Ace the free response questions on your AP Precalculus exam with practice FRQs graded by Kai. Choose your subject below.
Knowt can make mistakes. Consider checking important information.
The best way to get better at FRQs is practice. Browse through dozens of practice AP Precalculus FRQs to get ready for the big day.
Absolute Extrema and Local Extrema of a Polynomial
Consider the polynomial function $$p(x)= (x-3)^2*(x+3)$$.
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
Analyzing a Rational Function with a Hole
Consider the rational function $$R(x)= \frac{x^2-4}{x^2-x-6}$$.
Analyzing an Odd Polynomial Function
Consider the function $$p(x)= x^3 - 4*x$$. Investigate its properties by answering the following par
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
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
Carrying Capacity in Population Models
A rational function $$P(t) = \frac{50*t}{t + 10}$$ is used to model a population approaching its car
Comparative Analysis of Polynomial and Rational Functions
A function is defined piecewise by $$ f(x)=\begin{cases} x^2-4 & \text{if } x\le2, \\ \frac{x^2-4}{x
Comparing Polynomial and Rational Function Models
Two models are proposed to describe a data set. Model A is a polynomial function given by $$f(x)= 2*
Composite Function Analysis in Environmental Modeling
Environmental data shows the concentration (in mg/L) of a pollutant over time (in hours) as given in
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
Composite Function Transformations
Consider the polynomial function $$f(x)= x^2-4$$. A new function is defined by $$g(x)= \ln(|f(x)+5|)
Constructing a Function Model from Experimental Data
An engineer collects data on the stress (in MPa) experienced by a material under various applied for
Construction of a Polynomial Model
A company’s quarterly profit (in thousands of dollars) over five quarters is given in the table belo
Continuous Piecewise Function Modification
A company models its profit $$P(x)$$ (in thousands of dollars) with the piecewise function: $$ P(x)=
Cubic Polynomial Analysis
Consider the cubic polynomial function $$f(x) = 2*x^3 - 3*x^2 - 12*x + 8$$. Analyze the function as
Determining Domain and Range of a Transformed Rational Function
Consider the function $$g(x)= \frac{x^2 - 9}{x-3}$$. Answer the following:
Determining Function Behavior from a Data Table
A function $$f(x)$$ is represented by the table below: | x | f(x) | |-----|------| | -3 | 10 |
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 |
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
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
Examining End Behavior of Polynomial Functions
Consider the polynomial function $$f(x)= -3*x^4 + 2*x^3 - x + 7$$. Answer the following parts.
Expansion Using the Binomial Theorem in Forecasting
In a business forecast, the expression $$(x + 5)^4$$ is used to model compound factors affecting rev
Exploring Polynomial Function Behavior
Consider the polynomial function $$f(x)= 2*(x-1)^2*(x+2)$$, which is used to model a physical trajec
Factoring and Dividing Polynomial Functions
Engineers are analyzing the stress on a structural beam, modeled by the polynomial function $$P(x)=
Function Model Construction from Data Set
A data set shows how a quantity V changes over time t as follows: | Time (t) | Value (V) | |-------
Geometric Series Model in Area Calculations
An architect designs a sequence of rectangles such that each rectangle's area is 0.8 times the area
Graph Analysis and Identification of Discontinuities
A function is defined by $$r(x)=\frac{(x-1)(x+1)}{(x-1)(x+2)}$$ and is used to model a physical phen
Graphical Interpretation of Inverse Functions from a Data Table
A table below represents selected values of a polynomial function $$f(x)$$: | x | f(x) | |----|---
Interpreting Transformations of Functions
The parent function is $$f(x)= x^2$$. A transformed function is given by $$g(x)= -3*(x+2)^2+5$$. Ans
Intersection of Functions in Supply and Demand
Consider two functions that model supply and demand in a market. The supply function is given by $$f
Inverse Analysis Involving Multiple Transformations
Consider the function $$f(x)= 5 - 2*(x+3)^2$$. Answer the following questions regarding its inverse
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
Inverse Analysis of a Quartic Polynomial Function
Consider the quartic function $$f(x)= (x-1)^4 + 2$$. Answer the following questions concerning its i
Inverse Analysis of a Reciprocal Function
Consider the function $$f(x)= \frac{1}{x+2} + 3$$. Answer the following questions regarding its inve
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
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
Inversion of a Polynomial Ratio Function
Consider the function $$f(x)=\frac{x^2-1}{x+2}$$. Answer the following questions regarding its inver
Investigating a Real-World Polynomial Model
A physicist models the vertical trajectory of a projectile by the quadratic function $$h(t)= -5*t^2+
Investigating End Behavior of a Polynomial Function
Consider the polynomial function $$f(x)= -4*x^4+ x^3+ 2*x^2-7*x+1$$.
