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 End Behavior of a Polynomial

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

Average Rate of Change and Tangent Lines

For the function $$f(x)= x^3 - 6*x^2 + 9*x + 4$$, consider the relationship between secant (average

Average Rate of Change in a Quadratic Model

Let $$h(x)= x^2 - 4*x + 3$$ represent a model for a certain phenomenon. Calculate the average rate o

Binomial Theorem Expansion

Use the Binomial Theorem to expand the expression $$ (x + 2)^4 $$. Explain your steps in detail.

Carrying Capacity in Population Models

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

Comparative Analysis of 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

Composite Function Transformations

Consider the polynomial function $$f(x)= x^2-4$$. A new function is defined by $$g(x)= \ln(|f(x)+5|)

Constructing a Function Model from Experimental Data

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

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 from Graph Behavior

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

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

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

Designing a Rational Function to Meet Given Criteria

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

Determining Degree from Discrete Data

Below is a table representing the output values of a polynomial function for equally-spaced input va

Determining Function Behavior from a Data Table

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

Determining Polynomial Degree from Finite Differences

A function $$f(x)$$ is defined on equally spaced values of $$x$$, with the following data: | x | f(

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 |

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

Estimating Polynomial Degree from Finite Differences

The following table shows the values of a function $$f(x)$$ at equally spaced values of $$x$$: | x

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 End Behavior and Leading Coefficients

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

Factoring and Zero Multiplicity

Consider the polynomial $$p(x)= (x - 1)^2*(x+2)^3*(x-4)$$. Answer the following parts.

Finding and Interpreting Inflection Points

Consider the polynomial function $$f(x)= x^3 - 6*x^2 + 9*x + 1$$. Answer the following parts.

Geometric Series Model in Area Calculations

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

Graph 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 Polynomial Function with Multiple Turning Points

Consider the function $$f(x)= (x-2)^3 - 3*(x-2) + 1$$. Answer the following about its invertibility

Inverse Analysis of a Transformed Quadratic Function

Consider the function $$f(x)= -3*(x-2)^2 + 7$$ with a domain restriction that ensures one-to-one beh

Local and Global Extrema in a Polynomial Function

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

Logarithmic and Exponential Equations with Rational Functions

A process is modeled by the function $$F(x)= \frac{3*e^{2*x} - 5}{e^{2*x}+1}$$, where x is measured

Logarithmic 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 Population Growth with a Polynomial Function

A population of a certain species in a controlled habitat is modeled by the cubic function $$P(t)= -

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

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 Analysis

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

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 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

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

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 Inequalities Analysis

Solve the inequality $$\frac{x^2-4}{x+1} \ge 0$$ and represent the solution on a number line.

Real-World Modeling: Population Estimation

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

Regression Model Selection for Experimental Data

Experimental data was collected, and the following table represents the relationship between a contr

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^

Signal Strength Transformation Analysis

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

Slant Asymptote Determination for a Rational Function

Determine the slant (oblique) asymptote of the rational function $$r(x)= \frac{2*x^2 + 3*x - 5}{x -

Solving a 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 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

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

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 a Logarithmic Function from Data

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

Analyzing Exponential Function Behavior from a Graph

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

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

An arithmetic sequence is defined as an ordered list of numbers with a constant difference between c

Arithmetic Sequence Derived from Logarithms

Consider the exponential function $$f(x) = 10 \cdot 2^x$$. A new dataset is formed by taking the com

Arithmetic Sequence in Savings

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

Bacterial Growth Modeling

A biologist is studying a rapidly growing bacterial culture. The number of bacteria at time $$t$$ (i

Bacterial Population Growth Model

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

Cell Division Pattern

A culture of cells undergoes division such that the number of cells doubles every hour. The initial

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

Competing Exponential Cooling Models

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

Composite Function and Its Inverse

Let \(f(x)=3\cdot2^{x}\) and \(g(x)=x-1\). Consider the composite function \(h(x)=f(g(x))\). (a) Wr

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: 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

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

Compound Interest and Exponential Equations

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

Connecting Exponential Functions with Geometric Sequences

An exponential function $$f(x) = 5 \cdot 3^x$$ can also be interpreted as a geometric sequence where

Domain, Range, and Inversion of Logarithmic Functions

Consider the logarithmic function \(f(x)=\log_{2}(x-3)\). (a) Determine the domain and range of \(f

Exploring Logarithmic Scales: pH and Hydrogen Ion Concentration

In chemistry, the pH of a solution is defined by the relation $$pH = -\log([H^+])$$, where $$[H^+]$$

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 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 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 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 $$g_7=256$$.

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

Geometric Investment Growth

An investor places $$1000$$ dollars into an account that grows following a geometric sequence model.

