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AP Precalculus Free Response Questions

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  • Unit 1: Polynomial and Rational Functions (65)
  • Unit 2: Exponential and Logarithmic Functions (57)
  • Unit 3: Trigonometric and Polar Functions (58)
  • Unit 4: Functions Involving Parameters, Vectors, and Matrices (70)
Unit 1: Polynomial and Rational Functions

Analyzing an Odd Polynomial Function

Consider the function $$p(x)= x^3 - 4*x$$. Investigate its properties by answering the following par

Easy

Analyzing Concavity and Points of Inflection for a Polynomial Function

Consider the function $$f(x)= x^3-3*x^2+2*x$$. Although points of inflection are typically determine

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

Average Rate of Change and Tangent Lines

For the function $$f(x)= x^3 - 6*x^2 + 9*x + 4$$, consider the relationship between secant (average

Medium

Comparative Analysis of Even and Odd Polynomial Functions

Consider the functions $$f(x)= x^4 - 4*x^2 + 3$$ and $$g(x)= x^3 - 2*x$$. Answer the following parts

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

Comparing Polynomial and Rational Function Models

Two models are proposed to describe a data set. Model A is a polynomial function given by $$f(x)= 2*

Medium

Complex Zeros and Conjugate Pairs

Consider the polynomial $$p(x)= x^4 + 4*x^3 + 8*x^2 + 8*x + 4$$. Answer the following parts.

Hard

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

Construction of a Polynomial Model

A company’s quarterly profit (in thousands of dollars) over five quarters is given in the table belo

Medium

Cubic Polynomial Analysis

Consider the cubic polynomial function $$f(x) = 2*x^3 - 3*x^2 - 12*x + 8$$. Analyze the function as

Medium

Data Analysis with Polynomial Interpolation

A scientist measures the decay of a radioactive substance at different times. The following table sh

Hard

Degree Determination from Finite Differences

A researcher records the size of a bacterial colony at equal time intervals, obtaining the following

Easy

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 Domain and Range of a Transformed Rational Function

Consider the function $$g(x)= \frac{x^2 - 9}{x-3}$$. Answer the following:

Medium

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

Discontinuities in a Rational Model Function

Consider the function $$p(x)=\frac{(x-3)(x+1)}{x-3}$$, defined for all $$x$$ except when $$x=3$$. Ad

Easy

Engineering Curve Analysis: Concavity and Inflection

An engineering experiment recorded the deformation of a material, modeled by a function whose behavi

Easy

Expansion Using the Binomial Theorem in Forecasting

In a business forecast, the expression $$(x + 5)^4$$ is used to model compound factors affecting rev

Easy

Exploring Domain Restrictions via Inverse Functions in a Quadratic Model

Consider the quadratic function $$f(x)= -x^2 + 6*x - 8$$. Answer the following questions regarding i

Medium

Exploring Polynomial Function Behavior

Consider the polynomial function $$f(x)= 2*(x-1)^2*(x+2)$$, which is used to model a physical trajec

Easy

Function Simplification and Graph Analysis

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

Easy

Function Transformations and Parent Functions

The parent function is $$f(x)= x^2$$. Consider the transformed function $$g(x)= -3*(x-4)^2 + 5$$. An

Easy

Impact of Multiplicity on Graph Behavior

Consider the function $$f(x)= (x - 2)^2*(x + 1)$$. Examine how the multiplicity of each zero affects

Medium

Interpreting Transformations of Functions

The parent function is $$f(x)= x^2$$. A transformed function is given by $$g(x)= -3*(x+2)^2+5$$. Ans

Easy

Intersection of Functions in Supply and Demand

Consider two functions that model supply and demand in a market. The supply function is given by $$f

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 Polynomial Function with Multiple Turning Points

Consider the function $$f(x)= (x-2)^3 - 3*(x-2) + 1$$. Answer the following about its invertibility

Hard

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 of a Complex Rational Function

