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AP Precalculus Free Response Questions

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  • Unit 1: Polynomial and Rational Functions (68)
  • Unit 2: Exponential and Logarithmic Functions (56)
  • Unit 3: Trigonometric and Polar Functions (66)
  • Unit 4: Functions Involving Parameters, Vectors, and Matrices (60)
Unit 1: Polynomial and Rational Functions

Absolute Extrema and Local Extrema of a Polynomial

Consider the polynomial function $$p(x)= (x-3)^2*(x+3)$$.

Medium

Analysis of a Quartic Function

Consider the quartic function $$h(x)= x^4 - 4*x^2 + 3$$. Answer the following:

Hard

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 an Odd Polynomial Function

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

Easy

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

Application of the Binomial Theorem

Expand the expression $$(x+3)^5$$ using the Binomial Theorem and answer the following parts.

Easy

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

Binomial Theorem Expansion

Use the Binomial Theorem to expand the expression $$ (x + 2)^4 $$. Explain your steps in detail.

Easy

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

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

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 Analysis with Rational and Polynomial Functions

Consider the functions $$f(x)= \frac{x+2}{x-1}$$ and $$g(x)= x^2 - 3*x + 4$$. Let the composite func

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

Cubic Polynomial Analysis

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

Medium

Designing a Piecewise Function for a Temperature Model

A city experiences distinct temperature patterns during the day. A proposed model is as follows: for

Hard

Determining the Degree of a Polynomial from Data

A table of values is given below for a function $$f(x)$$ measured at equally spaced x-values: | x |

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

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 Application: Stress Analysis Model

In a stress testing experiment, the stress $$S(x)$$ on a component (in appropriate units) is modeled

Medium

Engineering Curve Analysis: Concavity and Inflection

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

Easy

Estimating Polynomial Degree from Finite Differences

The following table shows the values of a function $$f(x)$$ at equally spaced values of $$x$$: | x

Easy

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

Exploring Asymptotic Behavior in a Sales Projection Model

A sales projection model is given by $$P(x)=\frac{4*x-2}{x-1}$$, where $$x$$ represents time in year

Hard

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

Exponential Equations and Logarithm Applications in Decay Models

A radioactive substance decays according to the model $$A(t)= A_0*e^{-0.3*t}$$. A researcher analyze

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

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

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

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 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 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

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

Inversion of a Polynomial Ratio Function

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

Hard

Investigating a Real-World Polynomial Model

A physicist models the vertical trajectory of a projectile by the quadratic function $$h(t)= -5*t^2+

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 with a Polynomial Function: Optimization

A company’s profit (in thousands of dollars) is modeled by the polynomial function $$P(x)= -2*x^3+12

Medium

Modeling with Rational Functions

A chemist models the concentration of a reactant over time with the rational function $$C(t)= \frac{

Medium

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

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 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 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

Polynomial Division in Limit Evaluation

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

Hard

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 Long Division and Slant Asymptote

Consider the rational function $$R(x)= \frac{2*x^3 - 3*x^2 + 4*x - 5}{x^2 - 1}$$. Answer the followi

Hard

Polynomial Long Division and Slant Asymptotes

Consider the rational function $$R(x)= \frac{2*x^3+3*x^2-5*x+4}{x^2-1}$$.

Hard

Polynomial Model from Temperature Data

A researcher records the ambient temperature over time and obtains the following data: | Time (hr)

Medium

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 Graph and Asymptote Identification

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

Hard

Rational Function: Machine Efficiency Ratios

A machine's efficiency is modeled by the rational function $$E(x) = \frac{x^2 - 9}{x^2 - 4*x + 3}$$,

Medium

Rational Inequalities and Test Intervals

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

Medium

Real-World Inverse Function: Modeling a Reaction Process

The function $$f(x)=\frac{50}{x+2}+3$$ models the average concentration (in moles per liter) of a su

Medium

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

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

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 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 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

Using the Binomial Theorem for Polynomial Expansion

A scientist is studying the expansion of the polynomial expression $$ (1+2*x)^5$$, which is related

Easy

Zero Finding and Sign Charts

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

Easy
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 from a Graph

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

Easy

Arithmetic Savings Plan

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

Easy

Comparing Arithmetic and Exponential Models in Population Growth

Two neighboring communities display different population growth patterns. Community A increases by a

Hard

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 Exponential-Logarithmic Functions

Let f(x) = log₃(x) and g(x) = 2·3ˣ. Analyze the following compositions.

