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AP Precalculus Free Response Questions

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  • Unit 1: Polynomial and Rational Functions (64)
  • Unit 2: Exponential and Logarithmic Functions (58)
  • Unit 3: Trigonometric and Polar Functions (62)
  • Unit 4: Functions Involving Parameters, Vectors, and Matrices (66)
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 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 a Rational Function with a Hole

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

Medium

Analyzing an Odd Polynomial Function

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

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

Average Rate of Change in Rational Functions

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

Medium

Carrying Capacity in Population Models

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

Easy

Comparative Analysis of Polynomial and Rational Functions

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

Medium

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

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

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

Continuous Piecewise Function Modification

A company models its profit $$P(x)$$ (in thousands of dollars) with the piecewise function: $$ P(x)=

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

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 Function Behavior from a Data Table

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

Easy

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

Evaluating Limits and Discontinuities in a Rational Function

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

Medium

Examining End Behavior of Polynomial Functions

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

Easy

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 Polynomial Function Behavior

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

Easy

Factoring and Dividing Polynomial Functions

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

Hard

Function Model Construction from Data Set

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

Medium

Geometric Series Model in Area Calculations

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

Easy

Graph Analysis and Identification of Discontinuities

A function is defined by $$r(x)=\frac{(x-1)(x+1)}{(x-1)(x+2)}$$ and is used to model a physical phen

Medium

Graphical Interpretation of Inverse Functions from a Data Table

A table below represents selected values of a polynomial function $$f(x)$$: | x | f(x) | |----|---

Easy

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 Involving Multiple Transformations

Consider the function $$f(x)= 5 - 2*(x+3)^2$$. Answer the following questions regarding its inverse

Medium

Inverse Analysis of a Modified Rational Function

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

Extreme

Inverse Analysis of a Quartic Polynomial Function

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

Hard

Inverse Analysis of a Reciprocal Function

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

Medium

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

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 End Behavior of a Polynomial Function

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

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

Local and Global Extrema in a Polynomial Function

Consider the polynomial function $$f(x)= x^3 - 6*x^2 + 9*x + 15$$. Determine its local and global ex

Hard

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 Equation Solving in a Financial Model

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

Medium

Modeling a Real-World Scenario with a Rational Function

A biologist is studying the concentration of a nutrient in a lake. The concentration (in mg/L) is mo

Easy

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

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 Inverse Variation: A Rational Function

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

Easy

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

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 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 Transformation Challenge

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

Easy

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 Inequalities and Test Intervals

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

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

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 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

Solving Polynomial Inequalities

Consider the polynomial $$p(x)= x^3 - 5*x^2 + 6*x$$. Answer the following parts.

Medium

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 Complex Conjugates in Polynomial Functions

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

Easy

Zeros and Factorization Analysis

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

Medium
Unit 2: Exponential and Logarithmic Functions

Analyzing a Logarithmic Function from Data

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

Medium

Analyzing Exponential Function Behavior 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

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

Bacterial Growth Model

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

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

Comparing Linear and Exponential Revenue Models

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

Medium

Competing Exponential Cooling Models

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

Hard

Composite Functions with Exponential and Logarithmic Elements

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

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

Composition of Exponential and Logarithmic Functions

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

Medium

Compound Interest and Financial Growth

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

Easy

Compound Interest Model with Regular Deposits

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

Hard

Determining an Exponential Model from Data

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

Medium

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

Exponential Decay in Pollution Reduction

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

Medium

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

Exponential Inequality Solution

Solve the inequality $$5^(2*x - 1) < 3·5^(x)$$ for x.

Hard

Fitting a Logarithmic Model to Sales Data

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

Hard

Fractal Pattern Growth

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

Medium

Geometric Sequence in Compound Interest

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

Easy

Graphical Analysis of Inverse Functions

Given the exponential function f(x) = 2ˣ + 3, analyze its inverse function.

Medium

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 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

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 Cost Function in Production

A company’s cost function is given by $$C(x)= 50+ 10\log_{2}(x)$$, where $$x>0$$ represents the numb

Medium

Logarithmic Equation and Extraneous Solutions

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

Hard

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 Inequalities

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

Easy

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

Modeling Bacterial Growth with Exponential Functions

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

Medium

Parameter Sensitivity in Exponential Functions

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

Hard

pH and Logarithmic Functions

The pH of a solution is defined by $$pH = -\log_{10}[H^+]$$, where $$[H^+]$$ represents the hydrogen

Medium

Piecewise Exponential and Logarithmic Function Discontinuities

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

Hard

Piecewise Exponential-Log Function in Light Intensity Modeling

A scientist models the intensity of light as a function of distance using a piecewise function: $$

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 with an Immigration Factor

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

Hard

Radioactive Decay Modeling

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

Medium

Savings Account Growth: Arithmetic vs Geometric Sequences

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

Hard

Shifted Exponential Function Analysis

Consider the exponential function $$f(x) = 4e^x$$. A transformed function is defined by $$g(x) = 4e^

Medium

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

Temperature Decay Modeled by a Logarithmic Function

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

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

Easy

Traveling Sales Discount Sequence

A traveling salesman offers discounts on his products following a geometric sequence. The initial pr

Easy

Weekly Population Growth Analysis

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

Hard
Unit 3: Trigonometric and Polar Functions

Analysis of a Bridge Suspension Vibration Pattern

After an impact, engineers recorded the vertical displacement (in meters) of a suspension bridge, mo

Medium

Analysis of a Limacon

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

Extreme

Analyzing a Rose Curve

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

Medium

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

Application of Trigonometric Sum Identities

Utilize trigonometric sum identities to simplify and solve expressions.