Consider the function $$f(x)=\frac{3*x+2}{2*x-1}$$. Answer the following questions regarding its inv

Medium

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

Investigation of Refund Policy via Piecewise Continuous Functions

A retail store's refund policy is modeled by $$ R(x)=\begin{cases} 10-x & \text{for } x<5, \\ a*x+b

Easy

Loan Payment Model using Rational Functions

A bank uses the rational function $$R(x) = \frac{2*x^2 - 3*x - 5}{x - 2}$$ to model the monthly inte

Hard

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 Population Growth with a Polynomial Function

A population of a certain species in a controlled habitat is modeled by the cubic function $$P(t)= -

Medium

Modeling Vibration Data with a Cubic Function

A sensor records vibration data over time, and the data appears to be modeled by a cubic function of

Hard

Multivariable Rational Function: Zeros and Discontinuities

A pollutant concentration is modeled by $$C(x)= \frac{(x-3)*(x+2)}{(x-3)*(x-4)}$$, where x represent

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

Optimizing Production Using a Polynomial Model

A factory's production cost (in thousands of dollars) is modeled by the function $$C(x)= 0.02*x^3 -

Hard

Piecewise Financial Growth Model

A company’s quarterly growth rate is modeled using a piecewise function. For $$0 \le x \le 4$$, the

Extreme

Piecewise Function Analysis

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

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 Interpolation and Finite Differences

A quadratic function is used to model the height of a projectile. The following table gives the heig

Easy

Polynomial Transformation Challenge

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

Easy

Population Growth Modeling with a Polynomial Function

A regional population (in thousands) is modeled by a polynomial function $$P(t)$$, where $$t$$ repre

Medium

Rate of Change in a Quadratic Function

Consider the quadratic function $$f(x)= 2*x^2 - 4*x + 1$$. Answer the following parts regarding its

Medium

Rational Function Analysis for Signal Processing

A signal processing system is modeled by the rational function $$R(x)= \frac{2*x^2 - 3*x - 5}{x^2 -

Medium

Rational Function and Slant Asymptote Analysis

A study of speed and fuel efficiency is modeled by the function $$F(x)= \frac{3*x^2+2*x+1}{x-1}$$, w

Hard

Rational Function Asymptotes and Holes

Consider the rational function $$r(x)=\frac{x^2 - 4}{x^2 - x - 6}$$. Analyze the function according

Medium

Rational Function Graph and Asymptote Identification

Given the rational function $$R(x)= \frac{x^2 - 4}{x^2 - x - 6}$$, answer the following parts:

Hard

Rational Inequalities Analysis

Solve the inequality $$\frac{x^2-4}{x+1} \ge 0$$ and represent the solution on a number line.

Medium

Real-World Inverse Function: Temperature Conversion

The function $$f(x)= \frac{9}{5}*x + 32$$ converts a temperature in degrees Celsius to degrees Fahre

Easy

Return to a Rational Expression under Transformation

Consider the function $$f(x)=\frac{(x-2)(x+3)}{(x-2)(x-5)}$$, defined for $$x\neq2,5$$. Answer the f

Hard

Revenue Function Transformations

A company models its revenue with a polynomial function $$f(x)$$. It is known that $$f(x)$$ has x-in

Medium

Revenue Modeling with a Polynomial Function

A small theater's revenue from ticket sales is modeled by the polynomial function $$R(x)= -0.5*x^3 +

Medium

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

Transformation and Inversions of a Rational Function

A manufacturer models the cost per unit with the function $$C(x)= \frac{5*x+20}{x-2}$$, where x is t

Hard

Transformation in Composite Functions

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

Easy

Transformation of a Parabola

Starting with the parent function $$f(x)=x^2$$, a new function is defined by $$g(x) = -2*(x+3)^2 + 4

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

Zeros and End Behavior in a Higher-Degree Polynomial

Consider the polynomial $$P(x)= (x+1)^2 (x-2)^3 (x-5)$$. Answer the following parts.

