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AP Precalculus Free Response Questions

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  • Unit 1: Polynomial and Rational Functions (74)
  • Unit 2: Exponential and Logarithmic Functions (61)
  • Unit 3: Trigonometric and Polar Functions (55)
  • Unit 4: Functions Involving Parameters, Vectors, and Matrices (60)
Unit 1: Polynomial and Rational Functions

Analysis of a Quartic Function

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

Hard

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

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 End Behavior of Polynomial Functions

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

Easy

Application of the Binomial Theorem

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

Easy

Behavior Analysis of a Rational Function with Cancelled Factors

Consider the function $$f(x)=\frac{x^2-16}{x-4}$$. Analyze the behavior of the function at the point

Easy

Characterizing End Behavior and Asymptotes

A rational function modeling a population is given by $$R(x)=\frac{3*x^2+2*x-1}{x^2-4}$$. Analyze th

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 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

Constructing a Rational Function Model with Asymptotic Behavior

An engineer is modeling the concentration of a pollutant over time with a rational function. The fun

Hard

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

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 Function Behavior from a Data Table

A function $$f(x)$$ is represented by the table below: | x | f(x) | |-----|------| | -3 | 10 |

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

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

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

End Behavior of a Quartic Polynomial

Consider the quartic polynomial function $$f(x) = -3*x^4 + 5*x^3 - 2*x^2 + x - 7$$. Analyze the end

Easy

Engineering Application: Stress Analysis Model

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

Medium

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

Factoring and Dividing Polynomial Functions

Engineers are analyzing the stress on a structural beam, modeled by the polynomial function $$P(x)=

Hard

Factoring and Zero Multiplicity

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

Graph Interpretation and Log Transformation

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

Medium

Graphical Analysis of Inverse Function for a Linear Transformation

Consider the function $$f(x)=4*(x+1)-5$$. Answer the following questions regarding the transformatio

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 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 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

Linear Function Inverse Analysis

Consider the function $$f(x) = 2*x + 3$$. Answer the following questions concerning its inverse func

Easy

Logarithmic and Exponential Equations with Rational Functions

A process is modeled by the function $$F(x)= \frac{3*e^{2*x} - 5}{e^{2*x}+1}$$, where x is measured

Extreme

Logarithmic Linearization in Exponential Growth

An ecologist is studying the growth of a bacterial population in a laboratory experiment. The popula

Easy

Marketing Analysis Using Piecewise Polynomial Function

A firm's sales function is modeled by $$ S(x)=\begin{cases} -x^2+6*x & \text{for } x\le3, \\ 2*x+3 &

Easy

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

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

Piecewise Polynomial and Rational Function Analysis

A traffic flow model is described by the piecewise function $$f(t)= \begin{cases} a*t^2+b*t+c & \tex

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

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

Hard

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 Model Construction and Interpretation

A company’s profit (in thousands of dollars) over time t (in months) is modeled by the quadratic fun

Easy

Polynomial Model from Temperature Data

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

Medium

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

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 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 Inverse Analysis

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

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 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: 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

Real-World Modeling: Population Estimation

A biologist models the population of a species over time $$t$$ (in years) with the polynomial functi

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

Roller Coaster Track Polynomial Analysis

A section of a roller coaster track is modeled by a polynomial function $$h(x)$$ which gives the hei

Hard

Signal Strength Transformation Analysis

A satellite's signal strength is modeled by the function $$S(x) = 20*\sin(x)$$. A transformation is

Easy

Slant Asymptote Determination for a Rational Function

Determine the slant (oblique) asymptote of the rational function $$r(x)= \frac{2*x^2 + 3*x - 5}{x -

Medium

Solving a Polynomial Inequality

Solve the inequality $$x^3 - 4*x^2 + x + 6 \ge 0$$ and justify your solution.

