its displa

Hard

Daylight Hours Modeling

A city's daylight hours vary sinusoidally throughout the year. It is observed that the maximum dayli

Medium

Equivalent Representations Using Pythagorean Identity

Using trigonometric identities, answer the following:

Medium

Exploring a Limacon

Consider the polar equation $$r=2+3\,\cos(\theta)$$.

Hard

Exploring Coterminal Angles and Periodicity

Analyze the concept of coterminal angles.

Easy

Exploring Inverse Trigonometric Functions

Consider the inverse sine function $$\arcsin(x)$$, defined for \(x\in[-1,1]\).

Easy

Exploring Limacons in Polar Coordinates

Consider the polar function $$r=2+3*\cos(θ)$$ which represents a limacon. Evaluate its key features

Hard

Exploring Rates of Change in Polar Functions

Given the polar function $$r(\theta) = 2 + \sin(\theta)$$, answer the following:

Hard

Graph Interpretation from Tabulated Periodic Data

A study recorded the oscillation of a pendulum over time. Data is provided in the table below showin

Medium

Graphical Reflection of Trigonometric Functions and Their Inverses

Consider the sine function and its inverse. The graph of an inverse function is the reflection of th

Easy

Inverse Tangent Composition and Domain

Consider the composite function $$f(x) = \arctan(\tan(x))$$.

Extreme

Inverse Trigonometric Function Analysis

Consider the function $$f(x) = 2*\sin(x)$$.

Medium

Modeling Daylight Hours with a Sinusoidal Function

A city's daylight hours throughout the year are periodic. At t = 0 months, the city experiences maxi

Medium

Modeling Daylight Hours with a Sinusoidal Function

A study in a northern city recorded the number of daylight hours over the course of one year. The ob

Medium

Period Detection and Frequency Analysis

An engineer analyzes a signal modeled by $$P(t)=6*\cos(5*(t-1))$$.

Medium

Periodic Phenomena: Seasonal Daylight Variation

A scientist is studying the variation in daylight hours over the course of a year in a northern regi

Medium

Polar Circle Graph

Consider the polar equation $$r = 4$$ which represents a circle.

Easy

Polar Coordinates and Graphing a Circle

Answer the following questions on polar coordinates:

Medium

Polar Function with Rate of Change Analysis

Given the polar function $$r(\theta)=2+\sin(\theta)$$, analyze its behavior.

Medium

Polar Interpretation of Periodic Phenomena

A meteorologist models wind speed variations with direction over time using a polar function of the

Hard

Proof and Application of Trigonometric Sum Identities

Trigonometric sum identities are a powerful tool in analyzing periodic phenomena.

Extreme

Real-World Modeling: Exponential Decay with Sinusoidal Variation

A river's water level is affected by tides and evaporation. It is modeled by the function $$L(t)=8*

Extreme

Sine and Cosine Graph Transformations

Consider the functions $$f(\theta)=\sin(\theta)$$ and $$g(\theta)=\sin(\theta+\frac{\pi}{3})$$, whic

Easy

Sinusoidal Function Transformation Analysis

Analyze the sinusoidal function given by $$g(\theta)=3*\sin\left(2*(\theta-\frac{\pi}{4})\right)-1$$

Medium

Sinusoidal Function Transformations in Signal Processing

A communications engineer is analyzing a signal modeled by the sinusoidal function $$f(x)=3*\cos\Big

Medium

Sinusoidal Transformation and Logarithmic Manipulation

An electronic signal is modeled by $$S(t)=5*\sin(3*(t-2))$$ and its decay is described by $$D(t)=\ln

Hard

Solving a System Involving Exponential and Trigonometric Functions

Consider the system of equations: $$ \begin{aligned} f(x)&=e^{-x}+\sin(x)=1, \\ g(x)&=\ln(2-x)+\co

Extreme

Solving a Trigonometric Equation with Sum and Difference Identities

Solve the equation $$\sin\left(x+\frac{\pi}{6}\right)=\cos(x)$$ for $$0\le x<2\pi$$.

Hard

Solving a Trigonometric Inequality

Solve the inequality $$\sin(x) > \frac{1}{2}$$ for $$x$$ in the interval $$[0, 2\pi]$$.

