AP Precalculus FRQ Room

Absolute Extrema and Local Extrema of a Polynomial

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

Analysis of a Rational Function with Quadratic Components

Analyze the rational function $$f(x)= \frac{x^2 - 9}{x^2 - 4*x + 3}$$ and determine its key features

Analyzing a Rational Function with 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 a Quadratic Model

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

Average Rate of Change in Rational Functions

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

Average Rate of Change of a Rational Function

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

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

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

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 Analysis with Rational and Polynomial Functions

Consider the functions $$f(x)= \frac{x+2}{x-1}$$ and $$g(x)= x^2 - 3*x + 4$$. Let the composite func

Composite Functions and Inverses

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

Concavity and Inflection Points of a Polynomial Function

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

Constructing a Function Model from Experimental Data

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

Constructing a 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

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

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 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(

Engineering Application: Stress Analysis Model

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

Expanding a Binomial: Application of the Binomial Theorem

Expand the expression $$ (x+2)^5 $$ using the Binomial Theorem and answer the following:

Exploring Domain Restrictions via Inverse Functions in a Quadratic Model

Consider the quadratic function $$f(x)= -x^2 + 6*x - 8$$. Answer the following questions regarding i

Exploring Polynomial Function Behavior

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

Finding and Interpreting Inflection Points

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

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 Involving Multiple Transformations

Consider the function $$f(x)= 5 - 2*(x+3)^2$$. Answer the following questions regarding its inverse

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

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

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 Vibration Data with a Cubic Function

A sensor records vibration data over time, and the data appears to be modeled by a cubic function of

Modeling with Inverse Variation: A Rational Function

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

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 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

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

Polynomial Long Division and Slant Asymptotes

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

Polynomial Model Construction and Interpretation

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

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 and Slant Asymptote Analysis

A study of speed and fuel efficiency is modeled by the function $$F(x)= \frac{3*x^2+2*x+1}{x-1}$$, w

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 Inverse Analysis

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

Rational Function: Machine Efficiency Ratios

A machine's efficiency is modeled by the rational function $$E(x) = \frac{x^2 - 9}{x^2 - 4*x + 3}$$,

Rational Inequalities and Test Intervals

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

Regression Model Selection for Experimental Data

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

Roller Coaster Track Polynomial Analysis

A section of a roller coaster track is modeled by a polynomial function $$h(x)$$ which gives the hei

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

Zeros and End Behavior in a Higher-Degree Polynomial

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

Acoustics and the Logarithmic Scale

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

Analyzing a Logarithmic Function from Data

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

Arithmetic Sequence Analysis

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

Comparing Arithmetic and Exponential Models in Population Growth

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

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

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 Exponential-Logarithmic Functions

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

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 and Transformation Functions

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

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

Compound Interest and Exponential Equations

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

Compound Interest and Financial Growth

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

Compound Interest vs. Simple Interest

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

Data Modeling: Exponential vs. Linear Models

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

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

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 Energy Release

Earthquake energy is modeled by the equation $$E = k\cdot 10^{1.5M}$$, where $$E$$ is the energy rel

Earthquake Magnitude and Logarithms

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

Environmental Pollution Decay

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

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: 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 Transformation

An exponential function is given by $$f(x) = 2 \cdot 3^x$$. Analyze the effects of various transform

Exponential Inequality Solution

Solve the inequality $$5^(2*x - 1) < 3·5^(x)$$ for x.

Fitting a Logarithmic Model to Sales Data

A company observes that its sales revenue (in thousands of dollars) based on advertising spend (in t

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

Graphical Analysis of Inverse Functions

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

Inverse and Domain of a Logarithmic Transformation

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

Inverse Function of an Exponential Function

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

Inverse Functions of Exponential and 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

Investment Scenario Convergence

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

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 Cost Function in Production

A company’s cost function is given by $$C(x)= 50+ 10\log_{2}(x)$$, where $$x>0$$ represents the numb

Logarithmic Equation and Extraneous Solutions

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

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

Modeling Bacterial Growth with Exponential Functions

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

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 Growth Inversion

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

Population Growth of Bacteria

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

Radioactive Decay Analysis

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

Radioactive Decay and Half-Life Estimation Through Data

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

Radioactive Decay Modeling

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

Real Estate Price Appreciation

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

Shifted Exponential Function Analysis

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

Solving Exponential Equations Using Logarithms

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

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

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

Wildlife Population Decline

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

Amplitude and Period Transformations

A Ferris wheel ride is modeled by a sinusoidal function. The ride has a maximum height of 75 ft and

Analysis of a Rose Curve

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

Analyzing a Rose Curve

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

Analyzing Damped Oscillations

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

Analyzing Phase Shifts in Sinusoidal Functions

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

Applying Sine and Cosine Sum Identities in Modeling

A researcher uses trigonometric sum identities to simplify complex periodic data. Consider the ident

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.

