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.
Analysis of a Rational Function with Factorable Denominator
A function is given by $$f(x)=\frac{x^2-5*x+6}{x^2-4}$$. Examine its domain and discontinuities.
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 an Odd Polynomial Function
Consider the function $$p(x)= x^3 - 4*x$$. Investigate its properties by answering the following par
Analyzing End Behavior of Polynomial Functions
Consider the polynomial function $$P(x)= -2*x^4 + 3*x^3 - x + 5$$. Answer the following parts:
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
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
Comparative Analysis of Even and Odd Polynomial Functions
Consider the functions $$f(x)= x^4 - 4*x^2 + 3$$ and $$g(x)= x^3 - 2*x$$. Answer the following parts
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.
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 Functions and Inverses
Let $$f(x)= 3*(x-2)^2+1$$.
Constructing a Piecewise Function from Data
A company’s production cost function changes slopes at a production level of 100 units. The cost (in
Continuous Piecewise Function Modification
A company models its profit $$P(x)$$ (in thousands of dollars) with the piecewise function: $$ P(x)=
Cubic Function Inverse Analysis
Consider the cubic function $$f(x) = x^3 - 6*x^2 + 9*x$$. Answer the following questions related to
Data Analysis with Polynomial Interpolation
A scientist measures the decay of a radioactive substance at different times. The following table sh
Degree Determination from Finite Differences
A researcher records the size of a bacterial colony at equal time intervals, obtaining the following
Designing a Piecewise Function for a Temperature Model
A city experiences distinct temperature patterns during the day. A proposed model is as follows: for
Determining Degree from Discrete Data
Below is a table representing the output values of a polynomial function for equally-spaced input va
End Behavior of a Quartic Polynomial
Consider the quartic polynomial function $$f(x) = -3*x^4 + 5*x^3 - 2*x^2 + x - 7$$. Analyze the end
Engineering Application: Stress Analysis Model
In a stress testing experiment, the stress $$S(x)$$ on a component (in appropriate units) is modeled
Estimating Polynomial Degree from Finite Differences
The following table shows the values of a function $$f(x)$$ at equally spaced values of $$x$$: | x
Evaluating Limits and Discontinuities in a Rational Function
Consider the rational function $$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$$. Answer the following parts.
Exploring Asymptotic Behavior in a Sales Projection Model
A sales projection model is given by $$P(x)=\frac{4*x-2}{x-1}$$, where $$x$$ represents time in year
Exploring the Effect of Multiplicities on Graph Behavior
Consider the polynomial function $$q(x)= (x-1)^3*(x+2)^2$$.
Factoring and Dividing Polynomial Functions
Engineers are analyzing the stress on a structural beam, modeled by the polynomial function $$P(x)=
Finding and Interpreting Inflection Points
Consider the polynomial function $$f(x)= x^3 - 6*x^2 + 9*x + 1$$. Answer the following parts.
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
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 $$f
Inverse Analysis of a Modified Rational Function
Consider the function $$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$$. Answer the following questions concerning its i
Inverse Analysis of an Even Function with Domain Restriction
Consider the function $$f(x)=x^2$$ defined on the restricted domain $$x \ge 0$$. Answer the followin
Inversion of a Polynomial Ratio Function
Consider the function $$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+
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
Marketing Analysis Using Piecewise Polynomial Function
A firm's sales function is modeled by $$ S(x)=\begin{cases} -x^2+6*x & \text{for } x\le3, \\ 2*x+3 &
Model Interpretation: End Behavior and Asymptotic Analysis
A chemical reaction's saturation level is modeled by the rational function $$S(t)= \frac{10*t+5}{t+3
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 Inverse Variation: A Rational Approach
A variable $$y$$ is inversely proportional to $$x$$. Data indicates that when $$x=4$$, $$y=2$$, and
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$$ of light as inversely proportional to the square of
Modeling with Rational Functions
A chemist models the concentration of a reactant over time with the rational function $$C(t)= \frac{
Optimizing Production Using a Polynomial Model
