AP Precalculus FRQ Room

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

Analyzing Concavity and Points of Inflection for a Polynomial Function

Consider the function $$f(x)= x^3-3*x^2+2*x$$. Although points of inflection are typically determine

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 Polynomial Functions

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

Application of the Binomial Theorem

Expand the expression $$(x+3)^5$$ using the Binomial Theorem and 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

Average Rate of Change of a Rational Function

For the rational function $$r(x)= \frac{4*x}{x+2}$$, answer the following:

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

Composite Functions and Inverses

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

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 Rational Function from Graph Behavior

An unknown rational function has a graph with a vertical asymptote at $$x=3$$, a horizontal asymptot

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 from Graphical Data

A function is represented by a graph with certain open and closed endpoints. A table of select input

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

Engineering Application: Stress Analysis Model

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

Engineering Curve Analysis: Concavity and Inflection

An engineering experiment recorded the deformation of a material, modeled by a function whose behavi

Evaluating Limits Involving Rational Expressions with Asymptotic Behavior

Consider the function $$f(x)=\frac{2*x^2-3*x-5}{x^2-1}$$. 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 End Behavior and Leading Coefficients

Consider the function $$f(x)= -3*x^5 + 4*x^3 - x + 7$$. Answer the following:

Function Model Construction from Data Set

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

Graph Interpretation and Log Transformation

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

Impact of Multiplicity on Graph Behavior

Consider the function $$f(x)= (x - 2)^2*(x + 1)$$. Examine how the multiplicity of each zero affects

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 Shifted Cubic Function

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

Inversion of a Polynomial Ratio Function

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

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

Linear Function Inverse Analysis

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

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

Manufacturing Efficiency Polynomial Model

A company's manufacturing efficiency is modeled by a polynomial function. The function, given by $$P

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 &

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

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 Analysis

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

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 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

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

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

Quadratic Function Inverse Analysis with Domain Restriction

Consider the function $$f(x) = x^2 - 4*x + 5$$. Assume that the domain of $$f$$ is restricted so tha

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 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 Graph and Asymptote Identification

Given the rational function $$R(x)= \frac{x^2 - 4}{x^2 - x - 6}$$, answer the following parts:

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 and Test Intervals

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

Real-World Inverse Function: Temperature Conversion

The function $$f(x)= \frac{9}{5}*x + 32$$ converts a temperature in degrees Celsius to degrees Fahre

Real-World Modeling: Population Estimation

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

Solving a Polynomial Inequality

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

Solving Polynomial Inequalities

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

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

Zeros and End Behavior in a Higher-Degree Polynomial

Consider the polynomial $$P(x)= (x+1)^2 (x-2)^3 (x-5)$$. Answer the following parts.

Acoustics and the Logarithmic Scale

The sound intensity level (in decibels) of a sound is given by the function $$f(x)=10*\log_{10}(x)$$

Analyzing a Logarithmic Function

Consider the logarithmic function $$f(x)= \log_{3}(x-2) + 1$$.

Analyzing Exponential Function Behavior from a Graph

An exponential function is depicted in the graph provided. Analyze the key features of the function.

Analyzing Social Media Popularity with Logarithmic Growth

A social media analyst is studying the early-stage growth of a new account's followers. Initially, t

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

Cellular Data Usage Trend

A telecommunications company records monthly cellular data usage (in MB) that appears to grow expone

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 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 Functions: Shifting and Scaling in Log and Exp

Consider the functions $$f(x)=2*e^(x-3)$$ and $$g(x)=\ln(x)+4$$.

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 Exponential Equations

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

Compound Interest vs. Simple Interest

A financial analyst is comparing two interest methods on an initial deposit of $$10000$$ dollars. On

Exploring the Properties of Exponential Functions

Analyze the exponential function $$f(x)= 4 * 2^x$$.