Easy
Unit 2: Exponential and Logarithmic Functions

Acoustics and the Logarithmic Scale

The sound intensity level (in decibels) of a sound is given by the function $$f(x)=10*\log_{10}(x)$$

Medium

Analyzing Exponential Function Behavior from a Graph

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

Easy

Arithmetic Sequence Analysis

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

Easy

Arithmetic Sequence Analysis

Consider an arithmetic sequence with initial term $$a_0$$ and common difference $$d$$. Analyze the c

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

Bacterial Growth Model

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

Medium

Bacterial Population Growth Model

A certain bacterium population doubles every 3 hours. At time $$t = 0$$ hours the population is $$50

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

Comparing Linear and Exponential Revenue Models

A company is forecasting its revenue growth using two models based on different assumptions. Initial

Medium

Composite Function Analysis: Identity and Inverses

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

Medium

Composite Functions and Their Inverses

For the functions $$f(x) = 2^x$$ and $$g(x) = \log_2(x)$$, analyze their composite functions.

Easy

Composite Sequences: Combining Geometric and Arithmetic Models in Production

A factory’s monthly production is influenced by two factors. There is a fixed increase in production

Extreme

Compound Interest and Financial Growth

An investment account earns compound interest annually. An initial deposit of $$P = 1000$$ dollars i

Easy

Compound Interest with Periodic Deposits

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

Medium

Connecting Exponential Functions with Geometric Sequences

An exponential function $$f(x) = 5 \cdot 3^x$$ can also be interpreted as a geometric sequence where

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, Range, and Inversion of Logarithmic Functions

Consider the logarithmic function \(f(x)=\log_{2}(x-3)\). (a) Determine the domain and range of \(f

Easy

Earthquake Magnitude and Energy Release

Earthquake energy is modeled by the equation $$E = k\cdot 10^{1.5M}$$, where $$E$$ is the energy rel

Medium

Environmental Pollution Decay

The concentration of a pollutant in a lake decays exponentially due to natural processes. The concen

Medium

Exploring the Properties of Exponential Functions

Analyze the exponential function $$f(x)= 4 * 2^x$$.

Easy

Exponential Decay and Log Function Inverses in Pharmacokinetics

In a pharmacokinetics study, the concentration of a drug in a patient’s bloodstream is observed to d

Medium

Exponential Function from Data Points

An exponential function of the form f(x) = a·bˣ passes through the points (2, 12) and (5, 96).

Hard

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

Financial Growth: Savings Account with Regular Deposits

A savings account starts with an initial balance of $$1000$$ dollars and earns compound interest at

Hard

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

Geometric Sequence in Compound Interest

An investment grows according to a geometric sequence. The initial investment is $$1000$$ dollars an

Easy

Inverse Functions in Exponential Contexts

Consider the function $$f(x)= 5^x + 3$$. Analyze its inverse function.

Medium

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 a Composite Function

Let $$f(x)= e^x$$ and $$g(x)= \ln(x) + 3$$.

Hard

Inverse of an Exponential Function

Let f(x) = 5·e^(2*x) - 3. Find the inverse function f⁻¹(x) and verify your answer by composing f and

Easy

Inverse Relationship Verification

Given f(x) = 3ˣ - 4 and g(x) = log₃(x + 4), verify that g is the inverse of f.

Hard

Investment Growth via Sequences

A financial planner is analyzing two different investment strategies starting with an initial deposi

Medium

Logarithmic Equation and Extraneous Solutions

Solve the logarithmic equation $$log₂(x - 1) + log₂(3*x + 2) = 3$$.

Hard

Logarithmic Function Analysis

Consider the logarithmic function $$f(x) = 3 + 2·log₅(x - 1)$$.