Geometric Sequence and Exponential Modeling

A geometric sequence can be viewed as an exponential function defined by a constant ratio. The table

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 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 in Exponential Contexts

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

Inverse Functions of Exponential and Log Functions

Let \(f(x)=4\cdot3^{x}\) and \(g(x)=\log_{3}(x/4)\). (a) Show that \(f(g(x))=x\) for all \(x\) in t

Inverse Functions of Exponential and Logarithmic Forms

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

Loan Payment and Arithmetico-Geometric Sequence

A borrower takes a loan of $$10,000$$ dollars. The loan accrues a monthly interest of 1% and the bor

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 Function Analysis

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

Logarithmic Transformation and Composition of Functions

Let $$f(x)= \log_3(x)$$ and $$g(x)= 2^x$$. Using these functions, answer the following:

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

Natural Logarithms in Continuous Growth

A population grows continuously according to the function $$P(t) = P_0e^{kt}$$. At \(t = 0\), \(P(0)

Parameter Sensitivity in Exponential Functions

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

pH and Logarithmic Functions

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

pH Measurement and Inversion

A researcher uses the function $$f(x)=-\log_{10}(x)+7$$ to measure the pH of a solution, where $$x$$

Population Demographics Model

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

Population Growth of Bacteria

A bacterial colony doubles in size every hour, so that its size follows a geometric sequence. Recall

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 and Logarithmic Inversion

A radioactive substance decays such that its mass halves every 8 years. At time \(t=0\), the substan

Radioactive Decay Problem

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

Real Estate Price Appreciation

A real estate property appreciates according to an exponential model and receives an additional fixe

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) + \

Telephone Call Data Analysis on Semi-Log Plot

A telecommunications company records the number of calls received each hour. The data suggest an exp

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

Transformation of an Exponential Function

Consider the basic exponential function $$f(x)= 2^x$$. A transformed function is given by $$g(x)= 3\

Transformations in Logarithmic Functions

Given \(f(x)=\log_{3}(x)\), consider the transformed function \(g(x)=-2\log_{3}(2x-6)+4\). (a) Dete

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

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 $$r=3*\sin(3\theta)$$.

Analysis of Reciprocal Trigonometric Functions

Examine the properties of the reciprocal trigonometric functions $$\csc(θ)$$, $$\sec(θ)$$, and $$\co

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

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 the Tangent Function

Consider the tangent function $$T(x)=\tan(x)$$.

Calculating the Area Enclosed by a Polar Curve

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

Conversion Between Rectangular and Polar Coordinates

Convert the given points between rectangular and polar coordinate systems and discuss the relationsh

Converting and Graphing Polar Equations

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

Coterminal Angles and Trigonometric Evaluations

Consider the angle $$750^\circ$$. Using properties of coterminal angles and the unit circle, answer

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

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 $$r=2+3\,\cos(\theta)$$.

Graph Analysis of a Polar Function

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

Graph Transformations: Sine and Cosine Functions

The functions $$f(\theta)=\sin(\theta)$$ and $$g(\theta)=\cos(\theta)$$ are related through a phase

Graphing a Limacon

Given the polar equation $$r=2+3*\cos(\theta)$$, analyze and graph the corresponding limacon.

Graphing Sine and Cosine Functions from the Unit Circle

Using information from special right triangles, answer the following:

Graphing the Tangent Function with Asymptotes

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

Identity Verification

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

Interpreting Trigonometric Data Models

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

Inverse Trigonometric Function Analysis

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

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

Phase Shift and Frequency Analysis

Analyze the function $$f(x)=\cos\Bigl(4\bigl(x-\frac{\pi}{8}\bigr)\Bigr)$$.

Piecewise Trigonometric Function and Continuity Analysis

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

Polar Coordinates and Graphing a Circle

Answer the following questions on polar coordinates:

Polar Coordinates: Converting and Graphing

Given the rectangular coordinate point $$(3, -3\sqrt{3})$$, convert and analyze its polar representa

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

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

Rate of Change in Polar Functions

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

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*

Rewriting and Graphing a Composite Trigonometric Function

Given the function $$f(x)=\cos(x)+\sin(x)$$, transform it into the form $$R*\cos(x-\phi)$$.

Seasonal Demand Modeling

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

Seasonal Temperature Modeling

A city's average temperature over the year is modeled by a cosine function. The following table show

Secant Function and Its Transformations

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

Secant, Cosecant, and Cotangent Functions Analysis

Consider the reciprocal trigonometric functions. Answer the following:

Sine and Cosine Graph Transformations

Consider the functions $$f(\theta)=\sin(\theta)$$ and $$g(\theta)=\sin(\theta+\frac{\pi}{3})$$, whic

Sinusoidal Data Analysis

An experimental setup records data that follows a sinusoidal pattern. The table below gives the disp

Sinusoidal Function and Its Inverse

Consider the function $$f(x)=2*\sin(x)+1$$ defined on the restricted domain $$\left[-\frac{\pi}{2},\

Sinusoidal Function Transformation Analysis

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

Sinusoidal Transformations

The function $$g(x) = 2*\cos(3*(x - \frac{\pi}{4})) - 1$$ is a transformed cosine wave.

