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 Removable Discontinuities in an Experiment

In a chemical reaction process, the rate of reaction is modeled by $$R(x)=\frac{x^2-4}{x-2}$$ for $$

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 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 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

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 Polynomial and Rational Functions

A function is defined piecewise by $$ f(x)=\begin{cases} x^2-4 & \text{if } x\le2, \\ \frac{x^2-4}{x

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 Transformations

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

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

Continuous Piecewise Function Modification

A company models its profit $$P(x)$$ (in thousands of dollars) with the piecewise function: $$ P(x)=

Degree Determination from Finite Differences

A researcher records the size of a bacterial colony at equal time intervals, obtaining the following

Discontinuity Analysis in a Rational Function with High Degree

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

End Behavior of a Quartic Polynomial

Consider the quartic polynomial function $$f(x) = -3*x^4 + 5*x^3 - 2*x^2 + x - 7$$. Analyze the end

Engineering Application: Stress Analysis Model

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

Evaluating Limits and Discontinuities in a Rational Function

Consider the rational function $$f(x)=\frac{x^2-4}{x-2}$$, which is defined for all real $$x$$ excep

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:

Exploring the Effect of Multiplicities on Graph Behavior

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

Exponential Equations and Logarithm Applications in Decay Models

A radioactive substance decays according to the model $$A(t)= A_0*e^{-0.3*t}$$. A researcher analyze

Function Model Construction from Data Set

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

Function Simplification and Graph Analysis

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

Function Transformations and Parent Functions

The parent function is $$f(x)= x^2$$. Consider the transformed function $$g(x)= -3*(x-4)^2 + 5$$. An

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

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

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

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

Modeling with Inverse Variation: A Rational Function

A physics experiment models the intensity $$I$$ of light as inversely proportional to the square of

Modeling with Rational Functions

A chemist models the concentration of a reactant over time with the rational function $$C(t)= \frac{

Office Space Cubic Function Optimization

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

Parameter Identification in a Rational Function Model

A rational function modeling a certain phenomenon is given by $$r(x)= \frac{k*(x - 2)}{x+3}$$, where

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 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 Model Construction and Interpretation

A company’s profit (in thousands of dollars) over time t (in months) is modeled by the quadratic fun

Polynomial Model from Temperature Data

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

Polynomial Transformation Challenge

Consider the function transformation given by $$g(x)= -2*(x+1)^3 + 3$$. Answer each part that follow

Projectile Motion Analysis

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

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

Consider the rational function $$r(x)=\frac{x^2 - 4}{x^2 - x - 6}$$. Analyze the function according

Rational Function Asymptotes and Holes

A machine’s efficiency is modeled by the rational function $$R(x)= \frac{x^2 - 4}{x^2 - x - 6}$$, wh

Rational Inequalities Analysis

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

Return to a Rational Expression under Transformation

Consider the function $$f(x)=\frac{(x-2)(x+3)}{(x-2)(x-5)}$$, defined for $$x\neq2,5$$. Answer the f

Solving a Polynomial Inequality

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

Solving a System of Equations: Polynomial vs. Rational

Consider the system of equations where $$f(x)= x^2 - 1$$ and $$g(x)= \frac{2*x}{x+2}$$. Answer the f

Zeros and Complex Conjugates in Polynomial Functions

A polynomial function of degree 4 is known to have real zeros at $$x=1$$ and $$x=-2$$, and two non-r

Zeros and End Behavior in a Higher-Degree Polynomial

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

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.

Arithmetic Savings Plan

A person decides to save money every month, starting with an initial deposit of $$50$$ dollars, with

Arithmetic Sequence Analysis

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

Arithmetic Sequence Derived from Logarithms

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

Bacterial Growth Model

In a laboratory experiment, a bacteria colony doubles every 3 hours. The initial count is $$500$$ ba

Bacterial Growth Modeling

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

Comparing Arithmetic and Exponential Models in Population Growth

Two neighboring communities display different population growth patterns. Community A increases by a

Comparing Linear and Exponential Growth Models

A company is analyzing its profit growth using two distinct models: an arithmetic model given by $$P

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 Exponential-Logarithmic Functions

Let f(x) = log₃(x) and g(x) = 2·3ˣ. Analyze the following compositions.

Composite Function Analysis: Identity and Inverses

Let $$f(x)= 2^x$$ and $$g(x)= \log_2(x)$$.

Composite Functions 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 $$g(x)=\ln(x)+4$$.