Converting and Graphing Polar Equations

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

Coordinate Conversion

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

Damped Oscillations: Combining Sinusoidal Functions and Geometric Sequences

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

Daylight Variation Model

A company models the variation in daylight hours over a year using the function $$D(t) = 10*\sin\Big

Evaluating Sine and Cosine Values Using Special Triangles

Using the properties of special triangles, answer the following:

Exploring Inverse Trigonometric Functions

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

Exploring Rates of Change in Polar Functions

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

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 Interpretation from Tabulated Periodic Data

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

Graphing and Analyzing a Transformed Sine Function

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

Graphing 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 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:

Interpreting Trigonometric Data Models

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

Inverse Tangent Composition and Domain

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

Inverse Trigonometric Analysis

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

Inverse Trigonometric Function Analysis

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

Limacons and Cardioids

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

Modeling Daylight Hours with a Sinusoidal Function

A study in a northern city recorded the number of daylight hours over the course of one year. The ob

Modeling Daylight Hours with a Sinusoidal Function

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

Modeling Daylight Hours with a Sinusoidal Function

A city's daylight hours vary seasonally and are modeled by $$D(t)=11+1.5\sin\left(\frac{2\pi}{365}(t

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 Temperature Variation Model

A town's temperature is modeled by the function $$T(t)=10*\cos(\frac{\pi}{12}*(t-6))+20$$, where t r

Polar Coordinates Conversion

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

Polar Function with Rate of Change Analysis

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

Polar Graphs: Conversion and Analysis

Analyze the polar equation $$r=4*\cos(\theta)+3$$.

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

Reciprocal Trigonometric Functions

Consider the function $$f(x)=\sec(x)=\frac{1}{\cos(x)}\).

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 Transformations

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

Solving a Basic Trigonometric Equation

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

Solving a Trigonometric Equation

Solve the trigonometric equation $$2*\sin(\theta)+\sqrt{3}=0$$ for all solutions in the interval $$[

Solving Trigonometric Equations in a Specified Interval

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

Tangent Function and Asymptotes

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

Trigonometric Inequality Solution

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

Understanding Coterminal Angles Through Art Installation

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

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 Vector Directions and Transformations

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

Analyzing a Piecewise Function Representing a Linear Transformation

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

Average Rate of Change in Parametric Motion

For the parametric functions $$x(t) = t^3 - 3*t + 2$$ and $$y(t) = 2*t^2 - t$$ defined for $$t \in [

Circular Motion Parametrization

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

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{

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 Inverse Calculation

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

Discontinuity Analysis in an Implicitly Defined Function

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

Discontinuity in a Function Modeling Transition between States

A system's state is modeled by the function $$S(x)=\begin{cases} \frac{x^2-16}{x-4} & \text{if } x \

Displacement and Average Velocity from a Vector-Valued Function

A particle’s position is given by the vector-valued function $$p(t)=\langle 2*t, t^2 - 4*t + 3 \ran

Evaluating a Piecewise Function in a Vector Context

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

Evaluating Limits and Discontinuities in a Parameter-Dependent Function

For the function $$g(t)=\begin{cases} \frac{2*t^2 - 8}{t-2} & \text{if } t \neq 2, \\ 6 & \text{if }

Exponential Parametric Function and its Inverse

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

FRQ 2: Circular Motion and Parameterization

Consider a particle moving along a circular path represented by the parametric function $$f(t)=(x(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 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 11: Matrix Inversion and Determinants

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

FRQ 13: Area Determined by a Matrix's Determinant

Vectors $$\textbf{v}=\langle4,3\rangle$$ and $$\textbf{w}=\langle-2,5\rangle$$ form a parallelogram.

FRQ 14: Linear Transformation and Rotation Matrix

Consider the rotation matrix $$R=\begin{bmatrix}\cos(t) & -\sin(t)\\ \sin(t) & \cos(t)\end{bmatrix}$

FRQ 16: Inverse of a Linear Transformation

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

Graphical and Algebraic Analysis of a Function with a Removable Discontinuity

Consider the function $$g(x)=\begin{cases} \frac{\sin(x) - \sin(0)}{x-0} & \text{if } x \neq 0, \\ 1

Implicit Function Analysis

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

Inverse Analysis of a Quadratic Function

Consider the function $$f(x)=x^2-4$$ defined for $$x\geq0$$. Analyze the function and its inverse.

Inverse Analysis of a Rational Function

Consider the function $$f(x)=\frac{2*x+3}{x-1}$$. Analyze the properties of this function and its in

Inverse and Determinant of a 2×2 Matrix

Consider the matrix $$C=\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$$. Answer the following parts.

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:

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 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 Multiplication and Properties

Let $$A=\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}$$ and $$B=\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmat

Matrix Multiplication Exploration

Let $$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ and $$B = \begin{pmatrix} 0 & -1 \\ 5 & 2 \

Matrix Transformation in Graphics

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

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 Equations and Inverses

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

Parametric Equations and Rates in a Biological Context

A bacteria colony in a Petri dish is observed to move in a periodic manner, with its position descri

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 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 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 a Parabola

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

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 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 $$

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)$

Parametrization of a Circle

The circle defined by $$x^2+y^2=25$$ represents all points at a distance of 5 from the origin.

Parametrization of a Parabola

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

Parametrization of an Ellipse for a Racetrack

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

Parametrizing a Linear Path: Car Motion

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

Parametrizing a Parabola

A parabola is defined parametrically by $$x(t)=t$$ and $$y(t)=t^2$$.

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 $

Population Transition Matrix Analysis

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

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

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

Transition Matrix and State Changes

Consider a system with two states modeled by the transition matrix $$M = \begin{pmatrix} 0.7 & 0.2 \

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

Vector Analysis in Projectile Motion

A soccer ball is kicked so that its velocity vector is given by $$\mathbf{v}=\langle5, 7\rangle$$ (i

Vector Components and Magnitude

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

Vector Operations in the Plane

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

Vector Operations in the Plane

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

Vector Scalar Multiplication

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