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

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

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

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
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

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 Sequence Analysis

An arithmetic sequence is defined as an ordered list of numbers with a constant difference between c

Easy

Arithmetic Sequence Analysis

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

Easy

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

Bacterial Growth Model and Inverse Function

A bacterial culture grows according to the function $$f(x)=500*2^(x/3)$$, where $$x$$ is time in hou

Medium

Base Transformation and End Behavior

Consider the functions \(f(x)=2^{x}\) and \(g(x)=5\cdot2^{(x+3)}-7\). (a) Express the function \(f(

Medium

Cell Division Pattern

A culture of cells undergoes division such that the number of cells doubles every hour. The initial

Easy

Comparing Arithmetic and Exponential Models in Population Growth

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

Hard

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 Growth Models

A company is analyzing its profit growth using two distinct models: an arithmetic model given by $$P

Medium

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 with Exponential and Logarithmic Elements

Given the functions $$f(x)= \ln(x)$$ and $$g(x)= e^x$$, analyze their compositions.

Easy

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

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

Composition and Transformation Functions

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

Hard

Compound Interest and Exponential Equations

An investment account is compounded continuously with an initial balance of $$1000$$ and an annual i

Medium

Compound Interest and Financial Growth

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

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

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

Exploring Logarithmic Scales: pH and Hydrogen Ion Concentration

In chemistry, the pH of a solution is defined by the relation $$pH = -\log([H^+])$$, where $$[H^+]$$

Medium

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 Decay in Pollution Reduction

The concentration of a pollutant in a lake decreases exponentially according to the model $$f(t)= a\

Medium

Exponential Decay: Modeling Half-Life

A radioactive substance decays with a half-life of 5 years. At \(t = 10\) years, the mass of the sub

Hard

Exponential Equations via Logarithms

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

Hard

Exponential Growth from Percentage Increase

A process increases by 8% per unit time. Write an exponential function that models this growth.

Easy

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 Construction

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

Easy

Inverse Function of an Exponential Function

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

Hard

Inverse Functions in Exponential Contexts

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

Medium

Inverse Functions of Exponential and Logarithmic Forms

Consider the exponential function $$f(x) = 2 \cdot 3^x$$. Answer the following parts.

Medium

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

Investment Growth: Compound Interest

An investor deposits an initial amount \(P\) dollars in a savings account that compounds interest an

Medium

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

Logarithmic Function Analysis

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

Medium

Logarithmic Function and Properties

Consider the logarithmic function $$g(x) = \log_3(x)$$ and analyze its properties.

Medium

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 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

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 Modeling

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

Medium

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 Data Analysis

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

Medium

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 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 Cooling Model

An object cooling in a room follows Newton’s Law of Cooling. The temperature of the object is modele

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

Medium

Transformations of Exponential Functions

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

Easy

Transformed Exponential Equation

Solve the exponential equation $$5 \cdot (1.2)^{(x-3)} = 20$$.

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

Tumor Growth with Time Dilation Effects

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

Extreme

Using Exponential Product Property in Function Analysis

Consider the function $$f(x)= 3^x * 2^{2x}.$$

Easy
Unit 3: Trigonometric and Polar Functions

Amplitude and Period Transformations

A Ferris wheel ride is modeled by a sinusoidal function. The ride has a maximum height of 75 ft and

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

Analyzing Sinusoidal Variation in Daylight Hours

A researcher models daylight hours over a year with the function $$D(t) = 5 + 2.5*\sin((2\pi/365)*(t

Medium

Analyzing the Tangent Function

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

Easy

Application of Trigonometric Sum Identities

Utilize trigonometric sum identities to simplify and solve expressions.

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

Calculating the Area Enclosed by a Polar Curve

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

Hard

Combining Logarithmic and Trigonometric Equations

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

Hard

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

Evaluating Inverse Trigonometric Functions

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

Easy

Exploring Inverse Trigonometric Functions

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

Easy

Exploring the Pythagorean Identity

The Pythagorean identity $$\sin^2(θ)+\cos^2(θ)=1$$ is fundamental in trigonometry. Use this identity

Easy

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

Graphing a Limacon

Given the polar equation $$r=2+3*\cos(\theta)$$, analyze and graph the corresponding limacon.

Hard

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 Polar Circles and Roses

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

Medium

Graphing the Tangent Function with Asymptotes

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

Hard

Inverse Trigonometric Function Analysis

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

Easy

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 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 Daylight Variation

A coastal city records its daylight hours over the year. A sinusoidal model of the form $$D(t)=A*\si

Medium

Modeling Seasonal Temperature Data with Sinusoidal Functions

A sinusoidal pattern is observed in average monthly temperatures. Refer to the provided temperature

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

Pendulum Motion and Periodic Phenomena

A pendulum's angular displacement from the vertical is observed to follow a periodic pattern. Refer

Medium

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

Phase Shift Analysis in Sinusoidal Functions

A sinusoidal function describing a physical process is given by $$f(\theta)=5*\sin(\theta-\phi)+2$$.