Solving a System Involving Exponential and Trigonometric Functions

Consider the system of equations: $$ \begin{aligned} f(x)&=e^{-x}+\sin(x)=1, \\ g(x)&=\ln(2-x)+\co

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 a Trigonometric Inequality

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

Solving Trigonometric Equations in a Specified Interval

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

Solving Trigonometric Equations in a Survey

In a survey, participants' responses are modeled using trigonometric equations. Solve the following

Special Triangles and Trigonometric Values

Utilize the properties of special triangles to evaluate trigonometric functions.

Special Triangles and Unit Circle Coordinates

Consider the actual geometric constructions of the special triangles used within the unit circle, sp

Tangent Function and Asymptotes

Examine the function $$f(\theta)=\tan(\theta)$$ defined on the interval $$\left(-\frac{\pi}{2}, \fra

Tangent Function Shift

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

Understanding Coterminal Angles Through Art Installation

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

Analysis of a Particle's Parametric Path

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

Analysis of Vector Directions and Transformations

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

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 $$

Circular Motion Parametrization

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

Composition of Linear Transformations

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

Determinant and Area of a Parallelogram

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

Exponential Parametric Function and its Inverse

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

Ferris Wheel Motion

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

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 9: Vectors in Motion and Velocity

A particle's position is described by the vector-valued function $$p(t)=\langle2*t-1, t^2+1\rangle$$

FRQ 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 15: Composition of Linear Transformations

Consider two linear transformations represented by the matrices $$A=\begin{bmatrix}2 & 0\\1 & 3\end{

FRQ 16: Inverse of a Linear Transformation

Let the transformation be given by the matrix $$T=\begin{bmatrix}5 & 2\\3 & 1\end{bmatrix}$$.

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 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

Investigating a Rational Piecewise Function with a Jump Discontinuity

Let $$f(x)=\begin{cases} \frac{x^2 - 4x + 3}{x-1} & \text{if } x < 3, \\ 2x - 1 & \text{if } x \geq

Linear Parametric Motion Modeling

A car travels along a straight path, and its position in the plane is given by the parametric equati

Linear Transformation Evaluation

Given the transformation matrix $$T = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$$, answer the fo

Linear Transformations in the Plane

A linear transformation $$L$$ from $$\mathbb{R}^2$$ to $$\mathbb{R}^2$$ is defined by $$L(x,y)=(2x-y

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 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 Discontinuities in a Function Representing Planar Motion

A car's horizontal motion is modeled by the function $$x(t)=\begin{cases} \frac{t^2-1}{t-1} & \text{

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

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

Modified Circular Motion: Transformation Effects

Consider the parametric equations $$x(t)=2+4\cos(t)$$ and $$y(t)=-3+4\sin(t)$$ which describe a curv

Parabolic and Elliptical Parametric Representations

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

Parabolic Motion in a Parametric Framework

A projectile is launched with its motion described by the equations $$x(t)=4*t$$ and $$y(t)=-4.9*t^2

Parametric Equations and Inverses

A curve is defined parametrically by $$x(t)=t+2$$ and $$y(t)=3*t-1$$.

Parametric Function Modeling and Discontinuity Analysis

A particle moves in the plane with its horizontal position described by the piecewise function $$x(t

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 Hyperbola

For the hyperbola given by $$\frac{x^2}{9}-\frac{y^2}{4}=1$$:

Parametric Representation of a Line: Motion of a Car

A car travels in a straight line from point A = (2, -1) to point B = (10, 7) at a constant speed. (

Parametric Representation of an Ellipse

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

Parametric Representation 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 circle of radius 5 is modeled by the parametric equations $$x(t)= 5\cos(t)$$ and $$y(t)= 5\sin(t)$

Parametrically Defined Circular Motion

A particle moves along a circle of radius 2 with parametric equations $$x(t)=2*cos(t)$$ and $$y(t)=2

Parametrization of an Ellipse

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

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

Piecewise Function and Discontinuities

Consider the function $$f(x)=\begin{cases} \frac{x^2 - 1}{x-1} & \text{if } x \neq 1, \\ 3 & \text{i

Population Transition Matrix Analysis

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

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

Position and Velocity Vectors

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

Rate of Change Analysis in Parametric Motion

A particle’s movement is described by the parametric equations $$x(t)=t^3-6*t+4$$ and $$y(t)=2*t^2-t

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 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

Trigonometric Function Analysis

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

Vector Components and Magnitude

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

Vector Operations in the Plane

Let the vectors be given by $$\mathbf{u}=\langle 3,-4\rangle$$ and $$\mathbf{v}=\langle -2,5\rangle$

Vector-Valued Functions: Position and Velocity

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

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$