A factory's production cost (in thousands of dollars) is modeled by the function $$C(x)= 0.02*x^3 -
Piecewise Financial Growth Model
A company’s quarterly growth rate is modeled using a piecewise function. For $$0 \le x \le 4$$, the
Piecewise Function and Domain Restrictions
A temperature function is defined as $$ T(x)=\begin{cases} \frac{x^2-25}{x-5} & x<5, \\ 3*x-10 & x\g
Piecewise Function Construction for Utility Rates
A utility company charges for electricity according to the following scheme: For usage $$u$$ (in kWh
Piecewise Function without a Calculator
Let the function $$f(x)=\begin{cases} x^2-1 & \text{for } x<2, \\ \frac{x^2-4}{x-2} & \text{for } x\
Polynomial End Behavior and Zeros Analysis
A polynomial function is given by $$f(x)= 2*x^4 - 3*x^3 - 12*x^2$$. This function models a physical
Polynomial Interpolation and Finite Differences
A quadratic function is used to model the height of a projectile. The following table gives the heig
Polynomial Long Division and Slant Asymptote
Perform polynomial long division on the function $$f(x)= \frac{3*x^3 - 2*x^2 + 4*x - 5}{x^2 - 1}$$,
Polynomial Model Construction and Interpretation
A company’s profit (in thousands of dollars) over time t (in months) is modeled by the quadratic fun
Polynomial Transformation Challenge
Consider the function transformation given by $$g(x)= -2*(x+1)^3 + 3$$. Answer each part that follow
Population Growth Modeling with a Polynomial Function
A regional population (in thousands) is modeled by a polynomial function $$P(t)$$, where $$t$$ repre
Product Revenue Rational Model
A company’s product revenue (in thousands of dollars) is modeled by the rational function $$R(x)= \f
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 Function Asymptotes and Holes
Consider the rational function $$r(x)=\frac{x^2 - 4}{x^2 - x - 6}$$. Analyze the function according
Rational Function Asymptotes and Holes
A machine’s efficiency is modeled by the rational function $$R(x)= \frac{x^2 - 4}{x^2 - x - 6}$$, wh
Rational Function: Machine Efficiency Ratios
A machine's efficiency is modeled by the rational function $$E(x) = \frac{x^2 - 9}{x^2 - 4*x + 3}$$,
Rational Inequalities Analysis
Solve the inequality $$\frac{x^2-4}{x+1} \ge 0$$ and represent the solution on a number line.
Rational Inequalities and Test Intervals
Solve the inequality $$\frac{x-3}{(x+2)(x-1)} < 0$$. Answer the following parts.
Return to a Rational Expression under Transformation
Consider the function $$f(x)=\frac{(x-2)(x+3)}{(x-2)(x-5)}$$, defined for $$x\neq2,5$$. Answer the f
Revenue Modeling with a Polynomial Function
A small theater's revenue from ticket sales is modeled by the polynomial function $$R(x)= -0.5*x^3 +
Roller Coaster Curve Analysis
A roller coaster's vertical profile is modeled by the polynomial function $$f(x)= -0.05*x^3 + 1.2*x^
Solving a Polynomial Inequality
Solve the inequality $$x^3 - 4*x^2 + x + 6 \ge 0$$ and justify your solution.
Transformation and Inversions of a Rational Function
A manufacturer models the cost per unit with the function $$C(x)= \frac{5*x+20}{x-2}$$, where x is t
Use of Logarithms to Solve for Exponents in a Compound Interest Equation
An investment of $$1000$$ grows continuously according to the formula $$I(t)=1000*e^{r*t}$$ and doub
Using the Binomial Theorem for Polynomial Expansion
A scientist is studying the expansion of the polynomial expression $$ (1+2*x)^5$$, which is related
Zero Finding and Sign Charts
Consider the function $$p(x)= (x-2)(x+1)(x-5)$$.
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), $$x=1$
Arithmetic Sequence Analysis
An arithmetic sequence is defined as an ordered list of numbers with a constant difference between c
Arithmetic Sequence Analysis
Consider an arithmetic sequence with initial term $$a_0 = 5$$ and constant difference $$d$$. Given t
Bacterial Growth Model
In a laboratory experiment, a bacteria colony doubles every 3 hours. The initial count is $$500$$ ba
Bacterial Growth Modeling
A biologist is studying a rapidly growing bacterial culture. The number of bacteria at time $$t$$ (i
Bacterial Growth: Arithmetic vs Exponential Models
A laboratory study records the growth of a bacterial culture at regular one‐hour intervals. The data
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(
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 $$P
Comparing Linear and Exponential Revenue Models
A company is forecasting its revenue growth using two models based on different assumptions. Initial
Composite Exponential-Logarithmic Functions
Let f(x) = log₃(x) and g(x) = 2·3ˣ. Analyze the following compositions.