Loan Payment Model using Rational Functions
A bank uses the rational function $$R(x) = \frac{2*x^2 - 3*x - 5}{x - 2}$$ to model the monthly inte
Local and Global Extrema in a Polynomial Function
Consider the polynomial function $$f(x)= x^3 - 6*x^2 + 9*x + 15$$. Determine its local and global ex
Logarithmic and Exponential Equations with Rational Functions
A process is modeled by the function $$F(x)= \frac{3*e^{2*x} - 5}{e^{2*x}+1}$$, where x is measured
Logarithmic Equation Solving in a Financial Model
An investor compares two savings accounts. Account A grows continuously according to the model $$A(t
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
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
Modeling with a Polynomial Function: Optimization
A company’s profit (in thousands of dollars) is modeled by the polynomial function $$P(x)= -2*x^3+12
Modeling with Inverse Variation: A Rational Function
A physics experiment models the intensity $$I$$ of light as inversely proportional to the square of
Multivariable Rational Function: Zeros and Discontinuities
A pollutant concentration is modeled by $$C(x)= \frac{(x-3)*(x+2)}{(x-3)*(x-4)}$$, where x represent
Piecewise Polynomial and Rational Function Analysis
A traffic flow model is described by the piecewise function $$f(t)= \begin{cases} a*t^2+b*t+c & \tex
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
Polynomial Model Construction and Interpretation
A company’s profit (in thousands of dollars) over time t (in months) is modeled by the quadratic fun
Polynomial Transformation Challenge
Consider the function transformation given by $$g(x)= -2*(x+1)^3 + 3$$. Answer each part that follow
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
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
Rational Inequalities and Test Intervals
Solve the inequality $$\frac{x-3}{(x+2)(x-1)} < 0$$. Answer the following parts.
Revenue Function Transformations
A company models its revenue with a polynomial function $$f(x)$$. It is known that $$f(x)$$ has x-in
Slant Asymptote Determination for a Rational Function
Determine the slant (oblique) asymptote of the rational function $$r(x)= \frac{2*x^2 + 3*x - 5}{x -
Solving a System of Equations: Polynomial vs. Rational
Consider the system of equations where $$f(x)= x^2 - 1$$ and $$g(x)= \frac{2*x}{x+2}$$. Answer the f
Solving Polynomial Inequalities
Consider the polynomial $$p(x)= x^3 - 5*x^2 + 6*x$$. Answer the following parts.
Transformation of a Parabola
Starting with the parent function $$f(x)=x^2$$, a new function is defined by $$g(x) = -2*(x+3)^2 + 4
Trigonometric Function Analysis and Identity Verification
Consider the trigonometric function $$g(x)= 2*\tan(3*x-\frac{\pi}{4})$$, where $$x$$ is measured in
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
Zeros and Factorization Analysis
A fourth-degree polynomial $$Q(x)$$ is known to have zeros at $$x=-3$$ (with multiplicity 2), $$x=1$
Analyzing a Logarithmic Function from Data
A scientist proposes a logarithmic model for a process given by $$f(x)= \log_2(x) + 1$$. The observe
Analyzing Exponential Function Behavior from a Graph
An exponential function is depicted in the graph provided. Analyze the key features of the function.