Medium

Logarithmic Function and Inversion

Given the function $$f(x)= \log_3(x-2)+4$$, perform an analysis to determine its domain, prove it is

Medium

Logarithmic Function with Scaling and Inverse

Consider the function $$f(x)=\frac{1}{2}\log_{10}(x+4)+3$$. Analyze its monotonicity, find the inver

Easy

Logarithmic Transformation and Composition of Functions

Let $$f(x)= \log_3(x)$$ and $$g(x)= 2^x$$. Using these functions, answer the following:

Hard

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

Population Growth Inversion

A town's population grows according to the function $$f(t)=1200*(1.05)^(t)$$, where $$t$$ is the tim

Medium

Population Growth of Bacteria

A bacterial colony doubles in size every hour, so that its size follows a geometric sequence. Recall

Medium

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

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

Semi-Log Plot Data Analysis

A set of experimental data representing bacterial concentration (in CFU/mL) over time (in days) is g

Medium

Solving Logarithmic Equations and Checking Domain

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

Hard

System of Exponential Equations

Solve the following system of equations: $$2\cdot 2^x + 3\cdot 3^y = 17$$ $$2^x - 3^y = 1$$.

Medium

Transformation of Exponential Functions

Consider the exponential function $$f(x)= 3 * 5^x$$. A new function $$g(x)$$ is defined by applying

Medium

Transformations of Exponential Functions

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

Medium

Transformations of Exponential Functions

Consider the exponential function $$f(x)= 7 * e^{0.3x}$$. Investigate its transformations.

Easy

Tumor Growth with Time Dilation Effects

A medical researcher is studying the growth of a tumor, which is modeled by the exponential function

Extreme

Weekly Population Growth Analysis

A species exhibits exponential growth in its weekly population. If the initial population is $$2000$

Hard

Wildlife Population Decline

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

Easy
Unit 3: Trigonometric and Polar Functions

Analysis of a Cotangent Function

Consider the function $$f(\theta)=\cot(\theta)$$ defined on the interval \(\theta\in(0,\pi)\).

Medium

Analysis of a Limacon

Consider the polar function $$r(\theta) = 2 + 3*\cos(\theta)$$.

Extreme

Analysis of Reciprocal Trigonometric Functions

Examine the properties of the reciprocal trigonometric functions $$\csc(θ)$$, $$\sec(θ)$$, and $$\co

Hard

Average Rate of Change in a Polar Function

Given the polar function $$r(\theta) = 5*\sin(2*\theta) + 7$$ over the interval $$\theta \in [0, \fr

Medium

Average Rate of Change in a Polar Function

Consider the polar function $$r=f(θ)=3+2*\sin(θ)$$, which models a periodic phenomenon in polar coor

Medium

Combining Logarithmic and Trigonometric Equations

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

Hard

Concavity in the Sine Function

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

Medium

Conversion between Rectangular and Polar Coordinates

Given the point in rectangular coordinates $$(-3, 3\sqrt{3})$$, perform the following tasks.

Medium

Conversion Between Rectangular and Polar Coordinates

A point A in the Cartesian plane is given by $$(-3, 3\sqrt{3})$$.

Hard

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

Daily Temperature Fluctuations

The table below shows the recorded temperature (in $$^{\circ}\text{F}$$) at various times during the

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 Variation Model

A company models the variation in daylight hours over a year using the function $$D(t) = 10*\sin\Big

Medium

Determining Phase Shifts and Amplitude Changes

A wave function is modeled by $$W(\theta)=7*\cos(4*(\theta-c))+d$$, where c and d are unknown consta

Hard

Exploring Coterminal Angles and Periodicity

Analyze the concept of coterminal angles.

Easy

Exploring Rates of Change in Polar Functions

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

Hard

Graph Transformations of Sinusoidal Functions

Consider the sinusoidal function $$f(x) = 3*\sin\Bigl(2*(x - \frac{\pi}{4})\Bigr) - 1$$.

