AP Precalculus FRQ Room

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 a Hole

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

Analyzing a Rational Function with Asymptotes

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

Analyzing Concavity 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 a Polynomial

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

Analyzing End Behavior of Polynomial Functions

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

Average Rate of Change in Rational Functions

Let $$h(x)= \frac{3}{x-1}$$ represent the speed (in km/h) of a vehicle as a function of a variable x

Behavior Analysis of a Rational Function with Cancelled Factors

Consider the function $$f(x)=\frac{x^2-16}{x-4}$$. Analyze the behavior of the function at the point

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

Comparing Polynomial and Rational Function Models

Two models are proposed to describe a data set. Model A is a polynomial function given by $$f(x)= 2*

Constructing a Piecewise Function from Data

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

Constructing a Rational Function Model with Asymptotic Behavior

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

Construction of a Polynomial Model

A company’s quarterly profit (in thousands of dollars) over five quarters is given in the table belo

Continuous Piecewise Function Modification

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

Cubic Polynomial Analysis

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

Degree Determination from Finite Differences

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

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 |

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

Engineering Curve Analysis: Concavity and Inflection

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

Examining End Behavior of Polynomial Functions

Consider the polynomial function $$f(x)= -3*x^4 + 2*x^3 - x + 7$$. Answer the following parts.

Exploring Asymptotic Behavior in a Sales Projection Model

A sales projection model is given by $$P(x)=\frac{4*x-2}{x-1}$$, where $$x$$ represents time in year

Exploring End Behavior and Leading Coefficients

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

Exploring Polynomial Function Behavior

Consider the polynomial function $$f(x)= 2*(x-1)^2*(x+2)$$, which is used to model a physical trajec

Factoring and Zero Multiplicity

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

Function Model Construction from Data Set

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

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

Geometric Series Model in Area Calculations

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

Graphical Analysis of Inverse Function for a Linear Transformation

Consider the function $$f(x)=4*(x+1)-5$$. Answer the following questions regarding the transformatio

Impact of Multiplicity on Graph Behavior

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

Interpreting Transformations of Functions

The parent function is $$f(x)= x^2$$. A transformed function is given by $$g(x)= -3*(x+2)^2+5$$. Ans

Intersection of Functions in Supply and Demand

Consider two functions that model supply and demand in a market. The supply function is given by $$f

Inverse Analysis of a Shifted Cubic Function

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

Inverse of a Complex Rational Function

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

Investigating a Real-World Polynomial Model

A physicist models the vertical trajectory of a projectile by the quadratic function $$h(t)= -5*t^2+

Investigating End Behavior of a Polynomial Function

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

Investigation of Refund Policy via Piecewise Continuous Functions

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

Linear Function Inverse Analysis

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

Marketing Analysis Using Piecewise Polynomial Function

A firm's sales function is modeled by $$ S(x)=\begin{cases} -x^2+6*x & \text{for } x\le3, \\ 2*x+3 &

Model Interpretation: End Behavior and Asymptotic Analysis

A chemical reaction's saturation level is modeled by the rational function $$S(t)= \frac{10*t+5}{t+3

Modeling 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 Inverse Variation: A Rational Function

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

Multivariable Rational Function: Zeros and Discontinuities

A pollutant concentration is modeled by $$C(x)= \frac{(x-3)*(x+2)}{(x-3)*(x-4)}$$, where x represent

Office Space Cubic Function Optimization

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

Optimizing Production Using a Polynomial Model

A factory's production cost (in thousands of dollars) is modeled by the function $$C(x)= 0.02*x^3 -

Piecewise Function Analysis

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

Piecewise Function Construction for Utility Rates

A utility company charges for electricity according to the following scheme: For usage $$u$$ (in kWh

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 Interpolation and Curve Fitting

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

Polynomial Long Division and Slant Asymptote

Perform polynomial long division on the function $$f(x)= \frac{3*x^3 - 2*x^2 + 4*x - 5}{x^2 - 1}$$,

Polynomial Long Division and Slant Asymptote

Consider the function $$P(x)= \frac{2*x^3 - 3*x^2 + x - 5}{x-2}$$. Answer the following parts.