Arithmetic Savings Plan
A person decides to save money every month, starting with an initial deposit of $$50$$ dollars, with
Arithmetic Sequence Analysis
An arithmetic sequence is defined as an ordered list of numbers with a constant difference between c
Arithmetic Sequence Analysis
Consider an arithmetic sequence with initial term $$a_0$$ and common difference $$d$$. Analyze the c
Bacterial Growth Model
In a laboratory experiment, a bacteria colony doubles every 3 hours. The initial count is $$500$$ ba
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(
Cell Division Pattern
A culture of cells undergoes division such that the number of cells doubles every hour. The initial
Comparing Arithmetic and Exponential Models in Population Growth
Two neighboring communities display different population growth patterns. Community A increases by a
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
Comparing Linear and Exponential Growth Models
A company is analyzing its profit growth using two distinct models: an arithmetic model given by $$P
Comparing Linear and Exponential Revenue Models
A company is forecasting its revenue growth using two models based on different assumptions. Initial
Competing Exponential Cooling Models
Two models are proposed for the cooling of an object. Model A is $$T_A(t) = T_env + 30·e^(-0.5*t)$$
Composite Functions with Exponential and Logarithmic Elements
Given the functions $$f(x)= \ln(x)$$ and $$g(x)= e^x$$, analyze their compositions.
Composite Sequences: Combining Geometric and Arithmetic Models in Production
A factory’s monthly production is influenced by two factors. There is a fixed increase in production
Composition of Exponential and Logarithmic Functions
Consider the functions $$f(x)= \log_5\left(\frac{x}{2}\right)$$ and $$g(x)= 10\cdot 5^x$$. Answer th
Compound Interest and Financial Growth
An investment account earns compound interest annually. An initial deposit of $$P = 1000$$ dollars i
Compound Interest Model with Regular Deposits
An account offers an annual interest rate of 5% compounded once per year. In addition to an initial
Determining an Exponential Model from Data
An outbreak of a virus produced the following data: | Time (days) | Infected Count | |-------------
Earthquake Magnitude and Energy Release
Earthquake energy is modeled by the equation $$E = k\cdot 10^{1.5M}$$, where $$E$$ is the energy rel
Exponential Decay in Pollution Reduction
The concentration of a pollutant in a lake decreases exponentially according to the model $$f(t)= a\
Exponential Equations via Logarithms
Solve the exponential equation $$3 * 2^(2*x) = 6^(x+1)$$.
Exponential Growth from Percentage Increase
A process increases by 8% per unit time. Write an exponential function that models this growth.
Exponential Inequality Solution
Solve the inequality $$5^(2*x - 1) < 3·5^(x)$$ for x.
Fitting a Logarithmic Model to Sales Data
A company observes that its sales revenue (in thousands of dollars) based on advertising spend (in t
Fractal Pattern Growth
A fractal pattern is generated such that after its initial creation, each iteration adds an area tha
Geometric Sequence in Compound Interest
An investment grows according to a geometric sequence. The initial investment is $$1000$$ dollars an
Graphical Analysis of Inverse Functions
Given the exponential function f(x) = 2ˣ + 3, analyze its inverse function.
Inverse Function of an Exponential Function
Consider the function $$f(x)= 3\cdot 2^x + 4$$.
Inverse Functions in Exponential Contexts
Consider the function $$f(x)= 5^x + 3$$. Analyze its inverse function.
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
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
Logarithmic Analysis of Earthquake Intensity
The magnitude of an earthquake on the Richter scale is determined using a logarithmic function. Cons
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
Logarithmic Equation and Extraneous Solutions
Solve the logarithmic equation $$log₂(x - 1) + log₂(3*x + 2) = 3$$.
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
Logarithmic Inequalities
Solve the inequality $$\log_{2}(x-1) > 3$$.