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 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 Function Transformations

Given the exponential function f(x) = 4ˣ, describe the transformation that produces the function g(x

Exponential Inequalities

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

Finding the Inverse of an Exponential Function

Given the exponential function $$f(x)= 4\cdot e^{0.5*x} - 3,$$ find the inverse function $$f^{-1}(

Fractal Pattern Growth

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

General Exponential Equation Solving

Solve the equation $$2^{x}+2^{x+1}=48$$. (a) Factor the equation by rewriting \(2^{x+1}\) in terms

Inverse and Domain of a Logarithmic Transformation

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

Inverse Functions in Exponential Contexts

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

Inverse Functions of Exponential and Logarithmic Forms

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

Inverse of a Composite Function

Let $$f(x)= e^x$$ and $$g(x)= \ln(x) + 3$$.

Investment Growth via Sequences

A financial planner is analyzing two different investment strategies starting with an initial deposi

Investment Scenario Convergence

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

Logarithmic Equation and Extraneous Solutions

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

Logarithmic Function and Properties

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

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 Validation and Error Analysis in Exponential Trends

During a chemical reaction, a set of experimental data appears to follow an exponential trend when p

Parameter Sensitivity in Exponential Functions

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

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 Demographics Model

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

Radioactive Decay Analysis

A radioactive substance decays exponentially over time according to the function $$f(t) = a * b^t$$,

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 Modeling

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

Solving Exponential Equations Using Logarithms

Solve the exponential equation $$5\cdot2^{(x-2)}=40$$. (a) Isolate the exponential term and solve f

Solving Exponential Equations Using Logarithms

Solve for $$x$$ in the exponential equation $$2*3^(x)=54$$.

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

System of Exponential Equations

Solve the following system of equations: $$2\cdot 2^x + 3\cdot 3^y = 17$$ $$2^x - 3^y = 1$$.

Transformations of Exponential Functions

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

Transformations of Exponential Functions

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

Transformed Exponential Equation

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

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

Traveling Sales Discount Sequence

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

Tumor Growth with Time Dilation Effects

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

Validating the Negative Exponent Property

Demonstrate the negative exponent property using the expression $$b^{-3}$$.

Weekly Population Growth Analysis

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

Analysis of a Limacon

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

Analysis of Rose Curves

A polar curve is given by the equation $$r=4*\cos(3*θ)$$ which represents a rose curve. Analyze the

Analyzing a Limacon

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

Analyzing a Rose Curve

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

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 Function Rate of Change

A sound wave is modeled by the function $$f(t)=4*\sin(\frac{\pi}{2}*(t-1))+5$$, where t is measured

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

Calculating the Area Enclosed by a Polar Curve

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

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 $$g(x)=\cos\Bigl(x-\frac{\pi}{2}\Bigr)$$.

Concavity in the Sine Function

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

Damped Oscillations: Combining Sinusoidal Functions and Geometric Sequences

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

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

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 Rates of Change in Polar Functions

Given the polar function $$r(\theta) = 2 + \sin(\theta)$$, answer the following:

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 and Analyzing a Transformed Sine Function

Consider the function $$f(x)=3\sin\left(2\left(x-\frac{\pi}{4}\right)\right)+1$$. Answer the followi

Graphing Polar Circles and Roses

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

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 a Sinusoidal Graph

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

Inverse Tangent Composition and Domain

Consider the composite function $$f(x) = \arctan(\tan(x))$$.

Inverse Trigonometric Analysis

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

Inverse Trigonometric Function Analysis

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

Inverse Trigonometric Function Analysis

Consider the function $$f(x) = 2*\sin(x)$$.

Limacons and Cardioids

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

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 Variation

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

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-

Pendulum Motion and Periodic Phenomena

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

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

Polar Coordinates Conversion

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

Polar Interpretation of Periodic Phenomena

A meteorologist models wind speed variations with direction over time using a polar function of the

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

For the polar function $$r(\theta)=4+\cos(\theta)$$, investigate its rate of change.

Rate of Change in Polar Functions

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

Reciprocal and Pythagorean Identities

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

Reciprocal Trigonometric Functions: Secant, Cosecant, and Cotangent

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

Roses and Limacons in Polar Graphs

Consider the polar curves described below and answer the following:

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

Secant, Cosecant, and Cotangent Functions Analysis

Consider the reciprocal trigonometric functions. 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

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 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 Equations

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

Solving Trigonometric Equations

Solve the trigonometric equation $$\sin(\theta) + \sqrt{3}*\cos(\theta)=1$$.