Composite Function Analysis: Identity and Inverses
Let $$f(x)= 2^x$$ and $$g(x)= \log_2(x)$$.
Composite Functions and Their Inverses
For the functions $$f(x) = 2^x$$ and $$g(x) = \log_2(x)$$, analyze their composite functions.
Composite Functions Involving Exponential and Logarithmic Functions
Let $$f(x) = e^x$$ and $$g(x) = \ln(x)$$. Explore the compositions of these functions and their rela
Composite Sequences: Combining Geometric and Arithmetic Models in Production
A factory’s monthly production is influenced by two factors. There is a fixed increase in production
Compound Interest and Exponential Equations
An investment account is compounded continuously with an initial balance of $$1000$$ and an annual i
Compound Interest vs. Simple Interest
A financial analyst is comparing two interest methods on an initial deposit of $$10000$$ dollars. On
Compound Interest with Periodic Deposits
An investor opens an account with an initial deposit of $$5000$$ dollars and adds an additional $$50
Connecting Exponential Functions with Geometric Sequences
An exponential function $$f(x) = 5 \cdot 3^x$$ can also be interpreted as a geometric sequence where
Data Modeling: Exponential vs. Linear Models
A scientist collected data on the growth of a substance over time. The table below shows the measure
Earthquake Intensity on the Richter Scale
The Richter scale defines earthquake magnitude as \(M = \log_{10}(I/I_{0})\), where \(I/I_{0}\) is t
Earthquake Magnitude and Energy Release
Earthquake energy is modeled by the equation $$E = k\cdot 10^{1.5M}$$, where $$E$$ is the energy rel
Estimating Rates of Change from Table Data
A cooling object has its temperature recorded at various time intervals as shown in the table below:
Exploring the Properties of Exponential Functions
Analyze the exponential function $$f(x)= 4 * 2^x$$.
Exponential Decay and Half-Life
A radioactive substance decays according to an exponential decay function. The substance initially w
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)$$.
Exponential Function Transformation
An exponential function is given by $$f(x) = 2 \cdot 3^x$$. Analyze the effects of various transform
Exponential Function Transformations
Given the exponential function f(x) = 4ˣ, describe the transformation that produces the function g(x
Exponential Function with Compound Transformations and Its Inverse
Consider the function $$f(x)=2^(x-2)+3$$. Determine its invertibility, find its inverse function, an
Exponential Growth in a Bacterial Culture
A bacterial culture grows according to the model $$P(t) = P₀ · 2^(t/3)$$, where t (in hours) is the
Exponential Inequalities
Solve the inequality $$3 \cdot 2^x \le 48$$.
Exponential Inequality Solution
Solve the inequality $$5^(2*x - 1) < 3·5^(x)$$ for x.
Financial Growth: Savings Account with Regular Deposits
A savings account starts with an initial balance of $$1000$$ dollars and earns compound interest at
Finding Terms in a Geometric Sequence
A geometric sequence is known to satisfy $$g_3=16$$ and $$g_7=256$$.
Fitting a Logarithmic Model to Sales Data
A company observes that its sales revenue (in thousands of dollars) based on advertising spend (in t
Geometric Sequence Construction
Consider a geometric sequence where the first term is $$g_0 = 3$$ and the second term is $$g_1 = 6$$
Graphical Analysis of Inverse Functions
Given the exponential function f(x) = 2ˣ + 3, analyze its inverse function.
Inverse Relationship Verification
Given f(x) = 3ˣ - 4 and g(x) = log₃(x + 4), verify that g is the inverse of f.