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
Modeling Bacterial Growth with Exponential Functions
A research laboratory is tracking the growth of a bacterial culture. A graph showing experimental da
Parameter Sensitivity in Exponential Functions
Consider an exponential function of the form $$f(x) = a \cdot b^{c x}$$. Suppose two data points are
pH and Logarithmic Functions
The pH of a solution is defined by $$pH = -\log_{10}[H^+]$$, where $$[H^+]$$ represents the hydrogen
Piecewise Exponential and Logarithmic Function Discontinuities
Consider the function defined by $$ f(x)=\begin{cases} 2^x + 1, & x < 3,\\ 5, & x = 3,
Piecewise Exponential-Log Function in Light Intensity Modeling
A scientist models the intensity of light as a function of distance using a piecewise function: $$
Population Growth Inversion
A town's population grows according to the function $$f(t)=1200*(1.05)^(t)$$, where $$t$$ is the tim
Population Growth with an Immigration Factor
A city's population is modeled by an equation that combines exponential growth with a constant linea
Radioactive Decay Modeling
A radioactive substance decays with a half-life of $$5$$ years. A sample has an initial mass of $$80
Savings Account Growth: Arithmetic vs Geometric Sequences
An individual opens a savings account that incorporates both regular deposits and interest earnings.
Shifted Exponential Function Analysis
Consider the exponential function $$f(x) = 4e^x$$. A transformed function is defined by $$g(x) = 4e^
Solving Logarithmic Equations and Checking Domain
An engineer is analyzing a system and obtains the following logarithmic equation: $$\log_3(x+2) + \
Solving Logarithmic Equations with Extraneous Solutions
Solve the logarithmic equation $$\log_2(x - 1) + \log_2(2x) = \log_2(10)$$ and check for any extrane
Telephone Call Data Analysis on Semi-Log Plot
A telecommunications company records the number of calls received each hour. The data suggest an exp
Temperature Cooling Model
An object cooling in a room follows Newton’s Law of Cooling. The temperature of the object is modele
Temperature Decay Modeled by a Logarithmic Function
In an experiment, the temperature (in degrees Celsius) of an object decreases over time according to
Transformations of Exponential Functions
Consider the exponential function \(f(x)=3\cdot2^{x}\). (a) Determine the equation of the transform
Transformations of Exponential Functions
Consider the exponential function $$f(x) = 3 \cdot 2^x$$. This function is transformed to produce $$
Transformations of Exponential Functions
Consider the exponential function $$f(x)= 7 * e^{0.3x}$$. Investigate its transformations.
Traveling Sales Discount Sequence
A traveling salesman offers discounts on his products following a geometric sequence. The initial pr
Weekly Population Growth Analysis
A species exhibits exponential growth in its weekly population. If the initial population is $$2000$
Analysis of a Bridge Suspension Vibration Pattern
After an impact, engineers recorded the vertical displacement (in meters) of a suspension bridge, mo
Analysis of a Limacon
Consider the polar function $$r(\theta) = 2 + 3*\cos(\theta)$$.
Analyzing a Rose Curve
Consider the polar equation $$r=3\,\sin(2\theta)$$.
Analyzing Sinusoidal Variation in Daylight Hours
A researcher models daylight hours over a year with the function $$D(t) = 5 + 2.5*\sin((2\pi/365)*(t
Application of Trigonometric Sum Identities
Utilize trigonometric sum identities to simplify and solve expressions.
Applying Sine and Cosine Sum Identities in Modeling
A researcher uses trigonometric sum identities to simplify complex periodic data. Consider the ident
Calculating the Area Enclosed by a Polar Curve
Consider the polar curve $$r=2*\cos(θ)$$. Without performing any integral calculations, use symmetry
Combining Logarithmic and Trigonometric Equations
Consider a model where the amplitude of a cosine function is modulated by an exponential decay. The
Comparing Sinusoidal Functions
Consider the functions $$f(x)=\sin(x)$$ and $$g(x)=\cos\Bigl(x-\frac{\pi}{2}\Bigr)$$.
Concavity in the Sine Function
Consider the function $$h(x) = \sin(x)$$ defined on the interval $$[0, 2\pi]$$.
Converting and Graphing Polar Equations
Consider the polar equation $$r=2*\cos(\theta)$$.