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

Predator-Prey Dynamics as a Rational Function

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

Product Revenue Rational Model

A company’s product revenue (in thousands of dollars) is modeled by the rational function $$R(x)= \f

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 Analysis for Signal Processing

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

Rational Function Inverse Analysis

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

Rational Inequalities Analysis

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

Revenue Modeling with a Polynomial Function

A small theater's revenue from ticket sales is modeled by the polynomial function $$R(x)= -0.5*x^3 +

Signal Strength Transformation Analysis

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

Solving a 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

Transformation in Composite Functions

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

Use of Logarithms to Solve for Exponents in a Compound Interest Equation

An investment of $$1000$$ grows continuously according to the formula $$I(t)=1000*e^{r*t}$$ and doub

Using the Binomial Theorem for Polynomial Expansion

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

Zeros and Factorization Analysis

A fourth-degree polynomial $$Q(x)$$ is known to have zeros at $$x=-3$$ (with multiplicity 2), $$x=1$

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 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

Consider the function \(f(x)=5\cdot e^{-0.3\cdot x}+2\). (a) Determine the horizontal asymptote of

Analyzing Exponential Function Behavior from a Graph

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

Arithmetic Sequence Analysis

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

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

Cellular Data Usage Trend

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

Comparing Exponential and Linear Growth in Business

A company is analyzing its revenue over several quarters. They suspect that part of the growth is li

Comparing Linear and Exponential Growth Models

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

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 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 with Exponential and Logarithmic Elements

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

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 Log Functions

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

Compound Interest and Continuous Growth

A bank account grows continuously according to the formula $$A(t) = P\cdot e^{rt}$$, where $$P$$ is

Compound Interest vs. Simple Interest

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

Determining an Exponential Model from Data

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

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

Environmental Pollution Decay

The concentration of a pollutant in a lake decays exponentially due to natural processes. The concen

Estimating Rates of Change from Table Data

A cooling object has its temperature recorded at various time intervals as shown in the table below:

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 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 Function with Compound Transformations and Its Inverse

Consider the function $$f(x)=2^(x-2)+3$$. Determine its invertibility, find its inverse function, an

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}(

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 of an Exponential Function

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

Investment Growth: Compound Interest

An investor deposits an initial amount \(P\) dollars in a savings account that compounds interest an

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

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

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

Modeling Bacterial Growth with Exponential Functions

A research laboratory is tracking the growth of a bacterial culture. A graph showing experimental da

Natural Logarithms in Continuous Growth

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

pH and Logarithmic Functions

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

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 Growth Inversion

A town's population grows according to the function $$f(t)=1200*(1.05)^(t)$$, where $$t$$ is the tim

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 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

Real Estate Price Appreciation

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

Savings Account Growth: Arithmetic vs Geometric Sequences

An individual opens a savings account that incorporates both regular deposits and interest earnings.

Semi-Log Plot and Exponential Model

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

Semi-Log Plot Data Analysis

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

Solving Exponential Equations Using Logarithms

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

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

Transformation Effects on Exponential Functions

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

Transformations of Exponential Functions

Consider the 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

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 Cotangent Function

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

Analyzing the Tangent Function

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

Average Rate of Change in a Polar Function

Consider the polar function $$r=f(θ)=3+2*\sin(θ)$$, which models a periodic phenomenon in polar coor

Cardioid Polar Graphs

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

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

Composite Function Analysis with Polar and Trigonometric Elements

A radar system uses the polar function $$r(\theta)=5+2*\sin(\theta)$$ to model the distance to a tar

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

Coordinate Conversion

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

Daylight Hours Modeling

A city's daylight hours vary sinusoidally throughout the year. It is observed that the maximum dayli

Determining Phase Shifts and Amplitude Changes

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

Equivalent Representations Using Pythagorean Identity

Using trigonometric identities, answer the following:

Evaluating Inverse Trigonometric Functions

Inverse trigonometric functions such as $$\arcsin(x)$$ and $$\arccos(x)$$ have specific restricted d

Evaluating Sine and Cosine Using Special Triangles

Using knowledge of special right triangles, evaluate trigonometric functions.

Exploring Coterminal Angles and Periodicity

Analyze the concept of coterminal angles.

Exploring Rates of Change in Polar Functions

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

Exploring the Pythagorean Identity

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

Extracting Sinusoidal Parameters from Data

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

Graph Analysis of a Polar Function

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

Graph Interpretation from Tabulated Periodic Data

A study recorded the oscillation of a pendulum over time. Data is provided in the table below showin

Graph Transformations of Sinusoidal Functions

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

Graph Transformations: Sine and Cosine Functions

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

Graphing a Rose Curve

Consider the polar function $$r=4\cos(3\theta)$$ and analyze its properties.

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 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

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

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

Interpreting Trigonometric Data Models

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

Inverse Function Analysis

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

Inverse Trigonometric Analysis

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

Inverse Trigonometric Functions

Examine the inverse relationships for trigonometric functions over appropriate restricted domains.

Limacons and Cardioids

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

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 Heights with Periodic Data

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

Multiple Angle Equation

Solve the trigonometric equation $$2*\sin(2x) - \sqrt{3} = 0$$ for all $$x$$ in the interval $$[0, 2

Periodic Phenomena: Seasonal Daylight Variation

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

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 Circle Graph

Consider the polar equation $$r = 4$$ which represents a circle.