Composite Sequences: Combining Geometric and Arithmetic Models in Production

A factory’s monthly production is influenced by two factors. There is a fixed increase in production

Composition of Exponential and Log Functions

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

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

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

Compound Interest with Periodic Deposits

An investor opens an account with an initial deposit of $$5000$$ dollars and adds an additional $$50

Data Modeling: Exponential vs. Linear Models

A scientist collected data on the growth of a substance over time. The table below shows the measure

Earthquake Intensity 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, \

Experimental Data Modeling Using Semi-Log Plots

A set of experimental data regarding chemical concentration is given in the table below. The concent

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 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 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 Inequalities

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

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

Graphical Analysis of Inverse Functions

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

Inverse of a Composite Function

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

Inverse of an Exponential Function

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

Inverse Relationships in Exponential and Logarithmic Functions

Consider the functions \(f(x)=2^{(x-1)}+3\) and \(g(x)=\log_{2}(x-3)+1\). (a) Discuss under what co

Investment Growth via Sequences

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

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 with Scaling and Inverse

Consider the function $$f(x)=\frac{1}{2}\log_{10}(x+4)+3$$. Analyze its monotonicity, find the inver

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

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

Piecewise Exponential and Logarithmic Function Discontinuities

Consider the function defined by $$ f(x)=\begin{cases} 2^x + 1, & x < 3,\\ 5, & x = 3,

Population Demographics Model

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

Population Growth Modeling with Exponential Functions

A small town has its population recorded every 5 years, as shown in the table below: | Year | Popul

Population Growth with an Immigration Factor

A city's population is modeled by an equation that combines exponential growth with a constant linea

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 Model

A radioactive substance decays according to the function $$f(t)= a \cdot e^{-kt}$$. In an experiment

Radioactive Decay Modeling

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

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

Shifted Exponential Function Analysis

Consider the exponential function $$f(x) = 4e^x$$. A transformed function is defined by $$g(x) = 4e^

Shifted Exponential Function and Its Inverse

Consider the function $$f(x)=7-4*2^(x-3)$$. Determine its one-to-one nature, find its inverse functi

Solving Logarithmic Equations and Checking Domain

An engineer is analyzing a system and obtains the following logarithmic equation: $$\log_3(x+2) + \

Solving Logarithmic Equations with Extraneous Solutions

Solve the logarithmic equation $$\log_2(x - 1) + \log_2(2x) = \log_2(10)$$ and check for any extrane

Telephone Call Data Analysis on Semi-Log Plot

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

Temperature Decay Modeled by a Logarithmic Function

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

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 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.

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

Wildlife Population Decline

A wildlife population declines by 15% each year, forming a geometric sequence.

Analysis of a Limacon

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

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

Concavity in the Sine Function

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

Conversion between Rectangular and Polar Coordinates

Given the point in rectangular coordinates $$(-3, 3\sqrt{3})$$, perform the following tasks.

Conversion Between Rectangular and Polar Coordinates

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

Conversion Between Rectangular and Polar Coordinates

A point A in the Cartesian plane is given by $$(-3, 3\sqrt{3})$$.

Coordinate Conversion

Convert the point $$(-\sqrt{3}, 1)$$ from rectangular coordinates to polar coordinates, and then con

Coterminal Angles and the Unit Circle

Consider the angle $$\theta = \frac{5\pi}{3}$$ given in standard position.

Coterminal Angles and Trigonometric Evaluations

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

Daily Temperature Fluctuations

The table below shows the recorded temperature (in $$^{\circ}\text{F}$$) at various times during the

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

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 Rates of Change in Polar Functions

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

Graph Analysis of a Polar Function

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

Graph Transformations of Sinusoidal Functions

Consider the sinusoidal function $$f(x) = 3*\sin\Bigl(2*(x - \frac{\pi}{4})\Bigr) - 1$$.

Graphing a Limacon

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

Graphing a Transformed Sine Function

Analyze the function $$f(x)=3\,\sin\Bigl(2\bigl(x-\frac{\pi}{4}\bigr)\Bigr)-1$$ which is obtained fr

Graphing and Transforming a Function and Its Inverse

Examine the function $$f(x)=\cos(x)$$ defined on the interval $$[0,\pi]$$ and its inverse.

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

Consider the transformed tangent function $$g(\theta)=\tan(\theta-\frac{\pi}{4})$$.

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

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 city's daylight hours throughout the year are periodic. At t = 0 months, the city experiences maxi

Modeling Tidal Heights with Periodic Data

An oceanographer records tidal heights (in meters) over a 6-hour period. The following table gives t

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

Period Detection and Frequency Analysis

An engineer analyzes a signal modeled by $$P(t)=6*\cos(5*(t-1))$$.

Periodic Phenomena: Seasonal Daylight Variation

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

Piecewise Trigonometric Function and Continuity Analysis

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

Polar Coordinates Conversion

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

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

Polar Rate of Change

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

Polar Rose Analysis

Analyze the polar equation $$r = 2*\cos(3\theta)$$.