Daily Temperature Fluctuations
The table below shows the recorded temperature (in $$^{\circ}\text{F}$$) at various times during the
Daylight Hours Modeling
A city's daylight hours vary sinusoidally throughout the year. It is observed that the maximum dayli
Daylight Variation Model
A company models the variation in daylight hours over a year using the function $$D(t) = 10*\sin\Big
Exploring a Limacon
Consider the polar equation $$r=2+3\,\cos(\theta)$$.
Exploring Coterminal Angles and Periodicity
Analyze the concept of coterminal angles.
Exploring Inverse Trigonometric Functions
Consider the inverse sine function $$\arcsin(x)$$, defined for \(x\in[-1,1]\).
Exploring Limacons in Polar Coordinates
Consider the polar function $$r=2+3*\cos(θ)$$ which represents a limacon. Evaluate its key features
Graph Analysis of a Polar Function
The polar function $$r=4+3\sin(\theta)$$ is given, with the following data: | \(\theta\) (radians)
Graphing Polar Circles and Roses
Analyze the following polar equations: $$r=2$$ and $$r=3*\cos(2\theta)$$.
Graphing the Tangent Function with Asymptotes
The tangent function, $$f(\theta) = \tan(\theta)$$, exhibits vertical asymptotes where it is undefin
Identity Verification
Verify the following trigonometric identity using the sum formula for sine: $$\sin(\alpha+\beta) = \
Interpreting a Sinusoidal Graph
The graph provided displays a function of the form $$g(\theta)=a\sin[b(\theta-c)]+d$$. Use the graph
Inverse Trigonometric Functions
Examine the inverse relationships for trigonometric functions over appropriate restricted domains.
Modeling a Ferris Wheel's Motion Using Sinusoidal Functions
A Ferris wheel with a diameter of 10 meters rotates at a constant speed. The lowest point of the rid
Modeling Daylight Hours with a Sinusoidal Function
A study in a northern city recorded the number of daylight hours over the course of one year. The ob
Modeling Daylight Hours with a Sinusoidal Function
A city's daylight hours vary seasonally and are modeled by $$D(t)=11+1.5\sin\left(\frac{2\pi}{365}(t
Modeling Seasonal Temperature Data with Sinusoidal Functions
A sinusoidal pattern is observed in average monthly temperatures. Refer to the provided temperature
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
Modeling Tides with Sinusoidal Functions
Tidal heights at a coastal location are modeled by the function $$H(t)=2\,\sin\Bigl(\frac{\pi}{6}(t-
Period Detection and Frequency Analysis
An engineer analyzes a signal modeled by $$P(t)=6*\cos(5*(t-1))$$.
Periodic Phenomena: Seasonal Daylight Variation
A scientist is studying the variation in daylight hours over the course of a year in a northern regi
Phase Shift Analysis in Sinusoidal Functions
A sinusoidal function describing a physical process is given by $$f(\theta)=5*\sin(\theta-\phi)+2$$.
Phase Shift and Frequency Analysis
Analyze the function $$f(x)=\cos\Bigl(4\bigl(x-\frac{\pi}{8}\bigr)\Bigr)$$.
Phase Shifts and Reflections of Sine Functions
Analyze the relationship between the functions $$f(\theta)=\sin(\theta)$$ and $$g(\theta)=\sin(\thet
Piecewise Trigonometric Function and Continuity Analysis
Consider the piecewise defined function $$f(\theta)=\begin{cases}\frac{\sin(\theta)}{\theta} & ,\ \t
Polar Coordinates and Graphing a Circle
Answer the following questions on polar coordinates:
Polar Coordinates Conversion
Convert between Cartesian and polar coordinates and analyze related polar equations.
Polar Coordinates Conversion
Convert the rectangular coordinate point $$(-3,\,3\sqrt{3})$$ into polar form.
Polar Rose Analysis
Analyze the polar equation $$r = 2*\cos(3\theta)$$.
Probability and Trigonometry: Dartboard Game
A circular dartboard is divided into three regions by drawing two radii, forming sectors. One region
Proof and Application of Trigonometric Sum Identities
Trigonometric sum identities are a powerful tool in analyzing periodic phenomena.