Investment Growth via Sequences
A financial planner is analyzing two different investment strategies starting with an initial deposi
Log-Exponential Hybrid Function and Its Inverse
Consider the function $$f(x)=\log_3(8*3^(x)-5)$$. Analyze its domain, prove its one-to-one property,
Logarithmic Analysis of Earthquake Intensity
The magnitude of an earthquake on the Richter scale is determined using a logarithmic function. Cons
Logarithmic Cost Function in Production
A company’s cost function is given by $$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$$.
Logarithmic Function and Its Inverse
Let $$f(x)=\log_5(2x+3)-1$$. Analyze the function's one-to-one property and determine its inverse, i
Logarithmic Function with Scaling and Inverse
Consider the function $$f(x)=\frac{1}{2}\log_{10}(x+4)+3$$. Analyze its monotonicity, find the inver
Natural Logarithms in Continuous Growth
A population grows continuously according to the function $$P(t) = P_0e^{kt}$$. At \(t = 0\), \(P(0)
pH Measurement and Inversion
A researcher uses the function $$f(x)=-\log_{10}(x)+7$$ to measure the pH of a solution, where $$x$$
Piecewise Exponential and Logarithmic Function Discontinuities
Consider the function defined by $$ f(x)=\begin{cases} 2^x + 1, & x < 3,\\ 5, & x = 3,
Population Demographics Model
A small town’s population (measured in hundreds) is recorded over several time intervals. The data i
Population Growth Inversion
A town's population grows according to the function $$f(t)=1200*(1.05)^(t)$$, where $$t$$ is the tim
Profit Growth with Combined Models
A company's profit is modeled by a function that combines an arithmetic increase with exponential gr
Radioactive Decay and Exponential Functions
A sample of a radioactive substance is monitored over time. The decay in mass is recorded in the tab
Radioactive Decay and Logarithmic Inversion
A radioactive substance decays such that its mass halves every 8 years. At time \(t=0\), the substan
Radioactive Decay Model
A radioactive substance decays according to the function $$f(t)= a \cdot e^{-kt}$$. In an experiment
Radioactive Decay Modeling
A radioactive substance decays with a half-life of $$5$$ years. A sample has an initial mass of $$80
Shifted Exponential Function and Its Inverse
Consider the function $$f(x)=7-4*2^(x-3)$$. Determine its one-to-one nature, find its inverse functi
Solving Exponential Equations Using Logarithms
Solve for $$x$$ in the exponential equation $$2*3^(x)=54$$.
Solving Logarithmic Equations and Checking Domain
An engineer is analyzing a system and obtains the following logarithmic equation: $$\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
Transformation of Exponential Functions
Consider the exponential function $$f(x)= 3 * 5^x$$. A new function $$g(x)$$ is defined by applying
Transformations of Exponential Functions
Consider the exponential function \(f(x)=3\cdot2^{x}\). (a) Determine the equation of the transform
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
Using Exponential Product Property in Function Analysis
Consider the function $$f(x)= 3^x * 2^{2x}.$$
Amplitude and Period Transformations
A Ferris wheel ride is modeled by a sinusoidal function. The ride has a maximum height of 75 ft and
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 Limacon
Consider the polar function $$r=3+2\cos(\theta)$$.
Analyzing Damped Oscillations
A mass-spring system oscillates with damping according to the model $$y(t)=10*\cos(2*\pi*t)*e^{-0.5
Comparing Sinusoidal Function Models
Two models for daily illumination intensity are given by: $$I_1(t)=6*\sin\left(\frac{\pi}{12}(t-4)\r
Converting and Graphing Polar Equations
Consider the polar equation $$r=2*\cos(\theta)$$.
Converting Complex Numbers to Polar Form
Convert the complex number $$3-3*\text{i}$$ to polar form and use this representation to compute the
Coterminal Angles and the Unit Circle
Consider the angle $$\theta = \frac{5\pi}{3}$$ given in standard position.
Coterminal Angles and Unit Circle Analysis
Identify coterminal angles and determine the corresponding coordinates on the unit circle.
Daylight Variation Model
A company models the variation in daylight hours over a year using the function $$D(t) = 10*\sin\Big
Equivalent Representations Using Pythagorean Identity
Using trigonometric identities, answer the following:
Evaluating Inverse Trigonometric Functions
Inverse trigonometric functions such as $$\arcsin(x)$$ and $$\arccos(x)$$ have specific restricted d
Evaluating Sine and Cosine Using Special Triangles
Using knowledge of special right triangles, evaluate trigonometric functions.