Medium

Graphical Analysis of a Periodic Function

A periodic function is depicted in the graph provided. Analyze the function’s key features based on

Easy

Graphing a Rose Curve

Consider the polar function $$r=4\cos(3\theta)$$ and analyze its properties.

Medium

Graphing the Tangent Function and Analyzing Asymptotes

Consider the function $$y = \tan(x)$$. Answer the following:

Medium

Graphing the Tangent Function with Asymptotes

Consider the transformed tangent function $$g(\theta)=\tan(\theta-\frac{\pi}{4})$$.

Hard

Graphing the Tangent Function with Asymptotes

The tangent function, $$f(\theta) = \tan(\theta)$$, exhibits vertical asymptotes where it is undefin

Hard

Identity Verification

Verify the following trigonometric identity using the sum formula for sine: $$\sin(\alpha+\beta) = \

Easy

Interpreting Trigonometric Data Models

A set of experimental data capturing a periodic phenomenon is given in the table below. Use these da

Medium

Inverse Trigonometric Function Analysis

Consider the function $$f(x)=\sin(x)$$ defined on the interval $$\left[-\frac{\pi}{2},\frac{\pi}{2}\

Easy

Inverse Trigonometric Functions

Examine the inverse relationships for trigonometric functions over appropriate restricted domains.

Medium

Modeling a Ferris Wheel's Motion Using Sinusoidal Functions

A Ferris wheel with a diameter of 10 meters rotates at a constant speed. The lowest point of the rid

Easy

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

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 Tidal Heights with Periodic Data

An oceanographer records tidal heights (in meters) over a 6-hour period. The following table gives t

Hard

Modeling Tidal Motion with a Sinusoidal Function

A coastal town uses the model $$h(t)=4*\sin\left(\frac{\pi}{6}*(t-2)\right)+10$$ (with $$t$$ in hour

Medium

Modeling Tides with Sinusoidal Functions

Tidal heights at a coastal location are modeled by the function $$H(t)=2\,\sin\Bigl(\frac{\pi}{6}(t-

Easy

Periodic Temperature Variation Model

A town's temperature is modeled by the function $$T(t)=10*\cos(\frac{\pi}{12}*(t-6))+20$$, where t r

Easy

Polar Coordinates and Graphing a Circle

Answer the following questions on polar coordinates:

Medium

Polar Coordinates Conversion

Convert the rectangular coordinate point $$(-3,\,3\sqrt{3})$$ into polar form.

Medium

Polar Function with Rate of Change Analysis

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

Medium

Polar Rose Analysis

Analyze the polar equation $$r = 2*\cos(3\theta)$$.

Medium

Rate of Change in Polar Functions

For the polar function $$r(\theta)=4+\cos(\theta)$$, investigate its rate of change.

Medium

Rate of Change in Polar Functions

Consider the polar function $$r(\theta)=3+\sin(\theta)$$.

Hard

Seasonal Demand Modeling

A company's product demand follows a seasonal pattern modeled by $$D(t)=500+50\cos\left(\frac{2\pi}{

Medium

Secant Function and Its Transformations

Investigate the function $$f(\theta)=\sec(\theta)$$ and the transformation $$h(\theta)=2*\sec(\theta

Medium

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 Transformations

The function $$g(x) = 2*\cos(3*(x - \frac{\pi}{4})) - 1$$ is a transformed cosine wave.

Medium

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

Solve the trigonometric equation $$2*\sin(\theta)+\sqrt{3}=0$$ for all solutions in the interval $$[

Easy

Solving a Trigonometric Inequality

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

Easy

Solving Trigonometric Equations

A projectile is launched such that its launch angle satisfies the equation $$\sin(2*\theta)=0.5$$. A

Medium

Solving Trigonometric Equations

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

Hard

Solving Trigonometric Equations

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

Medium

Solving Trigonometric Equations in a Survey

In a survey, participants' responses are modeled using trigonometric equations. Solve the following

Easy

Tangent and Cotangent Equation

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

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 area experiences tides that follow a sinusoidal pattern described by $$T(t)=4+1.2\sin\left

Medium

Transformation and Reflection of a Cosine Function

Consider the function $$g(x) = -2*\cos\Bigl(\frac{1}{2}(x + \pi)\Bigr) + 3$$.