Rate of Change in Polar Functions
Consider the polar function $$r(\theta)=3+\sin(\theta)$$.
Reciprocal Trigonometric Functions: Secant, Cosecant, and Cotangent
Consider the functions $$f(\theta)=\sec(\theta)$$, $$g(\theta)=\csc(\theta)$$, and $$h(\theta)=\cot(
Rose Curve in Polar Coordinates
The polar function $$r(\theta) = 4*\cos(3*\theta)$$ represents a rose curve.
Roses and Limacons in Polar Graphs
Consider the polar curves described below and answer the following:
Roulette Wheel Outcomes and Angle Analysis
A casino roulette wheel is divided into 12 equal sectors. Answer the following:
Seasonal Demand Modeling
A company's product demand follows a seasonal pattern modeled by $$D(t)=500+50\cos\left(\frac{2\pi}{
Seasonal Temperature Modeling
A city's average temperature over the year is modeled by a cosine function. The following table show
Secant Function and Its Transformations
Investigate the function $$f(\theta)=\sec(\theta)$$ and the transformation $$h(\theta)=2*\sec(\theta
Sinusoidal Combination
Let $$f(x) = 3*\sin(x) + 2*\cos(x)$$.
Solving a Trigonometric Equation
Solve the trigonometric equation $$2*\sin(\theta)+\sqrt{3}=0$$ for all solutions in the interval $$[
Solving Trigonometric Equations
Solve the equation $$\sin(x)+\cos(x)=1$$ for \(0\le x<2\pi\).
Solving Trigonometric Equations in a Survey
In a survey, participants' responses are modeled using trigonometric equations. Solve the following
Solving Trigonometric Inequalities
Solve the inequality $$\sin(\theta)>\frac{1}{2}$$ for \(\theta\) in the interval [0, 2\pi].
Special Triangles and Unit Circle Coordinates
Consider the actual geometric constructions of the special triangles used within the unit circle, sp
Tangent Function and Asymptotes
Examine the function $$f(\theta)=\tan(\theta)$$ defined on the interval $$\left(-\frac{\pi}{2}, \fra
Tidal Patterns and Sinusoidal Modeling
A coastal area experiences tides that follow a sinusoidal pattern described by $$T(t)=4+1.2\sin\left
Transformations of Sinusoidal Functions
Consider the function $$y = 3*\sin(2*(x - \pi/4)) - 1$$. Answer the following:
Understanding Coterminal Angles and Their Applications
Coterminal angles are important in trigonometry as they represent angles with the same terminal side
Unit Circle and Special Triangles
Consider the unit circle and the properties of special right triangles. Answer the following for a 4
Verification and Application of Trigonometric Identities
Consider the sine addition identity $$\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\b
Analysis of a Particle's Parametric Path
A particle moves in the plane with parametric equations $$x(t)=t^2 - 3*t + 2$$ and $$y(t)=4*t - t^2$
Analysis of Vector Directions and Transformations
Given the vectors $$\mathbf{a}=\langle -1,2\rangle$$ and $$\mathbf{b}=\langle 4,3\rangle$$, perform
Analyzing a Piecewise Function Representing a Linear Transformation
Let $$T(x)=\begin{cases} \frac{2x-4}{x-2} & \text{if } x \neq 2, \\ 3 & \text{if } x=2 \end{cases}$$
Analyzing the Composition of Two Matrix Transformations
Let matrices be given by $$A=\begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}$$ and $$B=\begin{pmatrix}2 & 0
Composition of Transformations and Inverses
Let $$A=\begin{bmatrix}2 & 3\\ 1 & 4\end{bmatrix}$$ and consider the linear transformation $$L(\vec{
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
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
Determinant and Inverse Calculation
Given the matrix $$C = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$$, answer the following:
Discontinuity Analysis in a Function Modeling Particle Motion
A particle’s position along a line is given by the piecewise function: $$s(t)=\begin{cases} \frac{t^
Discontinuity in a Function Modeling Transition between States
A system's state is modeled by the function $$S(x)=\begin{cases} \frac{x^2-16}{x-4} & \text{if } x \
Eliminating the Parameter in an Implicit Function
A curve is defined by the parametric equations $$x(t)=t+1$$ and $$y(t)=t^2-1$$.