Polar to Cartesian Conversion for a Circle

Consider the polar equation $$r=6\cos(\theta)$$.

Probability and Trigonometry: Dartboard Game

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

Rose Curve in Polar Coordinates

The polar function $$r(\theta) = 4*\cos(3*\theta)$$ represents a rose curve.

Roses and Limacons in Polar Graphs

Consider the polar curves described below and answer the following:

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:

Solving a Basic Trigonometric Equation

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

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 a Trigonometric Inequality

Solve the inequality $$\sin(x)>\frac{1}{2}$$ for \(0\le x<2\pi\).

Solving Trigonometric Equations

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

Solving Trigonometric Equations in a Specified Interval

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

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 and Asymptotes

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

Tangent Function Shift

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

Tidal Motion Analysis

A coastal region's tidal heights are modeled by a sinusoidal function $$f(t) = A * \sin(b*(t - c)) +

Tide Height Model: Using Sine Functions

A coastal region experiences tides that follow a sinusoidal pattern. A table of tide heights (in fee

Understanding Coterminal Angles Through Art Installation

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

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 \(x\).

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

Analysis of a Particle's Parametric Path

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

Analysis of a Vector-Valued Position Function

Consider the vector-valued function $$\mathbf{p}(t) = \langle 2*t + 1, 3*t - 2 \rangle$$ representin

Analysis of Vector Directions and Transformations

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

Analyzing the Composition of Two Matrix Transformations

Let matrices be given by $$A=\begin{pmatrix}1 & 2\\0 & 1\end{pmatrix}$$ and $$B=\begin{pmatrix}2 & 0

Area of a Parallelogram Using Determinants

Given the vectors $$u=\langle 3, 5 \rangle$$ and $$v=\langle -2, 4 \rangle$$: (a) Write the 2×2 mat

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)

Composite Functions Involving Parametric and Matrix Transformations

A particle’s motion is initially modeled by the parametric function $$f(t)= \langle e^{0.1*t}, \ln(t

Composition of Linear Transformations

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

Composition of Transformations and Inverses

Let $$A=\begin{bmatrix}2 & 3\\ 1 & 4\end{bmatrix}$$ and consider the linear transformation $$L(\vec{

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 Limits in a Parametrically Defined Motion Scenario

A particle’s motion is given by the parametric equations: $$x(t)=\begin{cases} \frac{t^2-9}{t-3} & \

Exponential 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

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

Finding Angle Between Vectors

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

FRQ 2: Circular Motion and Parameterization

Consider a particle moving along a circular path represented by the parametric function $$f(t)=(x(t)

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 6: Implicit Function to Parametric Representation

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

FRQ 8: Vector Analysis - Dot Product and Angle

Given the vectors $$\textbf{u}=\langle3,4\rangle$$ and $$\textbf{v}=\langle-2,5\rangle$$, analyze th

FRQ 11: Matrix Inversion and Determinants

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

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

FRQ 20: Advanced Parametric and Matrix Modeling

A particle moves according to $$f(t)=(3*\cos(t)-t, 3*\sin(t)+2)$$ for time t. A transformation is ap

Implicitly Defined Circle

Consider the implicitly defined function given by $$x^2+y^2=16$$, which represents a circle.

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

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

Inverse Matrix with a Parameter

Consider the 2×2 matrix $$A=\begin{pmatrix} a & 2 \\ 3 & 4 \end{pmatrix}.$$ (a) Express the deter

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 Transformation and its Effect on Geometric Shapes

A linear transformation in \(\mathbb{R}^2\) is represented by the matrix $$M=\begin{pmatrix} 2 & 0 \

Linear Transformation Composition

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

Linear Transformation Evaluation

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

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 Methods for Solving Linear Systems

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

Matrix Modeling of Department Transitions

A company’s employee transitions between two departments are modeled by the matrix $$M=\begin{pmatri

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 Linear Transformations

Consider the matrices $$A= \begin{pmatrix} 1 & 3 \\ 2 & -1 \end{pmatrix}$$ and $$B= \begin{pmatrix}

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 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$

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 of an Ellipse

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

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:

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 $

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

Position and Velocity Vectors

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

Projectile Motion: Parabolic Path

A projectile is launched so that its motion is modeled by the parametric equations $$x(t)=t$$ and $

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}

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

Transition Matrix in Markov Chains

A system transitions between two states according to the matrix $$M= \begin{pmatrix} 0.7 & 0.3 \\ 0.

Vector Components and Magnitude

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

Vector Operations in the Plane

Let $$\vec{u}= \langle 3, -2 \rangle$$ and $$\vec{v}= \langle -1, 4 \rangle$$. Perform the following

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