Medium

Phase Shift and Frequency Analysis

Analyze the function $$f(x)=\cos\Bigl(4\bigl(x-\frac{\pi}{8}\bigr)\Bigr)$$.

Medium

Phase Shifts and Reflections of Sine Functions

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

Easy

Polar Coordinates Conversion

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

Medium

Polar Rate of Change

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

Medium

Proof and Application of Trigonometric Sum Identities

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

Extreme

Reciprocal Trigonometric Functions: Secant, Cosecant, and Cotangent

Consider the functions $$f(\theta)=\sec(\theta)$$, $$g(\theta)=\csc(\theta)$$, and $$h(\theta)=\cot(

Extreme

Roulette Wheel Outcomes and Angle Analysis

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

Hard

Secant, Cosecant, and Cotangent Functions Analysis

Consider the reciprocal trigonometric functions. Answer the following:

Hard

Sinusoidal Data Analysis

An experimental setup records data that follows a sinusoidal pattern. The table below gives the disp

Medium

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

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

Medium

Solving Trigonometric Equations in a Survey

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

Easy

Solving Trigonometric Inequalities

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

Medium

Tangent Function Shift

Consider the function $$f(x) = \tan\left(x - \frac{\pi}{6}\right)$$.

Medium

Tidal Motion Analysis

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

Medium

Transformations of Sinusoidal Functions

Consider the function $$y = 3*\sin(2*(x - \pi/4)) - 1$$. Answer the following:

Medium

Understanding Coterminal Angles and Their Applications

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

Easy

Understanding Coterminal Angles Through Art Installation

An artist designing a circular mural plans to use repeating motifs based on angles. Answer the follo

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

Analysis of Vector Directions and Transformations

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

Hard

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

Composition of Linear Transformations

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

Medium

Determinant and Inverse Calculation

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

Easy

Discontinuity in a Function Modeling Transition between States

A system's state is modeled by the function $$S(x)=\begin{cases} \frac{x^2-16}{x-4} & \text{if } x \

Medium

Dot Product, Projection, and Angle Calculation

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

Medium

Eliminating the Parameter

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

Hard

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

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

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 6: Implicit Function to Parametric Representation

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

Hard

FRQ 8: Vector Analysis - Dot Product and Angle

Given the vectors $$\textbf{u}=\langle3,4\rangle$$ and $$\textbf{v}=\langle-2,5\rangle$$, analyze th

Medium

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 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

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

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 Rational Function

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

Medium

Inverse Function and Transformation Mapping

Given the function $$f(x)=\frac{x+2}{3}$$, analyze its invertibility and the relationship between th

Easy

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 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 Area Scaling

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

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

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 Modeling of Department Transitions

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

Extreme

Matrix Multiplication and Properties

Let $$A=\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}$$ and $$B=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmat

Hard

Matrix Representation of Linear Transformations

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

Medium

Modeling Discontinuities in a Function Representing Planar Motion

A car's horizontal motion is modeled by the function $$x(t)=\begin{cases} \frac{t^2-1}{t-1} & \text{

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

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 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 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

A parabola is given by the equation $$y=x^2-2*x+1$$. A parametric representation for this parabola i

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

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 Table and Graph Analysis

Consider the parametric function $$f(t)= (x(t), y(t))$$ where $$x(t)= t^2$$ and $$y(t)= 2*t$$ for $$

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

Parametrically Defined Circular Motion

A particle moves along a circle of radius 2 with parametric equations $$x(t)=2*cos(t)$$ and $$y(t)=2

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

Population Transition Matrix Analysis

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

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

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

Rate of Change Analysis in Parametric Motion

A particle’s movement is described by the parametric equations $$x(t)=t^3-6*t+4$$ and $$y(t)=2*t^2-t

Hard

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

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 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 and Dot Product

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

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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Need to review before working on AP Precalculus FRQs?

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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.