AP Precalculus FRQ Room

Analysis of a Quartic Function

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

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 a Rational Function with Asymptotes

Consider the rational function $$R(x)= \frac{(x-2)(x+3)}{(x-1)(x+4)}$$. Answer each part that follow

Analyzing Concavity in Polynomial Functions

A car’s displacement over time is modeled by the polynomial function $$f(x)= x^3 - 6*x^2 + 11*x - 6$

Analyzing End Behavior of a Polynomial

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

Application of the Binomial Theorem

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

Break-even Analysis via Synthetic Division

A company’s cost model is represented by the polynomial function $$C(x) = x^3 - 6*x^2 + 11*x - 6$$,

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

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

Concavity and Inflection Points of a Polynomial Function

For the function $$g(x)= x^3 - 3*x^2 - 9*x + 5$$, analyze the concavity and determine any inflection

Constructing a Function Model from Experimental Data

An engineer collects data on the stress (in MPa) experienced by a material under various applied for

Constructing a Piecewise Function from Data

A company’s production cost function changes slopes at a production level of 100 units. The cost (in

Constructing a Rational Function Model with Asymptotic Behavior

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

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 Function Inverse Analysis

Consider the cubic function $$f(x) = x^3 - 6*x^2 + 9*x$$. Answer the following questions related to

Cubic Polynomial Analysis

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

Designing a Rational Function to Meet Given Criteria

A mathematician wishes to construct a rational function R(x) that satisfies the following properties

Discontinuities in a Rational Model Function

Consider the function $$p(x)=\frac{(x-3)(x+1)}{x-3}$$, defined for all $$x$$ except when $$x=3$$. Ad

Discontinuity Analysis in a Rational Function with High Degree

Consider the function $$f(x)=\frac{x^3-8}{x^2-4}$$. Answer the following:

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 the Effect of Multiplicities on Graph Behavior

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

Function Simplification and Graph Analysis

Consider the function $$h(x)= \frac{x^2 - 4}{x-2}$$. Answer the following parts.

Graph Interpretation and Log Transformation

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

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 Quartic Polynomial Function

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

Inverse Analysis of a Shifted Cubic Function

Consider the function $$f(x)= (x-1)^3 + 4$$. Answer the following questions regarding its inverse.

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

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

Inverse of a Complex Rational Function

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

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

Investigation of Refund Policy via Piecewise Continuous Functions

A retail store's refund policy is modeled by $$ R(x)=\begin{cases} 10-x & \text{for } x<5, \\ a*x+b

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

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

Logarithmic Linearization in Exponential Growth

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

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 with Rational Functions

An experiment shows that the intensity of a light source varies inversely with the square of the dis

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

Office Space Cubic Function Optimization

An office building’s usable volume (in thousands of cubic feet) is modeled by the cubic function $$V

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 Curve Fitting

A set of three points, $$(1, 3)$$, $$(2, 8)$$, and $$(4, 20)$$, is known to lie on a quadratic funct

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 Long Division and Slant Asymptote

Consider the rational function $$F(x)= \frac{x^3 + 2*x^2 - 5*x + 1}{x - 2}$$. Answer the following p

Polynomial Long Division and Slant Asymptote

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

Polynomial Long Division and Slant Asymptotes

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

Polynomial Model from Temperature Data

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

Predator-Prey Dynamics as a Rational Function

An ecologist models the ratio of predator to prey populations with the rational function $$P(x) = \f

Projectile Motion Analysis

A projectile is launched so that its height (in meters) as a function of time (in seconds) is given

Rational Function Analysis for Signal Processing

A signal processing system is modeled by the rational function $$R(x)= \frac{2*x^2 - 3*x - 5}{x^2 -

Rational Function Inverse Analysis

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

Rational Inequalities and Test Intervals

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

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 +

Signal Strength Transformation Analysis

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

Solving a Logarithmic Equation with Polynomial Bases

Consider the equation $$\log_2(p(x)) = x + 1$$ where $$p(x)= x^2+2*x+1$$.

Solving Polynomial Inequalities

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

Transformation in Composite Functions

Let the parent function be $$f(x)= x^2$$ and consider the composite transformation given by $$g(x)=

Using the Binomial Theorem for Polynomial Expansion

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

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

Consider an arithmetic sequence with initial term $$a_0 = 5$$ and constant difference $$d$$. Given t

Arithmetic Sequence Analysis

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

Arithmetic Sequence in Savings

A student saves money every month and deposits a fixed additional amount each month, so that her sav

Bacterial Growth Model and Inverse Function

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

Bacterial Population Growth Model

A certain bacterium population doubles every 3 hours. At time $$t = 0$$ hours the population is $$50

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 Revenue Models

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

Composite Function Involving Exponential and Logarithmic Components

Consider the composite function defined by $$h(x) = \log_5(2\cdot 5^x + 3)$$. Answer the following p

Composite Functions with Exponential and Logarithmic Elements

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

Composition and Transformation Functions

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

Composition of Exponential and Logarithmic Functions

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

Compound Interest and Financial Growth

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

Determining an Exponential Model from Data

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

Earthquake Intensity and Logarithmic Function

The Richter scale measures earthquake intensity using a logarithmic function. Suppose the energy rel

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 Logarithms

The Richter scale is logarithmic and is used to measure earthquake intensity. The energy released, \

Exponential Decay and Half-Life

A radioactive substance decays according to an exponential decay function. The substance initially w

Exponential Decay and Log Function Inverses in Pharmacokinetics

In a pharmacokinetics study, the concentration of a drug in a patient’s bloodstream is observed to d

Exponential Decay: Modeling Half-Life

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

Exponential Equations via Logarithms

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

Exponential Function from Data Points

An exponential function of the form f(x) = a·bˣ passes through the points (2, 12) and (5, 96).