Medium

Trigonometric Inequality Solution

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

Easy

Unit Circle and Special Triangle Values

Using the unit circle and properties of special triangles, answer the following.

Easy

Unit Circle and Special Triangles

Consider the unit circle and the properties of special right triangles. Answer the following for a 4

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

Analyzing a Piecewise Function Representing a Linear Transformation

Let $$T(x)=\begin{cases} \frac{2x-4}{x-2} & \text{if } x \neq 2, \\ 3 & \text{if } x=2 \end{cases}$$

Easy

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

A projectile is launched and its motion is modeled by $$x(t)=3*t+1$$ and $$y(t)=16-4*t^2$$, where $$

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

Circular Motion and Transformation

The motion of a particle is given by the parametric equations $$x(t)=3\cos(t)$$ and $$y(t)=3\sin(t)$

Easy

Composite Functions Involving Parametric and Matrix Transformations

A particle’s motion is initially modeled by the parametric function $$f(t)= \langle e^{0.1*t}, \ln(t

Extreme

Composite Transformations in the Plane

Consider two linear transformations in $$\mathbb{R}^2$$: a rotation by 90° counterclockwise and a re

Easy

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 Applications in Area Computation

Vectors $$\mathbf{u}=\langle 5,2\rangle$$ and $$\mathbf{v}=\langle 1,4\rangle$$ form adjacent sides

Easy

Discontinuity Analysis in a Function Modeling Particle Motion

A particle’s position along a line is given by the piecewise function: $$s(t)=\begin{cases} \frac{t^

Medium

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

Eliminating the Parameter

Given the parametric equations $$x(t) = 2 + 3*t$$ and $$y(t) = 4 - t^2$$, answer the following:

Hard

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 and Discontinuities in a Parameter-Dependent Function

For the function $$g(t)=\begin{cases} \frac{2*t^2 - 8}{t-2} & \text{if } t \neq 2, \\ 6 & \text{if }

Easy

Ferris Wheel Motion

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

Medium

FRQ 1: Parametric Path and Motion Analysis

Consider the parametric function $$f(t)=(x(t),y(t))$$ defined by $$x(t)=t^2-4*t+3$$ and $$y(t)=2*t-1

Medium

FRQ 6: Implicit Function to Parametric Representation

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

Hard

FRQ 9: Vectors in Motion and Velocity

A particle's position is described by the vector-valued function $$p(t)=\langle2*t-1, t^2+1\rangle$$

Medium

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 18: Dynamic Systems and Transition Matrices

Consider a transition matrix modeling state changes given by $$M=\begin{bmatrix}0.7 & 0.3\\0.4 & 0.6

Hard

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

Graph Analysis of an Implicitly Defined Ellipse

A graph is produced for the implicitly defined ellipse given by $$\left(\frac{x}{2}\right)^2 + \lef

Medium

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

Graphical and Algebraic Analysis of a Function with a Removable Discontinuity

Consider the function $$g(x)=\begin{cases} \frac{\sin(x) - \sin(0)}{x-0} & \text{if } x \neq 0, \\ 1

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

Hyperbola Parametrization Using Trigonometric Functions

Consider the hyperbola defined by $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$. Answer the following:

Hard

Inverse and Determinant of a Matrix

Let the 2×2 matrix be given by $$A= \begin{pmatrix} a & 2 \\ 3 & 4 \end{pmatrix}$$. Answer the follo

Easy

Inverse Matrix with a Parameter

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

Medium

Inverse of a 2×2 Matrix

Consider the matrix $$A=\begin{bmatrix}2 & 5\\ 3 & 7\end{bmatrix}$$.