Medium

Composite Function Analysis: Identity and Inverses

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

Medium

Composite Functions: Shifting and Scaling in Log and Exp

Consider the functions $$f(x)=2*e^(x-3)$$ and $$g(x)=\ln(x)+4$$.

Medium

Composition of Exponential and Log Functions

Consider the functions $$f(x)=\ln(x)$$ and $$g(x)=2*e^(x)$$.

Medium

Composition of Exponential and Logarithmic Functions

Given two functions: $$f(x) = 3 \cdot 2^x$$ and $$g(x) = \log_2(x)$$, answer the following parts.

Easy

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 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

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 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 Transformation

An exponential function is given by $$f(x) = 2 \cdot 3^x$$. Analyze the effects of various transform

Medium

Exponential Inequalities

Solve the inequality $$3 \cdot 2^x \le 48$$.

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 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

Geometric Investment Growth

An investor places $$1000$$ dollars into an account that grows following a geometric sequence model.

Medium

Geometric Sequence and Exponential Modeling

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

Medium

Geometric Sequence Construction

Consider a geometric sequence where the first term is $$g_0 = 3$$ and the second term is $$g_1 = 6$$

Easy

Inverse and Domain of a Logarithmic Transformation

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

Medium

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 of an Exponential Function

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

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 Scenario Convergence

An investment yields returns modeled by the infinite geometric series $$S=500 + 500*r + 500*r^2 + \c

Easy

Loan Payment and Arithmetico-Geometric Sequence

A borrower takes a loan of $$10,000$$ dollars. The loan accrues a monthly interest of 1% and the bor

Hard

Log-Exponential Function and Its Inverse

For the function $$f(x)=\log_2(3^(x)-5)$$, determine the domain, prove it is one-to-one, find its in

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 Analysis

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

Medium

Logarithmic Inequalities

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

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

Model Error Analysis in Exponential Function Fitting

A researcher uses the exponential model $$f(t) = 100 \cdot e^{0.05t}$$ to predict a process. At \(t

Hard

Model Validation and Error Analysis in Exponential Trends

During a chemical reaction, a set of experimental data appears to follow an exponential trend when p

Hard

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

Population Growth of Bacteria

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

Medium

Radioactive Decay Analysis

A radioactive substance decays exponentially over time according to the function $$f(t) = a * b^t$$,

Easy

Radioactive Decay and Exponential Functions

A sample of a radioactive substance is monitored over time. The decay in mass is recorded in the tab

Medium

Radioactive Decay and Half-Life Estimation Through Data

A radioactive substance decays exponentially according to the function $$f(t)= a * b^t$$. The follow

Easy

Radioactive Decay Problem

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

Easy

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

Solving Exponential Equations Using Logarithms

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

Easy

Solving Exponential Equations Using Logarithms

Solve for $$x$$ in the exponential equation $$2*3^(x)=54$$.

Easy

Solving Logarithmic Equations and Checking Domain

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

Hard

Solving Logarithmic Equations with Extraneous Solutions

Solve the logarithmic equation $$\log_2(x - 1) + \log_2(2x) = \log_2(10)$$ and check for any extrane

Hard

Telephone Call Data Analysis on Semi-Log Plot

A telecommunications company records the number of calls received each hour. The data suggest an exp

Medium

Temperature Decay Modeled by a Logarithmic Function

In an experiment, the temperature (in degrees Celsius) of an object decreases over time according to

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 \(f(x)=3\cdot2^{x}\). (a) Determine the equation of the transform

Medium

Transformations of Exponential Functions

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

Easy

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 a Rose Curve

Examine the polar equation $$r=3*\sin(3\theta)$$.

Hard

Analyzing a Rose Curve

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

Medium

Application of Trigonometric Sum Identities

Utilize trigonometric sum identities to simplify and solve expressions.

Hard

Calculating the Area Enclosed by a Polar Curve

Consider the polar curve $$r=2*\cos(θ)$$. Without performing any integral calculations, use symmetry

Hard

Cardioid Polar Graphs

Consider the cardioid given by the polar equation $$r=1+\cos(\theta)$$.

Medium

Conversion Between Rectangular and Polar Coordinates

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

Hard

Coterminal Angles and the Unit Circle

Consider the angle $$\theta = \frac{5\pi}{3}$$ given in standard position.

Medium

Daylight Hours Modeling

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

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

Evaluating Inverse Trigonometric Functions

Inverse trigonometric functions such as $$\arcsin(x)$$ and $$\arccos(x)$$ have specific restricted d

Easy

Evaluating Sine and Cosine Values Using Special Triangles

Using the properties of special triangles, answer the following:

Easy

Exploring Coterminal Angles and Periodicity

Analyze the concept of coterminal angles.