Hard

Applying Sine and Cosine Sum Identities in Modeling

A researcher uses trigonometric sum identities to simplify complex periodic data. Consider the ident

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

Comparing Sinusoidal Functions

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

Easy

Concavity in the Sine Function

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

Medium

Converting and Graphing Polar Equations

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

Medium

Daily Temperature Fluctuations

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

Easy

Daylight Hours Modeling

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

Medium

Daylight Variation Model

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

Medium

Exploring a Limacon

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

Hard

Exploring Coterminal Angles and Periodicity

Analyze the concept of coterminal angles.

Easy

Exploring Inverse Trigonometric Functions

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

Easy

Exploring Limacons in Polar Coordinates

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

Hard

Graph Analysis of a Polar Function

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

Hard

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

Identity Verification

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

Easy

Interpreting a Sinusoidal Graph

The graph provided displays a function of the form $$g(\theta)=a\sin[b(\theta-c)]+d$$. Use the graph

Medium

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 vary seasonally and are modeled by $$D(t)=11+1.5\sin\left(\frac{2\pi}{365}(t

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

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

Period Detection and Frequency Analysis

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

Medium

Periodic Phenomena: Seasonal Daylight Variation

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

Medium

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

Piecewise Trigonometric Function and Continuity Analysis

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

Medium

Polar Coordinates and Graphing a Circle

Answer the following questions on polar coordinates:

Medium

Polar Coordinates Conversion

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

Medium

Polar Coordinates Conversion

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

Medium

Polar Rose Analysis

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

Medium

Probability and Trigonometry: Dartboard Game

A circular dartboard is divided into three regions by drawing two radii, forming sectors. One region

Extreme

Proof and Application of Trigonometric Sum Identities

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

Extreme

Rate of Change in Polar Functions

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

Hard

Reciprocal Trigonometric Functions: Secant, Cosecant, and Cotangent

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

Extreme

Rose Curve in Polar Coordinates

The polar function $$r(\theta) = 4*\cos(3*\theta)$$ represents a rose curve.

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 Function and Its Transformations

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

Medium

Sinusoidal Combination

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

Hard

Solving a Trigonometric Equation

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

Easy

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

Solving Trigonometric Inequalities

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

Medium

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

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

Unit Circle and Special Triangles

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

Easy

Verification and Application of Trigonometric Identities

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

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

Analysis of a Particle's Parametric Path

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

Medium

Analysis of Vector Directions and Transformations

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

Hard

Analyzing a Piecewise Function 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

Composition of Transformations and Inverses

Let $$A=\begin{bmatrix}2 & 3\\ 1 & 4\end{bmatrix}$$ and consider the linear transformation $$L(\vec{

Extreme

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

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 and Inverse Calculation

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

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 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

Eliminating the Parameter in an Implicit Function

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

Medium

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

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 Parametric Function and its Inverse

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

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 3: Linear Parametric Motion - Car Journey

A car travels along a linear path described by the parametric equations $$x(t)=3+2*t$$ and $$y(t)=4-

Easy

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 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 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

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

Hyperbola Parametrization Using Trigonometric Functions

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

Hard

Implicit Function Analysis

Consider the implicitly defined equation $$x^2 + y^2 - 4*x + 6*y - 12 = 0$$. Answer the following:

Easy

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 and Determinant of a 2×2 Matrix

Consider the matrix $$C=\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$$. Answer the following parts.

Easy

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 and Determinant of a Matrix

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

Easy

Inverse of a 2×2 Matrix

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

Medium

Investigating Inverse Transformations in the Plane

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

Medium

Linear Transformations via Matrices

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

Medium

Logarithmic and Exponential Parametric Functions

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

Medium

Matrices as 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 State Transitions

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

Extreme

Matrix Multiplication and Non-Commutativity

Let the matrices be defined as $$A=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ and $$B=\begin{pma

Medium

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

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 Linear Motion Using Parametric Equations

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

Easy

Parabolic and Elliptical Parametric Representations

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

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 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 an Ellipse

Consider the ellipse defined by $$\frac{x^2}{9}+\frac{y^2}{4}=1$$. A common parametrization uses $$x

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

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

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 Parabola

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

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

Position and Velocity Vectors

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

Easy

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

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

Tangent Line to a Parametric Curve

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

Medium

Transition 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

Uniform Circular Motion

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

Easy

Vector Addition and Scalar Multiplication

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

Medium

Vector Operations

Given the vectors $$u=\langle 3, -2 \rangle$$ and $$v=\langle -1, 4 \rangle$$, (a) Compute the magn

Easy

Vector Operations and Dot Product

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

Easy

Vector Operations in the Plane

Let $$\mathbf{u}=\langle3, -2\rangle$$ and $$\mathbf{v}=\langle -1, 4\rangle$$.

Medium

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

Vectors in the Context of Physics

A force vector applied to an object is given by $$\vec{F}=\langle 5, -7 \rangle$$ and the displaceme

Medium

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Need to review before working on AP Precalculus FRQs?

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Tips from Former AP Students

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