Medium

Inverses and Solving a Matrix Equation

Given the matrix $$D = \begin{pmatrix} -2 & 5 \\ 1 & 3 \end{pmatrix}$$, answer the following:

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 Transformation and Area Scaling

Consider the linear transformation L on \(\mathbb{R}^2\) defined by the matrix $$A= \begin{pmatrix}

Medium

Linear Transformation Composition

Consider two linear transformations with matrices $$A=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

Hard

Linear Transformations in the Plane

A linear transformation $$L$$ from $$\mathbb{R}^2$$ to $$\mathbb{R}^2$$ is defined by $$L(x,y)=(2x-y

Hard

Matrices as Models for Population Dynamics

A population of two species is modeled by the transition matrix $$P=\begin{pmatrix} 0.8 & 0.1 \\ 0.2

Hard

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 Representation of Linear Transformations

Consider the linear transformation defined by $$L(x,y)=(3*x-2*y, 4*x+y)$$.

Medium

Matrix Transformation of a Vector

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

Medium

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

Movement Analysis via Position Vectors

A particle is moving in the plane with its position given by the functions $$x(t)=2*t+1$$ and $$y(t)

Easy

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 and Rates in a Biological Context

A bacteria colony in a Petri dish is observed to move in a periodic manner, with its position descri

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 Motion Analysis Using Tabulated Data

A particle moves in the plane following a parametric function. The following table represents the pa

Medium

Parametric Representation of an Ellipse

An ellipse is defined by the equation $$\frac{(x-3)^2}{4} + \frac{(y+2)^2}{9} = 1.$$ (a) Write a

Easy

Parametric Representation of an Implicit Curve

The equation $$x^2+y^2-6*x+8*y+9=0$$ defines a curve in the plane. Analyze this curve.

Easy

Parametric Representation of an Implicitly Defined Function

Consider the implicitly defined curve $$x^2+y^2=16$$. A common parametric representation is given by

Easy

Parametric Representation on the Unit Circle and Special Angles

Consider the unit circle defined by the parametric equations $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$.

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

Particle Motion Through Position and Velocity Vectors

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

Medium

Particle Motion with Quadratic Parametric Functions

A particle moves in the plane according to the parametric equations $$x(t)=t^2$$ and $$y(t)=2*t$$. A

Easy

Piecewise Function and Discontinuities

Consider the function $$f(x)=\begin{cases} \frac{x^2 - 1}{x-1} & \text{if } x \neq 1, \\ 3 & \text{i

Easy

Population Transition Matrix Analysis

A population dynamics model is represented by the transition matrix $$T=\begin{pmatrix}0.7 & 0.2 \\

Medium

Properties of a Parametric Curve

Consider a curve defined parametrically by $$x(t)=t^3$$ and $$y(t)=t^2.$$ (a) Determine for which

Medium

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

Transformation Matrices in Computer Graphics

A transformation matrix $$A = \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$$ is applied to points in

Medium

Transition from Parametric to Explicit Function

A curve is defined by the parametric equations $$x(t)=\ln(t)$$ and $$y(t)=t+1$$, where $$t>0$$. Answ

Medium

Trigonometric Function Analysis

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

Medium

Uniform Circular Motion

A car is moving along a circular track of radius 10 meters. Its motion is described by the parametri

Easy

Vector Analysis in Projectile Motion

A soccer ball is kicked so that its velocity vector is given by $$\mathbf{v}=\langle5, 7\rangle$$ (i

Easy

Vector Operations

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

Easy

Vector Operations in the Plane

Let the vectors be given by $$\mathbf{u}=\langle 3,-4\rangle$$ and $$\mathbf{v}=\langle -2,5\rangle$

Easy

Vector Operations in the Plane

Let $$\vec{u}= \langle 3, -2 \rangle$$ and $$\vec{v}= \langle -1, 4 \rangle$$. Perform the following

Easy

Vector Scalar Multiplication

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

Easy

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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.