Hard

Solving a Trigonometric Inequality

Solve the inequality $$\sin(x)>\frac{1}{2}$$ for \(0\le x<2\pi\).

Easy

Solving Trigonometric Equations

Solve the equation $$\sin(x)+\cos(x)=1$$ for \(0\le x<2\pi\).

Medium

Solving Trigonometric Equations

Solve the trigonometric equation $$\sin(\theta) + \sqrt{3}*\cos(\theta)=1$$.

Hard

Solving Trigonometric Equations in a Specified Interval

Solve the given trigonometric equations within specified intervals and explain the underlying reason

Easy

Tangent and Cotangent Equation

Consider the trigonometric equation $$\tan(x) - \cot(x) = 0$$ for $$x$$ in the interval $$[0, 2\pi]$

Medium

Tangent Function and Asymptotes

Examine the function $$f(\theta)=\tan(\theta)$$ defined on the interval $$\left(-\frac{\pi}{2}, \fra

Medium

Tidal Motion Analysis

A coastal region's tidal heights are modeled by a sinusoidal function $$f(t) = A * \sin(b*(t - c)) +

Medium

Tidal Patterns and Sinusoidal Modeling

A coastal engineer models tide heights (in meters) as a function of time (in hours) using the sinuso

Medium

Understanding Coterminal Angles and Their Applications

Coterminal angles are important in trigonometry as they represent angles with the same terminal side

Easy

Using Trigonometric Sum and Difference Identities

Prove the identity $$\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$$ and apply it.

Hard
Unit 4: Functions Involving Parameters, Vectors, and Matrices

Acceleration in a Vector-Valued Function

Given the particle's position vector $$\mathbf{r}(t) = \langle t^2, t^3 - 3*t \rangle$$, answer the

Medium

Analysis of a Vector-Valued Position Function

Consider the vector-valued function $$\mathbf{p}(t) = \langle 2*t + 1, 3*t - 2 \rangle$$ representin

Easy

Analysis of Vector Directions and Transformations

Given the vectors $$\mathbf{a}=\langle -1,2\rangle$$ and $$\mathbf{b}=\langle 4,3\rangle$$, perform

Hard

Analyzing a Piecewise Function Involving Absolute Value and Removability

Consider the function $$F(x)=\begin{cases} \frac{|x-2|(x+1) - (x-2)(x+1)}{x-2} & \text{if } x \neq 2

Hard

Analyzing the Composition of Two Matrix Transformations

Let matrices be given by $$A=\begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}$$ and $$B=\begin{pmatrix}2 & 0

Medium

Average Rate of Change in Parametric Motion

For the parametric functions $$x(t) = t^3 - 3*t + 2$$ and $$y(t) = 2*t^2 - t$$ defined for $$t \in [

Medium

Composition of Linear Transformations

Let $$A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$$ and $$B = \begin{pmatrix} 3 & 0 \\ 1 & 2 \e

Medium

Computing Average Rate of Change in Parametric Functions

Consider a particle moving with its position given by $$x(t)=t^2 - 4*t + 3$$ and $$y(t)=2*t + 1$$. A

Medium

Determinant and Area of a Parallelogram

Given vectors $$\vec{u}=\langle 2, 3 \rangle$$ and $$\vec{v}=\langle -1, 4 \rangle$$, consider the 2

Medium

Determinant and Inverse Calculation

Given the matrix $$C = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$$, answer the following:

Easy

Discontinuity Analysis in an Implicitly Defined Function

Consider the circle defined by $$x^2+y^2=4$$. A piecewise function for $$y$$ is attempted as $$y(x)=

Medium

Dot Product, Projection, and Angle Calculation

Let $$\mathbf{u}=\langle4, 1\rangle$$ and $$\mathbf{v}=\langle2, 3\rangle$$.

Medium

Eliminating the Parameter in an Implicit Function

A curve is defined by the parametric equations $$x(t)=t+1$$ and $$y(t)=t^2-1$$.