Evaluating Sine and Cosine Values Using Special Triangles
Using the properties of special triangles, answer the following:
Exploring a Limacon
Consider the polar equation $$r=2+3\,\cos(\theta)$$.
Exploring Limacons in Polar Coordinates
Consider the polar function $$r=2+3*\cos(θ)$$ which represents a limacon. Evaluate its key features
Exploring the Pythagorean Identity
The Pythagorean identity $$\sin^2(θ)+\cos^2(θ)=1$$ is fundamental in trigonometry. Use this identity
Graph Analysis of a Polar Function
The polar function $$r=4+3\sin(\theta)$$ is given, with the following data: | \(\theta\) (radians)
Graph Interpretation from Tabulated Periodic Data
A study recorded the oscillation of a pendulum over time. Data is provided in the table below showin
Graphical Analysis of a Periodic Function
A periodic function is depicted in the graph provided. Analyze the function’s key features based on
Graphical Reflection of Trigonometric Functions and Their Inverses
Consider the sine function and its inverse. The graph of an inverse function is the reflection of th
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
Inverse Function Analysis
Given the function $$f(\theta)=2*\sin(\theta)+1$$, analyze its invertibility and determine its inver
Inverse Tangent Composition and Domain
Consider the composite function $$f(x) = \arctan(\tan(x))$$.
Limacon Analysis
Investigate the polar function $$r = 3 + 2*\cos(\theta)$$.
Limacons and Cardioids
Consider the polar function $$r=1+2*\cos(\theta)$$.
Modeling Daylight Hours with a Sinusoidal Function
A city's daylight hours throughout the year are periodic. At t = 0 months, the city experiences maxi
Modeling Seasonal Temperature Data with Sinusoidal Functions
A sinusoidal pattern is observed in average monthly temperatures. Refer to the provided temperature
Modeling Tidal Heights with Periodic Data
An oceanographer records tidal heights (in meters) over a 6-hour period. The following table gives t
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
Multiple Angle Equation
Solve the trigonometric equation $$2*\sin(2x) - \sqrt{3} = 0$$ for all $$x$$ in the interval $$[0, 2
Period Detection and Frequency Analysis
An engineer analyzes a signal modeled by $$P(t)=6*\cos(5*(t-1))$$.
Phase Shift and Frequency Analysis
Analyze the function $$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
Polar Coordinates Conversion
Convert between Cartesian and polar coordinates and analyze related polar equations.
Polar Rate of Change
Consider the polar function $$r = 3 + \sin(\theta)$$.
Probability and Trigonometry: Dartboard Game
A circular dartboard is divided into three regions by drawing two radii, forming sectors. One region
Rate of Change in Polar Functions
For the polar function $$r(\theta)=4+\cos(\theta)$$, investigate its rate of change.
Reciprocal Trigonometric Functions
Consider the function $$f(x)=\sec(x)=\frac{1}{\cos(x)}\).
Roses and Limacons in Polar Graphs
Consider the polar curves described below and answer the following:
Sinusoidal Data Analysis
An experimental setup records data that follows a sinusoidal pattern. The table below gives the disp
Sinusoidal Function Transformations in Signal Processing
A communications engineer is analyzing a signal modeled by the sinusoidal function $$f(x)=3*\cos\Big
Solving a Basic Trigonometric Equation
Solve the trigonometric equation $$2\cos(x)-1=0$$ for $$0 \le x < 2\pi$$.
Solving a Trigonometric Equation
Solve the trigonometric equation $$2*\cos(\theta) - 1 = 0$$ for $$\theta$$ in the interval $$[0, 2\p
Solving a Trigonometric Equation with Sum and Difference Identities
Solve the equation $$\sin\left(x+\frac{\pi}{6}\right)=\cos(x)$$ for $$0\le x<2\pi$$.
Solving Trigonometric Inequalities
Solve the inequality $$\sin(\theta)>\frac{1}{2}$$ for \(\theta\) in the interval [0, 2\pi].
Special Triangles and Trigonometric Values
Utilize the properties of special triangles to evaluate trigonometric functions.