Polar Coordinates and Graphing a Circle

Answer the following questions on polar coordinates:

Polar Function with Rate of Change Analysis

Given the polar function $$r(\theta)=2+\sin(\theta)$$, analyze its behavior.

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.

Reciprocal and Pythagorean Identities

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

Rewriting and Graphing a Composite Trigonometric Function

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

Rose Curve in Polar Coordinates

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

Seasonal Demand Modeling

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

Sinusoidal Combination

Let $$f(x) = 3*\sin(x) + 2*\cos(x)$$.

Sinusoidal Function and Its Inverse

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

Solving a Trigonometric Inequality

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

Solving Trigonometric Equations

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

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

Tidal Patterns and Sinusoidal Modeling

A coastal area experiences tides that follow a sinusoidal pattern described by $$T(t)=4+1.2\sin\left

Understanding Coterminal Angles and Their Applications

Coterminal angles are important in trigonometry as they represent angles with the same terminal side

Unit Circle and Special Triangles

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

Vibration Analysis

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

Advanced Matrix Modeling in Economic Transitions

An economic model is represented by a 3×3 transition matrix $$M=\begin{pmatrix}0.5 & 0.2 & 0.3\\0.1

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

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

Converting an Explicit Function to Parametric Form

The function $$f(x)=x^3-3*x+2$$ is given explicitly. One way to parametrize this function is by lett

Determinant and Inverse Calculation

Given the matrix $$C = \begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$$, answer the following:

Determinant Applications in Area Computation

Vectors $$\mathbf{u}=\langle 5,2\rangle$$ and $$\mathbf{v}=\langle 1,4\rangle$$ form adjacent sides

Discontinuity Analysis in a Function Modeling Particle Motion

A particle’s position along a line is given by the piecewise function: $$s(t)=\begin{cases} \frac{t^

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)=

Dot Product, Projection, and Angle Calculation

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

Eliminating the Parameter

Given the parametric equations $$x(t) = 2 + 3*t$$ and $$y(t) = 4 - t^2$$, answer the following:

Estimating a Definite Integral with a Table

The function x(t) represents the distance traveled (in meters) by an object over time, with the foll

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

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 4: Parametric Representation of a Parabola

The parabola given by $$y=(x-1)^2-2$$ can be represented parametrically as $$ (x(t), y(t)) = (t, (t-

FRQ 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 10: Unit Vectors and Direction

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

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 $$T=\begin{bmatrix}5 & 2\\3 & 1\end{bmatrix}$$.

FRQ 17: Matrix Representation of a Reflection

A reflection about the line \(y=x\) is given by the matrix $$Q=\begin{bmatrix}0 & 1\\1 & 0\end{bmatr

Graph Analysis of an Implicitly Defined Ellipse

A graph is produced for the implicitly defined ellipse given by $$\left(\frac{x}{2}\right)^2 + \lef

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

Implicit Function Analysis

Consider the implicitly defined equation $$x^2 + y^2 - 4*x + 6*y - 12 = 0$$. Answer the following:

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

Inverse of a 2×2 Matrix

Consider the matrix $$A=\begin{bmatrix}2 & 5\\ 3 & 7\end{bmatrix}$$.

Inverses and Solving a Matrix Equation

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

Linear Transformation Composition

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

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

Matrices as Representations of Rotation

Consider the matrix $$A=\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$$, which represents a rotation in

Matrix Modeling of Department Transitions

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

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)

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 with Variable Rates

A particle moves in the plane with its motion described by $$x(t)=4*t-t^2$$ and $$y(t)=t^2-2*t$$.

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.

Parametric Representation on the Unit Circle and Special Angles

Consider the unit circle defined by the parametric equations $$x(t)=\cos(t)$$ and $$y(t)=\sin(t)$$.

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 a Parabola

Given the explicit function $$y = 2*x^2 + 3*x - 1$$, answer the following:

Parametrizing a Linear Path: Car Motion

A car moves along a straight line from point $$A=(1,2)$$ to point $$B=(7,8)$$.

Particle Motion from Parametric Equations

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

Particle Motion with Quadratic Parametric Functions

A particle moves in the plane according to the parametric equations $$x(t)=t^2$$ and $$y(t)=2*t$$. A

Position and Velocity Vectors

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

Properties of a Parametric Curve

Consider a curve defined parametrically by $$x(t)=t^3$$ and $$y(t)=t^2.$$ (a) Determine for which

Rational Piecewise Function with Parameter Changes: Discontinuity Analysis

Let $$R(t)=\begin{cases} \frac{3t^2-12}{t-2} & \text{if } t\neq2, \\ 5 & \text{if } t=2 \end{cases}$

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.

Trigonometric Function Analysis

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

Uniform Circular Motion

A car is moving along a circular track of radius 10 meters. Its motion is described by the parametri

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$

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