Medium

Evaluating a Piecewise Function in a Vector Context

A vector-valued function is defined as $$\mathbf{p}(t)=\langle p_x(t),p_y(t) \rangle$$ where the hor

Medium

Evaluating Limits in a Parametrically Defined Motion Scenario

A particle’s motion is given by the parametric equations: $$x(t)=\begin{cases} \frac{t^2-9}{t-3} & \

Medium

Exponential Decay Modeled by Matrices

Consider a system where decay over time is modeled by the matrix $$M(t)= e^{-k*t}I$$, where I is the

Medium

Exponential Parametric Function and its Inverse

Consider the exponential function $$f(x)=e^{x}+2$$ defined for all real numbers. Analyze the functio

Medium

Ferris Wheel Motion

A Ferris wheel rotates counterclockwise with a center at $$ (2, 3) $$ and a radius of $$5$$. The whe

Medium

Finding Angle Between Vectors

Given vectors $$\mathbf{a}=\langle 1,2 \rangle$$ and $$\mathbf{b}=\langle 3,4 \rangle$$, determine t

Medium

FRQ 2: Circular Motion and Parameterization

Consider a particle moving along a circular path represented by the parametric function $$f(t)=(x(t)

Medium

FRQ 4: Parametric Representation of a Parabola

The parabola given by $$y=(x-1)^2-2$$ can be represented parametrically as $$ (x(t), y(t)) = (t, (t-

Medium

FRQ 5: Parametrically Defined Ellipse

An ellipse is described parametrically by $$x(t)=3*\cos(t)$$ and $$y(t)=2*\sin(t)$$ for $$t\in[0,2\p

Hard

FRQ 6: Implicit Function to Parametric Representation

Consider the implicitly defined circle $$x^2+y^2-6*x+8*y+9=0$$.

Hard

FRQ 10: Unit Vectors and Direction

Consider the vector $$\textbf{w}=\langle -5, 12 \rangle$$.

Easy

FRQ 12: Matrix Multiplication in Transformation

Let matrices $$A=\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$$ and $$B=\begin{bmatrix}0 & 1\\1 & 0\end{

Hard

FRQ 13: Area Determined by a Matrix's Determinant

Vectors $$\textbf{v}=\langle4,3\rangle$$ and $$\textbf{w}=\langle-2,5\rangle$$ form a parallelogram.

Medium

FRQ 15: Composition of Linear Transformations

Consider two linear transformations represented by the matrices $$A=\begin{bmatrix}2 & 0\\1 & 3\end{

Hard

FRQ 16: Inverse of a Linear Transformation

Let the transformation be given by the matrix $$T=\begin{bmatrix}5 & 2\\3 & 1\end{bmatrix}$$.

Hard

FRQ 17: Matrix Representation of a Reflection

A reflection about the line \(y=x\) is given by the matrix $$Q=\begin{bmatrix}0 & 1\\1 & 0\end{bmatr

Easy

FRQ 19: Parametric Functions and Matrix Transformation

A particle's motion is given by the parametric equations $$f(t)=(t, t^2)$$ for $$t\in[0,2]$$. A line

Hard

Graphical Analysis of Parametric Motion

A particle moves in the plane with its position defined by the functions $$x(t)= t^2 - 2*t$$ and $$y

Easy

Growth Models: Exponential and Logistic Equations

Consider a population growth model of the form $$P(t)= P_{0}*e^{r*t}$$ and a logistic model given by

Medium

Implicitly Defined Circle

Consider the implicitly defined function given by $$x^2+y^2=16$$, which represents a circle.

Easy

Inverse Analysis of a Rational Function

Consider the function $$f(x)=\frac{2*x+3}{x-1}$$. Analyze the properties of this function and its in

Medium

Inverse Matrix with a Parameter

Consider the 2×2 matrix $$A=\begin{pmatrix} a & 2 \\ 3 & 4 \end{pmatrix}.$$ (a) Express the deter

Medium

Investigating Inverse Transformations in the Plane

Consider the linear transformation defined by $$L(\mathbf{v})=\begin{pmatrix}2 & 1\\3 & 4\end{pmatri