Easy

Exploring Limacons in Polar Coordinates

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

Hard

Graph Analysis of a Polar Function

The polar function $$r=4+3\sin(\theta)$$ is given, with the following data: | \(\theta\) (radians)

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

Graph Transformations of Sinusoidal Functions

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

Medium

Graph Transformations: Sine and Cosine Functions

The functions $$f(\theta)=\sin(\theta)$$ and $$g(\theta)=\cos(\theta)$$ are related through a phase

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

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

Graphing a Transformed Sine Function

Analyze the function $$f(x)=3\,\sin\Bigl(2\bigl(x-\frac{\pi}{4}\bigr)\Bigr)-1$$ which is obtained fr

Medium

Graphing and Analyzing a Transformed Sine Function

Consider the function $$f(x)=3\sin\left(2\left(x-\frac{\pi}{4}\right)\right)+1$$. Answer the followi

Medium

Graphing Polar Circles and Roses

Analyze the following polar equations: $$r=2$$ and $$r=3*\cos(2\theta)$$.

Medium

Identity Verification

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

Easy

Inverse Function Analysis

Given the function $$f(\theta)=2*\sin(\theta)+1$$, analyze its invertibility and determine its inver

Easy

Inverse Tangent Composition and Domain

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

Extreme

Limacon Analysis

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

Medium

Limacons and Cardioids

Consider the polar function $$r=1+2*\cos(\theta)$$.

Hard

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 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

Modeling Daylight Hours with a Sinusoidal Function

A city's daylight hours vary seasonally and are modeled by $$D(t)=11+1.5\sin\left(\frac{2\pi}{365}(t

Medium

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

Multiple Angle Equation

Solve the trigonometric equation $$2*\sin(2x) - \sqrt{3} = 0$$ for all $$x$$ in the interval $$[0, 2

Medium

Period Detection and Frequency Analysis

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

Medium

Periodic Phenomena in Weather Patterns

A city's average daily temperature over the course of a year is modeled by a sinusoidal function. Th

Medium

Phase Shifts and Reflections of Sine Functions

Analyze the relationship between the functions $$f(\theta)=\sin(\theta)$$ and $$g(\theta)=\sin(\thet

Easy

Piecewise Trigonometric Function and Continuity Analysis

Consider the piecewise defined function $$f(\theta)=\begin{cases}\frac{\sin(\theta)}{\theta} & ,\ \t

Medium

Polar Coordinates Conversion

Convert between Cartesian and polar coordinates and analyze related polar equations.

Medium

Polar Graphs: Conversion and Analysis

Analyze the polar equation $$r=4*\cos(\theta)+3$$.

Hard

Polar Rate of Change

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

Medium

Polar Rose Analysis

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

Medium

Rate of Change in Polar Functions

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

Hard

Reciprocal and Pythagorean Identities

Verify the identity $$1+\cot^2(x)=\csc^2(x)$$ and use it to solve the related trigonometric equation

Easy

Reciprocal Trigonometric Functions

Consider the function $$f(x)=\sec(x)=\frac{1}{\cos(x)}\).

Medium

Rewriting and Graphing a Composite Trigonometric Function

Given the function $$f(x)=\cos(x)+\sin(x)$$, transform it into the form $$R*\cos(x-\phi)$$.

Hard

Roses and Limacons in Polar Graphs

Consider the polar curves described below and answer the following:

Hard

Roulette Wheel Outcomes and Angle Analysis

A casino roulette wheel is divided into 12 equal sectors. Answer the following:

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

Seasonal Temperature Modeling

A city's average temperature over the year is modeled by a cosine function. The following table show

Easy

Secant, Cosecant, and Cotangent Functions Analysis

Consider the reciprocal trigonometric functions. Answer the following:

Hard

Sinusoidal Combination

Let $$f(x) = 3*\sin(x) + 2*\cos(x)$$.

Hard

Sinusoidal Function Transformation Analysis

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

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 Basic Trigonometric Equation

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

Easy

Solving Trigonometric Equations in a Survey

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

Easy

Special Triangles and Unit Circle Coordinates

Consider the actual geometric constructions of the special triangles used within the unit circle, sp

Easy

Tangent Function and Asymptotes

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

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

Tide Height Model: Using Sine Functions

A coastal region experiences tides that follow a sinusoidal pattern. A table of tide heights (in fee

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

Using Trigonometric Sum and Difference Identities

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

Hard

Verification and Application of Trigonometric Identities

Consider the sine addition identity $$\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\b

Easy

Verifying a Trigonometric Identity

Demonstrate that the identity $$\sin^2(x)+\cos^2(x)=1$$ holds for all real numbers \(x\).