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 Trigonometric Values

Utilize the properties of special triangles to evaluate trigonometric functions.

Tangent and Cotangent Equation

Consider the trigonometric equation $$\tan(x) - \cot(x) = 0$$ for $$x$$ in the interval $$[0, 2\pi]$

Tangent Function Shift

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

Transformation and Reflection of a Cosine Function

Consider the function $$g(x) = -2*\cos\Bigl(\frac{1}{2}(x + \pi)\Bigr) + 3$$.

Unit Circle and Special Triangle Values

Using the unit circle and properties of special triangles, answer the following.

Unit Circle and Special Triangles

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

Verifying a Trigonometric Identity

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

Vibration Analysis

A mechanical system oscillates with displacement given by $$d(t) = 5*\cos(4t - \frac{\pi}{3})$$ (in

Acceleration in a Vector-Valued Function

Given the particle's position vector $$\mathbf{r}(t) = \langle t^2, t^3 - 3*t \rangle$$, answer the

Analyzing a Piecewise Function Involving Absolute Value and Removability

Consider the function $$F(x)=\begin{cases} \frac{|x-2|(x+1) - (x-2)(x+1)}{x-2} & \text{if } x \neq 2

Average Rate of Change in Parametric Motion

A projectile is launched and its motion is modeled by $$x(t)=3*t+1$$ and $$y(t)=16-4*t^2$$, where $$

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 [

Circular Motion and Transformation

The motion of a particle is given by the parametric equations $$x(t)=3\cos(t)$$ and $$y(t)=3\sin(t)$

Circular Motion Parametrization

Consider a particle moving along a circular path defined by the parametric equations $$x(t)= 5*\cos(

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)

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

Let two linear transformations in \(\mathbb{R}^2\) be represented by the matrices $$E=\begin{pmatrix

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

Discontinuity Analysis in an Implicitly Defined Function

Consider the circle defined by $$x^2+y^2=4$$. A piecewise function for $$y$$ is attempted as $$y(x)=

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 \

Dot Product, Projection, and Angle Calculation

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

Eliminating the Parameter in an Implicit Function

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

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} & \

Finding Angle Between Vectors

Given vectors $$\mathbf{a}=\langle 1,2 \rangle$$ and $$\mathbf{b}=\langle 3,4 \rangle$$, determine t

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 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 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

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 and Determinant of a Matrix

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

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 Matrix and Transformation of the Unit Square

Given the transformation matrix $$A=\begin{pmatrix}3 & 1 \\ 2 & 2\end{pmatrix}$$ applied to the unit

Linear Transformation and Area Scaling

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

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

Matrix Modeling in Population Dynamics

A biologist is studying a species with two age classes: juveniles and adults. The population dynamic

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 Transformation in Graphics

In computer graphics, images are often transformed using matrices. Consider the transformation matri

Modeling Linear Motion Using Parametric Equations

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

Modeling Particle Trajectory with Parametric Equations

A particle’s motion is described by the parametric equations $$x(t)=3*t+1$$ and $$y(t)=-2*t^2+8*t-1$

Modeling State Transitions with a Transition Matrix (Probability-Based Scenario)

A small business models its customer behavior between two states: Regular and Occasional. The transi

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 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 Representation of a Parabola

Consider the parabola defined by $$y= 2*x^2 + 3$$. Answer the following:

Parametric Representation of an Ellipse

An ellipse is defined by the equation $$\frac{(x-3)^2}{4} + \frac{(y+2)^2}{9} = 1.$$ (a) Write a

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 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 $$

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 $$\frac{(x-1)^2}{16}+\frac{(y+2)^2}{9}=1$$.

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 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

Population Transition Matrix Analysis

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

Resolving Discontinuities in an Elliptical Parameterization

An ellipse is parameterized by the following equations: $$x(\theta)=\begin{cases} 5\cos(\theta) & \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

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

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 Components and Magnitude

Given the vector $$\vec{v}=\langle 3, -4 \rangle$$:

Vector Operations

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

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