Medium

Linear Transformations via Matrices

A linear transformation \(L\) in \(\mathbb{R}^2\) is defined by $$L(x,y)=(3*x- y, 2*x+4*y)$$. This t

Medium

Logarithmic and Exponential Parametric Functions

A particle’s position is defined by the parametric equations $$x(t)= \ln(1+t)$$ and $$y(t)= e^{1-t}$

Medium

Matrices as Representations of Rotation

Consider the matrix $$A=\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$$, which represents a rotation in

Easy

Matrix Applications in State Transitions

In a system representing transitions between two states, the following transition matrix is used: $

Hard

Matrix Methods for Solving Linear Systems

Solve the system of linear equations below using matrix methods: $$2x+3y=7$$ $$4x-y=5$$

Easy

Matrix Modeling in Population Dynamics

A biologist is studying a species with two age classes: juveniles and adults. The population dynamic

Extreme

Matrix Modeling of State Transitions

In a two-state system, the transition matrix is given by $$T=\begin{pmatrix}0.8 & 0.2 \\ 0.3 & 0.7\e

Extreme

Matrix Transformation of a Vector

Let the transformation matrix be $$A=\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix},$$ and let the

Medium

Modeling State Transitions with a Transition Matrix (Probability-Based Scenario)

A small business models its customer behavior between two states: Regular and Occasional. The transi

Hard

Modified Circular Motion: Transformation Effects

Consider the parametric equations $$x(t)=2+4\cos(t)$$ and $$y(t)=-3+4\sin(t)$$ which describe a curv

Medium

Parabolic and Elliptical Parametric Representations

A parabola is given by the equation $$y=x^2-4*x+3$$.

Medium

Parabolic Motion in a Parametric Framework

A projectile is launched with its motion described by the equations $$x(t)=4*t$$ and $$y(t)=-4.9*t^2

Medium

Parametric Curve with Logarithmic and Exponential Components

A curve is described by the parametric equations $$x(t)= t + \ln(t)$$ and $$y(t)= e^{t} - 3$$ for t

Medium

Parametric Equations and Inverses

A curve is defined parametrically by $$x(t)=t+2$$ and $$y(t)=3*t-1$$.

Medium

Parametric Equations of an Ellipse

Consider the ellipse defined by $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. Answer the following:

Easy

Parametric Function and Its Inverse: Parabolic Function

Consider the function $$f(x)= (x-1)^2 + 2$$ for x \(\ge\) 1. (a) Provide a parametrization for the

Hard

Parametric Function Modeling and Discontinuity Analysis

A particle moves in the plane with its horizontal position described by the piecewise function $$x(t

Medium

Parametric Motion with Variable Rates

A particle moves in the plane with its motion described by $$x(t)=4*t-t^2$$ and $$y(t)=t^2-2*t$$.

Hard

Parametric Representation of a Hyperbola

For the hyperbola given by $$\frac{x^2}{9}-\frac{y^2}{4}=1$$:

Hard

Parametric Representation of a Line: Motion of a Car

A car travels in a straight line from point A = (2, -1) to point B = (10, 7) at a constant speed. (

Easy

Parametric Representation of a Parabola

Consider the parabola defined by $$y= 2*x^2 + 3$$. Answer the following:

Easy

Parametrizing a Linear Path: Car Motion

A car moves along a straight line from point $$A=(1,2)$$ to point $$B=(7,8)$$.

Easy

Parametrizing a Parabola

A parabola is defined parametrically by $$x(t)=t$$ and $$y(t)=t^2$$.

Easy

Particle Motion from Parametric Equations

A particle moves in the plane with position functions $$x(t)=t^2-2*t$$ and $$y(t)=4*t-t^2$$, where $

Medium

Planar Motion Analysis

A particle moves in the plane with parametric functions $$x(t)= 3*t - t^2$$ and $$y(t)= 4*t - 2*t^2$

Medium

Position and Velocity Vectors

For a particle with position $$\mathbf{p}(t)=\langle2*t+1, 3*t-2\rangle$$, where $$t$$ is in seconds

Easy

Rational Piecewise Function with Parameter Changes: Discontinuity Analysis

Let $$R(t)=\begin{cases} \frac{3t^2-12}{t-2} & \text{if } t\neq2, \\ 5 & \text{if } t=2 \end{cases}$