Easy

Vibration Analysis

A mechanical system oscillates with displacement given by $$d(t) = 5*\cos(4t - \frac{\pi}{3})$$ (in

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

Advanced Matrix Modeling in Economic Transitions

An economic model is represented by a 3×3 transition matrix $$M=\begin{pmatrix}0.5 & 0.2 & 0.3\\0.1

Extreme

Area of a Parallelogram Using Determinants

Given the vectors $$u=\langle 3, 5 \rangle$$ and $$v=\langle -2, 4 \rangle$$: (a) Write the 2×2 mat

Easy

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

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

Circular Motion Parametrization

Consider a particle moving along a circular path defined by the parametric equations $$x(t)= 5*\cos(

Medium

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

Composition of Linear Transformations

Let two linear transformations in \(\mathbb{R}^2\) be represented by the matrices $$E=\begin{pmatrix

Hard

Converting an Explicit Function to Parametric Form

The function $$f(x)=x^3-3*x+2$$ is given explicitly. One way to parametrize this function is by lett

Easy

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 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

Estimating a Definite Integral with a Table

The function x(t) represents the distance traveled (in meters) by an object over time, with the foll

Medium

Ferris Wheel Motion

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

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 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 14: Linear Transformation and Rotation Matrix

Consider the rotation matrix $$R=\begin{bmatrix}\cos(t) & -\sin(t)\\ \sin(t) & \cos(t)\end{bmatrix}$

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 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

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 Analysis of a Quadratic Function

Consider the function $$f(x)=x^2-4$$ defined for $$x\geq0$$. Analyze the function and its inverse.

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 and Determinant of a Matrix

Consider the matrix $$A=\begin{pmatrix}4 & 3 \\ 2 & 1\end{pmatrix}$$.

Easy

Inverse Matrix and Transformation of the Unit Square

Given the transformation matrix $$A=\begin{pmatrix}3 & 1 \\ 2 & 2\end{pmatrix}$$ applied to the unit

Extreme

Linear Parametric Motion Modeling

A car travels along a straight path, and its position in the plane is given by the parametric equati

Easy

Linear Transformation and its Effect on Geometric Shapes

A linear transformation in \(\mathbb{R}^2\) is represented by the matrix $$M=\begin{pmatrix} 2 & 0 \

Easy

Linear Transformation Composition

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

Hard

Linear Transformation Evaluation

Given the transformation matrix $$T = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$$, answer the fo

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

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 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 Modeling of Department Transitions

A company’s employee transitions between two departments are modeled by the matrix $$M=\begin{pmatri

Extreme

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

Modeling Linear Motion Using Parametric Equations

A car travels along a straight road. Its position in the plane is given by the parametric equations

Easy

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

Parametric Equations and Inverses

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

Medium

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 Motion Analysis Using Tabulated Data

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

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 Parabola

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

Easy

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

Parametrically Defined Circular Motion

A circle of radius 5 is modeled by the parametric equations $$x(t)= 5\cos(t)$$ and $$y(t)= 5\sin(t)$

Easy

Parametrization of a Circle

The circle defined by $$x^2+y^2=25$$ represents all points at a distance of 5 from the origin.

Easy

Parametrization of a Parabola

Given the explicit function $$y = 2*x^2 + 3*x - 1$$, answer the following:

Medium

Parametrization of an Ellipse for a Racetrack

A racetrack is shaped like the ellipse given by $$\frac{(x-1)^2}{16}+\frac{(y+2)^2}{9}=1$$.

Medium

Parametrizing a Linear Path: Car Motion

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

Easy

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

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 in Vector-Valued Functions

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

Easy

Reflection Transformation Using Matrices

A reflection over the line \(y=x\) in the plane can be represented by the matrix $$R=\begin{pmatrix}

Easy

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

Transition Matrix in Markov Chains

A system transitions between two states according to the matrix $$M= \begin{pmatrix} 0.7 & 0.3 \\ 0.

Medium

Vector Operations

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

Easy

Vectors in Polar and Cartesian Coordinates

A drone's position is described in polar coordinates by $$r(t)=5+t$$ and $$\theta(t)=\frac{\pi}{6}t$

Medium

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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.