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
Evaluating Limits and Discontinuities in a Parameter-Dependent Function
For the function $$g(t)=\begin{cases} \frac{2*t^2 - 8}{t-2} & \text{if } t \neq 2, \\ 6 & \text{if }
Evaluating Limits in a Parametrically Defined Motion Scenario
A particle’s motion is given by the parametric equations: $$x(t)=\begin{cases} \frac{t^2-9}{t-3} & \
Exponential Parametric Function and its Inverse
Consider the exponential function $$f(x)=e^{x}+2$$ defined for all real numbers. Analyze the functio
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 $$y(t)=2*t-1
FRQ 3: Linear Parametric Motion - Car Journey
A car travels along a linear path described by the parametric equations $$x(t)=3+2*t$$ and $$y(t)=4-
FRQ 5: Parametrically Defined Ellipse
An ellipse is described parametrically by $$x(t)=3*\cos(t)$$ and $$y(t)=2*\sin(t)$$ for $$t\in[0,2\p
FRQ 9: Vectors in Motion and Velocity
A particle's position is described by the vector-valued function $$p(t)=\langle2*t-1, t^2+1\rangle$$
FRQ 14: Linear Transformation and Rotation Matrix
Consider the rotation matrix $$R=\begin{bmatrix}\cos(t) & -\sin(t)\\ \sin(t) & \cos(t)\end{bmatrix}$
FRQ 15: Composition of Linear Transformations
Consider two linear transformations represented by the matrices $$A=\begin{bmatrix}2 & 0\\1 & 3\end{
Graphical Analysis of Parametric Motion
A particle moves in the plane with its position defined by the functions $$x(t)= t^2 - 2*t$$ and $$y
Graphical and Algebraic Analysis of a Function with a Removable Discontinuity
Consider the function $$g(x)=\begin{cases} \frac{\sin(x) - \sin(0)}{x-0} & \text{if } x \neq 0, \\ 1
Hyperbola Parametrization Using Trigonometric Functions
Consider the hyperbola defined by $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$. Answer the following:
Implicit Function Analysis
Consider the implicitly defined equation $$x^2 + y^2 - 4*x + 6*y - 12 = 0$$. Answer the following:
Inverse Analysis of a Quadratic Function
Consider the function $$f(x)=x^2-4$$ defined for $$x\geq0$$. Analyze the function and its inverse.
Inverse and Determinant of a 2×2 Matrix
Consider the matrix $$C=\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$$. Answer the following parts.
Inverse and Determinant of a Matrix
Let the 2×2 matrix be given by $$A= \begin{pmatrix} a & 2 \\ 3 & 4 \end{pmatrix}$$. Answer the follo
Inverse and Determinant of a Matrix
Consider the matrix $$A=\begin{pmatrix}4 & 3 \\ 2 & 1\end{pmatrix}$$.
Inverse of a 2×2 Matrix
Consider the matrix $$A=\begin{bmatrix}2 & 5\\ 3 & 7\end{bmatrix}$$.