Medium

Resolving Discontinuities in an Elliptical Parameterization

An ellipse is parameterized by the following equations: $$x(\theta)=\begin{cases} 5\cos(\theta) & \t

Easy

Rotation of a Force Vector

A force vector is given by \(\vec{F}= \langle 10, 5 \rangle\). This force is rotated by 30° counterc

Easy

Table-Driven Analysis of a Piecewise Defined Function

A researcher defines a function $$h(x)=\begin{cases} \frac{x^2 - 4}{x-2} & \text{if } x < 2, \\ x+3

Medium

Tangent Line to a Parametric Curve

Consider the parametric equations $$x(t)=t^2-3$$ and $$y(t)=2*t+1$$. (a) Compute the average rate o

Medium

Transition Matrix and State Changes

Consider a system with two states modeled by the transition matrix $$M = \begin{pmatrix} 0.7 & 0.2 \

Hard

Trigonometric Function Analysis

Consider the trigonometric function $$f(x)= 2*\tan(x - \frac{\pi}{6})$$. Without using a calculator,

Medium

Vector Addition and Scalar Multiplication

Consider the vectors $$\vec{u}=\langle 1, 3 \rangle$$ and $$\vec{v}=\langle -2, 4 \rangle$$:

Medium

Vector Components and Magnitude

Given the vector $$\vec{v}=\langle 3, -4 \rangle$$:

Easy

Vector Operations and Dot Product

Let $$\mathbf{u}=\langle 3,-1 \rangle$$ and $$\mathbf{v}=\langle -2,4 \rangle$$. Use these vectors t

Easy

Vector Scalar Multiplication

Given the vector $$\mathbf{w} = \langle -2, 5 \rangle$$ and the scalar $$k = -3$$, answer the follow

Easy

Vector-Valued Functions: Position and Velocity

A particle’s position is given by the vector-valued function $$\mathbf{p}(t)=\langle 2*t+1, t^2-3*t+

Medium

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Need to review before working on AP Precalculus FRQs?

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Tips from Former AP Students

FAQWe thought you might have some questions...
Where can I find practice free response questions for the AP Precalculus exam?
The free response section of each AP exam varies slightly, so you’ll definitely want to practice that before stepping into that exam room. Here are some free places to find practice FRQs :
  • Of course, make sure to run through College Board's past FRQ questions!
  • Once you’re done with those go through all the questions in the AP PrecalculusFree Response Room. You can answer the question and have it grade you against the rubric so you know exactly where to improve.
  • Reddit it also a great place to find AP free response questions that other students may have access to.
How do I practice for AP AP Precalculus Exam FRQs?
Once you’re done reviewing your study guides, find and bookmark all the free response questions you can find. The question above has some good places to look! while you’re going through them, simulate exam conditions by setting a timer that matches the time allowed on the actual exam. Time management is going to help you answer the FRQs on the real exam concisely when you’re in that time crunch.
What are some tips for AP Precalculus free response questions?
Before you start writing out your response, take a few minutes to outline the key points you want to make sure to touch on. This may seem like a waste of time, but it’s very helpful in making sure your response effectively addresses all the parts of the question. Once you do your practice free response questions, compare them to scoring guidelines and sample responses to identify areas for improvement. When you do the free response practice on the AP Precalculus Free Response Room, there’s an option to let it grade your response against the rubric and tell you exactly what you need to study more.
How do I answer AP Precalculus free-response questions?
Answering AP Precalculus free response questions the right way is all about practice! As you go through the AP AP Precalculus Free Response Room, treat it like a real exam and approach it this way so you stay calm during the actual exam. When you first see the question, take some time to process exactly what it’s asking. Make sure to also read through all the sub-parts in the question and re-read the main prompt, making sure to circle and underline any key information. This will help you allocate your time properly and also make sure you are hitting all the parts of the question. Before you answer each question, note down the key points you want to hit and evidence you want to use (where applicable). Once you have the skeleton of your response, writing it out will be quick, plus you won’t make any silly mistake in a rush and forget something important.