Exponential Function Transformations

Consider an exponential function defined by f(x) = a·bˣ. A graph of this function is provided in the

Exponential Inequalities

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

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 $$g_7=256$$.

Graphical Analysis of Inverse Functions

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

Inverse and Domain of a Logarithmic Transformation

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

Inverse Function of an Exponential Function

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

Inverse Functions of Exponential and Logarithmic Forms

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

Inverse of an Exponential Function

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

Inverse Relationship Verification

Given f(x) = 3ˣ - 4 and g(x) = log₃(x + 4), verify that g is the inverse of f.

Investment Scenario Convergence

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

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

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 Function Analysis

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

Logarithmic Function and Inversion

Given the function $$f(x)= \log_3(x-2)+4$$, perform an analysis to determine its domain, prove it is

Logarithmic Function and Properties

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

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

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 Demographics Model

A small town’s population (measured in hundreds) is recorded over several time intervals. The data i

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 Half-Life Estimation Through Data

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

Radioactive Decay Modeling

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

Radioactive Decay Modeling

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

Radioactive Decay Problem

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

Semi-Log Plot and Exponential Model

A researcher studies the concentration of a chemical over time using a semi-log plot, where the y-ax

Semi-Log Plot Data Analysis

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

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

Temperature Decay Modeled by a Logarithmic Function

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

Transformation Effects on Exponential Functions

Consider the function $$f(x) = 3 \cdot 2^x$$, which is transformed to $$g(x) = 3 \cdot 2^{(x+1)} - 4

Transformations of Exponential Functions

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

Transformations of Exponential Functions

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

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

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 Cotangent Function

Consider the function $$f(\theta)=\cot(\theta)$$ defined on the interval \(\theta\in(0,\pi)\).

Analysis of a Rose Curve

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

Analyzing Damped Oscillations

A mass-spring system oscillates with damping according to the model $$y(t)=10*\cos(2*\pi*t)*e^{-0.5

Analyzing Phase Shifts in Sinusoidal Functions

Investigate the function $$y=\sin\Big(2*(x-\frac{\pi}{3})\Big)+0.5$$ by identifying its transformati

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.

Calculating the Area Enclosed by a Polar Curve

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

Cardioid Polar Graphs

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

Combining Logarithmic and Trigonometric Equations

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

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

Comparing Sinusoidal Functions

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

Converting and Graphing Polar Equations

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

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.

Damped Oscillations: Combining Sinusoidal Functions and Geometric Sequences

A mass-spring system oscillates with decreasing amplitude following a geometric sequence. Its displa

Determining Phase Shifts and Amplitude Changes

A wave function is modeled by $$W(\theta)=7*\cos(4*(\theta-c))+d$$, where c and d are unknown consta

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 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 Inverse Trigonometric Functions

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

Exploring the Pythagorean Identity

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

Extracting Sinusoidal Parameters from Data

The function $$f(x)=a\sin[b(x-c)]+d$$ models periodic data, with the following values provided: | x

Graph Analysis of a Polar Function

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

Graphical Analysis of a Periodic Function

A periodic function is depicted in the graph provided. Analyze the function’s key features based on

Graphing Sine and Cosine Functions from the Unit Circle

Using information from special right triangles, answer the following:

Graphing the Tangent Function and Analyzing Asymptotes

Consider the function $$y = \tan(x)$$. Answer the following:

Graphing the Tangent Function with Asymptotes

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

Interpreting Trigonometric Data Models

A set of experimental data capturing a periodic phenomenon is given in the table below. Use these da

Inverse Function Analysis

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

Inverse Trigonometric Analysis

Consider the inverse sine function $$y = \arcsin(x)$$ which is used to determine angle measures from

Limacons and Cardioids

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

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 Tides with Sinusoidal Functions

Tidal heights at a coastal location are modeled by the function $$H(t)=2\,\sin\Bigl(\frac{\pi}{6}(t-

Periodic Phenomena: Seasonal Daylight Variation

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

Polar Coordinates Conversion

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

Probability and Trigonometry: Dartboard Game

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

Real-World Modeling: Exponential Decay with Sinusoidal Variation

A river's water level is affected by tides and evaporation. It is modeled by the function $$L(t)=8*

Reciprocal and Pythagorean Identities

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

Seasonal Demand Modeling

A company's product demand follows a seasonal pattern modeled by $$D(t)=500+50\cos\left(\frac{2\pi}{