Investigating Inverse Transformations in the Plane
Consider the linear transformation defined by $$L(\mathbf{v})=\begin{pmatrix}2 & 1\\3 & 4\end{pmatri
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
Logarithmic and Exponential Parametric Functions
A particle’s position is defined by the parametric equations $$x(t)= \ln(1+t)$$ and $$y(t)= e^{1-t}$
Matrices as Models for Population Dynamics
A population of two species is modeled by the transition matrix $$P=\begin{pmatrix} 0.8 & 0.1 \\ 0.2
Matrices as Representations of Rotation
Consider the matrix $$A=\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$$, which represents a rotation in
Matrix Applications in State Transitions
In a system representing transitions between two states, the following transition matrix is used: $
Matrix Modeling of State Transitions
In a two-state system, the transition matrix is given by $$T=\begin{pmatrix}0.8 & 0.2 \\ 0.3 & 0.7\e
Matrix Multiplication and Non-Commutativity
Let the matrices be defined as $$A=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ and $$B=\begin{pma
Matrix Multiplication and Properties
Let $$A=\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}$$ and $$B=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmat
Modeling Discontinuities in a Function Representing Planar Motion
A car's horizontal motion is modeled by the function $$x(t)=\begin{cases} \frac{t^2-1}{t-1} & \text{
Modeling Linear Motion Using Parametric Equations
A car travels along a straight road. Its position in the plane is given by the parametric equations
Parabolic and Elliptical Parametric Representations
A parabola is given by the equation $$y=x^2-4*x+3$$.
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
Parametric Equations and Inverses
A curve is defined parametrically by $$x(t)=t+2$$ and $$y(t)=3*t-1$$.
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. (
Parametric Representation of an Ellipse
Consider the ellipse defined by $$\frac{x^2}{9}+\frac{y^2}{4}=1$$. A common parametrization uses $$x
Parametric Representation on the Unit Circle and Special Angles
Consider the unit circle defined by the parametric equations $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$.
Parametrically Defined Circular Motion
A circle of radius 5 is modeled by the parametric equations $$x(t)= 5\cos(t)$$ and $$y(t)= 5\sin(t)$
Parametrically Defined Circular Motion
A particle moves along a circle of radius 2 with parametric equations $$x(t)=2*cos(t)$$ and $$y(t)=2
Parametrization of an Ellipse for a Racetrack
A racetrack is shaped like the ellipse given by $$\frac{(x-1)^2}{16}+\frac{(y+2)^2}{9}=1$$.
Parametrizing a Parabola
A parabola is defined parametrically by $$x(t)=t$$ and $$y(t)=t^2$$.
Particle Motion Through Position and Velocity Vectors
A particle’s position is given by the vector function $$\vec{p}(t)= \langle 3*t^2 - 2*t,\, t^3 \rang
Position and Velocity Vectors
For a particle with position $$\mathbf{p}(t)=\langle2*t+1, 3*t-2\rangle$$, where $$t$$ is in seconds
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
Rotation of a Force Vector
A force vector is given by \(\vec{F}= \langle 10, 5 \rangle\). This force is rotated by 30° counterc
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
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
Uniform Circular Motion
A car is moving along a circular track of radius 10 meters. Its motion is described by the parametri
Vector Addition and Scalar Multiplication
Consider the vectors $$\vec{u}=\langle 1, 3 \rangle$$ and $$\vec{v}=\langle -2, 4 \rangle$$:
Vector Operations
Given the vectors $$u=\langle 3, -2 \rangle$$ and $$v=\langle -1, 4 \rangle$$, (a) Compute the magn
Vector Operations and Dot Product
Let $$\mathbf{u}=\langle 3,-1 \rangle$$ and $$\mathbf{v}=\langle -2,4 \rangle$$. Use these vectors t
Vector Operations in the Plane
Let $$\mathbf{u}=\langle3, -2\rangle$$ and $$\mathbf{v}=\langle -1, 4\rangle$$.
Vector Operations in the Plane
Let the vectors be given by $$\mathbf{u}=\langle 3,-4\rangle$$ and $$\mathbf{v}=\langle -2,5\rangle$
Vector Operations in the Plane
Let $$\vec{u}= \langle 3, -2 \rangle$$ and $$\vec{v}= \langle -1, 4 \rangle$$. Perform the following
Vector Scalar Multiplication
Given the vector $$\mathbf{w} = \langle -2, 5 \rangle$$ and the scalar $$k = -3$$, answer the follow
Vectors in the Context of Physics
A force vector applied to an object is given by $$\vec{F}=\langle 5, -7 \rangle$$ and the displaceme
Everyone is relying on Knowt, and we never let them down.
We have over 5 million resources across various exams, and subjects to refer to at any point.
We’ve found the best flashcards & notes on Knowt.