Tangent Function and Asymptotes
Examine the function $$f(\theta)=\tan(\theta)$$ defined on the interval $$\left(-\frac{\pi}{2}, \fra
Tangent Function Shift
Consider the function $$f(x) = \tan\left(x - \frac{\pi}{6}\right)$$.
Tidal Motion Analysis
A coastal region's tidal heights are modeled by a sinusoidal function $$f(t) = A * \sin(b*(t - c)) +
Tidal Patterns and Sinusoidal Modeling
A coastal engineer models tide heights (in meters) as a function of time (in hours) using the sinuso
Tide Height Model: Using Sine Functions
A coastal region experiences tides that follow a sinusoidal pattern. A table of tide heights (in fee
Transformation and Reflection of a Cosine Function
Consider the function $$g(x) = -2*\cos\Bigl(\frac{1}{2}(x + \pi)\Bigr) + 3$$.
Transformations of Inverse Trigonometric Functions
Analyze the inverse trigonometric function $$g(x)=\arccos(x)$$ and its transformation into $$h(x)=2-
Transformations of Sinusoidal Functions
Consider the function $$y = 3*\sin(2*(x - \pi/4)) - 1$$. Answer the following:
Trigonometric Identities and Sum Formulas
Trigonometric identities are important for simplifying expressions that arise in wave interference a
Vibration Analysis
A mechanical system oscillates with displacement given by $$d(t) = 5*\cos(4t - \frac{\pi}{3})$$ (in
Analysis of a Function with Trigonometric Components and Discontinuities
Examine the function $$f(\theta)=\begin{cases} \frac{1-\cos(\theta)}{\theta} & \text{if } \theta \ne
Analysis of a Particle's Parametric Path
A particle moves in the plane with parametric equations $$x(t)=t^2 - 3*t + 2$$ and $$y(t)=4*t - t^2$
Analysis of a Vector-Valued Position Function
Consider the vector-valued function $$\mathbf{p}(t) = \langle 2*t + 1, 3*t - 2 \rangle$$ representin
Analysis of Vector Directions and Transformations
Given the vectors $$\mathbf{a}=\langle -1,2\rangle$$ and $$\mathbf{b}=\langle 4,3\rangle$$, perform
Average Rate of Change in Parametric Motion
For the parametric functions $$x(t) = t^3 - 3*t + 2$$ and $$y(t) = 2*t^2 - t$$ defined for $$t \in [
Composition of Linear Transformations
Consider two linear transformations represented by the matrices $$A= \begin{pmatrix} 1 & 2 \\ 0 & 1
Composition of Linear Transformations
Let $$L_1: \mathbb{R}^2 \to \mathbb{R}^2$$ be defined by $$L_1(x,y)=(x+y,\,2x-y)$$ and $$L_2: \mathb
Determinant and Inverse Calculation
Given the matrix $$C = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$$, answer the following:
Ferris Wheel Motion
A Ferris wheel rotates counterclockwise with a center at $$ (2, 3) $$ and a radius of $$5$$. The whe
FRQ 4: Parametric Representation of a Parabola
The parabola given by $$y=(x-1)^2-2$$ can be represented parametrically as $$ (x(t), y(t)) = (t, (t-
FRQ 12: Matrix Multiplication in Transformation
Let matrices $$A=\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$$ and $$B=\begin{bmatrix}0 & 1\\1 & 0\end{
FRQ 13: Area Determined by a Matrix's Determinant
Vectors $$\textbf{v}=\langle4,3\rangle$$ and $$\textbf{w}=\langle-2,5\rangle$$ form a parallelogram.
FRQ 15: Composition of Linear Transformations
Consider two linear transformations represented by the matrices $$A=\begin{bmatrix}2 & 0\\1 & 3\end{
FRQ 19: Parametric Functions and Matrix Transformation
A particle's motion is given by the parametric equations $$f(t)=(t, t^2)$$ for $$t\in[0,2]$$. A line
Graphical Analysis of Parametric Motion
A particle moves in the plane with its position defined by the functions $$x(t)= t^2 - 2*t$$ and $$y
Graphical and Algebraic Analysis of a Function with a Removable Discontinuity
Consider the function $$g(x)=\begin{cases} \frac{\sin(x) - \sin(0)}{x-0} & \text{if } x \neq 0, \\ 1
Growth Models: Exponential and Logistic Equations
Consider a population growth model of the form $$P(t)= P_{0}*e^{r*t}$$ and a logistic model given by
Inverse Analysis of a Quadratic Function
Consider the function $$f(x)=x^2-4$$ defined for $$x\geq0$$. Analyze the function and its inverse.