Secant Function and Its Transformations

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

Secant, Cosecant, and Cotangent Functions Analysis

Consider the reciprocal trigonometric functions. Answer the following:

Sinusoidal Function Transformation Analysis

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

Sinusoidal Function Transformations in Signal Processing

A communications engineer is analyzing a signal modeled by the sinusoidal function $$f(x)=3*\cos\Big

Sinusoidal Transformation and Logarithmic Manipulation

An electronic signal is modeled by $$S(t)=5*\sin(3*(t-2))$$ and its decay is described by $$D(t)=\ln

Solving a Basic Trigonometric Equation

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

Solving Trigonometric Equations

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

Solving Trigonometric Equations

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

Solving Trigonometric Equations in a Specified Interval

Solve the given trigonometric equations within specified intervals and explain the underlying reason

Solving Trigonometric Inequalities

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

Tangent Function Shift

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

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

Transformations of Sinusoidal Functions

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

Understanding Coterminal Angles Through Art Installation

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

Unit Circle and Special Triangles

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

Using Trigonometric Sum and Difference Identities

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

Verifying a Trigonometric Identity

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

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 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

Analyzing a Piecewise Function Representing a Linear Transformation

Let $$T(x)=\begin{cases} \frac{2x-4}{x-2} & \text{if } x \neq 2, \\ 3 & \text{if } x=2 \end{cases}$$

Complex Parametric and Matrix Analysis in Planar Motion

A particle moves in the plane according to the parametric equations $$x(t)=3\cos(t)+2*t$$ and $$y(t)

Composite Transformations in the Plane

Consider two linear transformations in $$\mathbb{R}^2$$: a rotation by 90° counterclockwise and a re

Composition of Linear Transformations

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

Composition of Linear Transformations

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

Composition of Linear Transformations

Given matrices $$A=\begin{pmatrix}2 & 0 \\ 0 & 3\end{pmatrix}$$ and $$B=\begin{pmatrix}0 & 1 \\ 1 &

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 Area of a Parallelogram

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

Evaluating a Piecewise Function in a Vector Context

A vector-valued function is defined as $$\mathbf{p}(t)=\langle p_x(t),p_y(t) \rangle$$ where the hor

Exponential Decay Modeled by Matrices

Consider a system where decay over time is modeled by the matrix $$M(t)= e^{-k*t}I$$, where I is the

Ferris Wheel Motion

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

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 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 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 6: Implicit Function to Parametric Representation

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

FRQ 10: Unit Vectors and Direction

Consider the vector $$\textbf{w}=\langle -5, 12 \rangle$$.

FRQ 11: Matrix Inversion and Determinants

Let matrix $$A=\begin{bmatrix}3 & 4\\2 & -1\end{bmatrix}$$.

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 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{

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 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 Matrix

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

Inverse Function and Transformation Mapping

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

Inverses and Solving a Matrix Equation

Given the matrix $$D = \begin{pmatrix} -2 & 5 \\ 1 & 3 \end{pmatrix}$$, answer the following:

Linear Transformation Composition

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

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

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

Matrix Applications in State Transitions

In a system representing transitions between two states, the following transition matrix is used: $

Matrix Methods for Solving Linear Systems

Solve the system of linear equations below using matrix methods: $$2x+3y=7$$ $$4x-y=5$$

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

Matrix Representation of Linear Transformations

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

Movement Analysis via Position Vectors

A particle is moving in the plane with its position given by the functions $$x(t)=2*t+1$$ and $$y(t)

Parameter Transition in a Piecewise-Defined Function

Consider the function $$g(t)=\begin{cases} \frac{t^3-1}{t-1} & \text{if } t \neq 1, \\ 5 & \text{if

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 Equations of an Ellipse

Consider the ellipse defined by $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. Answer the following:

Parametric Function and Its Inverse: Parabolic Function

Consider the function $$f(x)= (x-1)^2 + 2$$ for x \(\ge\) 1. (a) Provide a parametrization for the

Parametric Motion Analysis Using Tabulated Data

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

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 Implicit Curve

The equation $$x^2+y^2-6*x+8*y+9=0$$ defines a curve in the plane. Analyze this curve.

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

Consider the ellipse defined by $$\frac{(x-2)^2}{9} + \frac{(y+1)^2}{4} = 1$$. Answer the following:

Parametrization of an Ellipse for a Racetrack

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

Planar Motion Analysis

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

Position and Velocity in Vector-Valued Functions

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

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}$

Transformation Matrices in Computer Graphics

A transformation matrix $$A = \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$$ is applied to points in

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

Transition Matrices in Dynamic Models

A system with two states is modeled by the transition matrix $$T=\begin{bmatrix}0.8 & 0.3\\ 0.2 & 0.

Trigonometric Function Analysis

Consider the trigonometric function $$f(x)= 2*\tan(x - \frac{\pi}{6})$$. Without using a calculator,

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 Operations

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

Vector Operations

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

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 Scalar Multiplication

Given the vector $$\mathbf{w} = \langle -2, 5 \rangle$$ and the scalar $$k = -3$$, answer the follow

Vectors in Polar and Cartesian Coordinates

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