Inverse Analysis of a Rational Function
Consider the function $$f(x)=\frac{2*x+3}{x-1}$$. Analyze the properties of this function and its in
Inverse and Determinant of a 2×2 Matrix
Consider the matrix $$C=\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$$. Answer the following parts.
Inverses and Solving a Matrix Equation
Given the matrix $$D = \begin{pmatrix} -2 & 5 \\ 1 & 3 \end{pmatrix}$$, answer the following:
Linear Parametric Motion Modeling
A car travels along a straight path, and its position in the plane is given by the parametric equati
Logarithmic and Exponential Parametric Functions
A particle’s position is defined by the parametric equations $$x(t)= \ln(1+t)$$ and $$y(t)= e^{1-t}$
Matrices as Representations of Rotation
Consider the matrix $$A=\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$$, which represents a rotation in
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 Representation of Linear Transformations
Consider the linear transformation defined by $$L(x,y)=(3*x-2*y, 4*x+y)$$.
Matrix Transformation in Graphics
In computer graphics, images are often transformed using matrices. Consider the transformation matri
Matrix Transformation of a Vector
Let the transformation matrix be $$A=\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix},$$ and let the
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 Equations and Inverses
A curve is defined parametrically by $$x(t)=t+2$$ and $$y(t)=3*t-1$$.
Parametric Equations and Rates in a Biological Context
A bacteria colony in a Petri dish is observed to move in a periodic manner, with its position descri
Parametric Representation of a Hyperbola
For the hyperbola given by $$\frac{x^2}{9}-\frac{y^2}{4}=1$$:
Parametric Representation of a Parabola
Consider the parabola defined by $$y= 2*x^2 + 3$$. Answer the following:
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
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.
Parametric Representation of an Implicitly Defined Function
Consider the implicitly defined curve $$x^2+y^2=16$$. A common parametric representation is given by
Parametric Representation on the Unit Circle and Special Angles
Consider the unit circle defined by the parametric equations $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$.
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 $$
Parametrizing a Parabola
A parabola is defined parametrically by $$x(t)=t$$ and $$y(t)=t^2$$.
Particle Motion from Parametric Equations
A particle moves in the plane with position functions $$x(t)=t^2-2*t$$ and $$y(t)=4*t-t^2$$, where $
Particle Motion with Quadratic Parametric Functions
A particle moves in the plane according to the parametric equations $$x(t)=t^2$$ and $$y(t)=2*t$$. A
Position and Velocity Vectors
For a particle with position $$\mathbf{p}(t)=\langle2*t+1, 3*t-2\rangle$$, where $$t$$ is in seconds
Properties of a Parametric Curve
Consider a curve defined parametrically by $$x(t)=t^3$$ and $$y(t)=t^2.$$ (a) Determine for which
Rational Piecewise Function with Parameter Changes: Discontinuity Analysis
Let $$R(t)=\begin{cases} \frac{3t^2-12}{t-2} & \text{if } t\neq2, \\ 5 & \text{if } t=2 \end{cases}$
Reflection Transformation Using Matrices
A reflection over the line \(y=x\) in the plane can be represented by the matrix $$R=\begin{pmatrix}
Table-Driven Analysis of a Piecewise Defined Function
A researcher defines a function $$h(x)=\begin{cases} \frac{x^2 - 4}{x-2} & \text{if } x < 2, \\ x+3
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 Matrix in Markov Chains
A system transitions between two states according to the matrix $$M= \begin{pmatrix} 0.7 & 0.3 \\ 0.
Vector Analysis in Projectile Motion
A soccer ball is kicked so that its velocity vector is given by $$\mathbf{v}=\langle5, 7\rangle$$ (i
Vector Scalar Multiplication
Given the vector $$\mathbf{w} = \langle -2, 5 \rangle$$ and the scalar $$k = -3$$, answer the follow
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.