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 Quartic Function

Consider the quartic function $$h(x)= x^4 - 4*x^2 + 3$$. Answer the following:

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 an Odd Polynomial Function

Consider the function $$p(x)= x^3 - 4*x$$. Investigate its properties by answering the following par

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$

Application of the Binomial Theorem

Expand the expression $$(x+3)^5$$ using the Binomial Theorem and answer the following parts.

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

Binomial Theorem Expansion

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

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

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*

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 in Environmental Modeling

Environmental data shows the concentration (in mg/L) of a pollutant over time (in hours) as given in

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 Function Transformations

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

Cubic Polynomial Analysis

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

Designing a Piecewise Function for a Temperature Model

A city experiences distinct temperature patterns during the day. A proposed model is as follows: for

Determining the Degree of a Polynomial from Data

A table of values is given below for a function $$f(x)$$ measured at equally spaced x-values: | x |

Determining the Degree of a Polynomial via Differences

A function $$f(x)$$ is defined on equally spaced inputs and the following table gives selected value

Discontinuities in a Rational Model Function

Consider the function $$p(x)=\frac{(x-3)(x+1)}{x-3}$$, defined for all $$x$$ except when $$x=3$$. Ad

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

Estimating Polynomial Degree from Finite Differences

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

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 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 Polynomial Function Behavior

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

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

Finding and Interpreting Inflection Points

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

Geometric Series Model in Area Calculations

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

Impact of Multiplicity on Graph Behavior

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

Inverse Analysis of a Modified Rational Function

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

Inverse Analysis of a Polynomial Function with Multiple Turning Points

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

Inverse Analysis of a Transformed Quadratic Function

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

Inverse Function of a Rational Function with a Removable Discontinuity

Consider the function $$f(x)= \frac{x^2-4}{x-2}$$. Answer the following questions regarding its inve

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

Inversion of a Polynomial Ratio Function

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

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 Piecewise Behavior of a Function

A function is defined as follows: $$ f(x)=\begin{cases} \frac{x^2-9}{x-3} & x<3, \\ 2*x+1 & x\ge3

Logarithmic Equation Solving in a Financial Model

An investor compares two savings accounts. Account A grows continuously according to the model $$A(t

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 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

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 -

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 Financial Growth Model

A company’s quarterly growth rate is modeled using a piecewise function. For $$0 \le x \le 4$$, the

Piecewise Function Analysis

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

Piecewise Function and Domain Restrictions

A temperature function is defined as $$ T(x)=\begin{cases} \frac{x^2-25}{x-5} & x<5, \\ 3*x-10 & x\g

Polynomial Division in Limit Evaluation

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

Polynomial Interpolation and Finite Differences

A quadratic function is used to model the height of a projectile. The following table gives the heig

Polynomial Long Division and Slant Asymptote

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

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 from Temperature Data

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

Population Growth Modeling with a Polynomial Function

A regional population (in thousands) is modeled by a polynomial function $$P(t)$$, where $$t$$ repre

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 Graph and Asymptote Identification

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

Rational Function: Machine Efficiency Ratios

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

Rational Inequalities and Test Intervals

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

Real-World Inverse Function: Modeling a Reaction Process

The function $$f(x)=\frac{50}{x+2}+3$$ models the average concentration (in moles per liter) of a su

Revenue Function Transformations

A company models its revenue with a polynomial function $$f(x)$$. It is known that $$f(x)$$ has x-in

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 +

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 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

Transformation and Reflection of a Parent Function

Given the parent function $$f(x)= x^2$$, consider the transformed function $$g(x)= -3*(x+2)^2 + 5$$.

Transformation of a Parabola

Starting with the parent function $$f(x)=x^2$$, a new function is defined by $$g(x) = -2*(x+3)^2 + 4

Using the Binomial Theorem for Polynomial Expansion

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

Zero Finding and Sign Charts

Consider the function $$p(x)= (x-2)(x+1)(x-5)$$.

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

Comparing Arithmetic and Exponential Models in Population Growth

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

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: 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 $$g(x)=2*e^(x)$$.

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.

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

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

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 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 Transformation

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

Exponential Inequalities

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

Financial Growth: Savings Account with Regular Deposits

A savings account starts with an initial balance of $$1000$$ dollars and earns compound interest at

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 Investment Growth

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

Geometric Sequence and Exponential Modeling

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

Geometric Sequence Construction

Consider a geometric sequence where the first term is $$g_0 = 3$$ and the second term is $$g_1 = 6$$

Inverse and Domain of a Logarithmic Transformation

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

Inverse of an Exponential Function

Let f(x) = 5·e^(2*x) - 3. Find the inverse function f⁻¹(x) and verify your answer by composing f and

Inverse of an Exponential Function

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

Inverse Relationship Verification

Given f(x) = 3ˣ - 4 and g(x) = log₃(x + 4), verify that g is the inverse of f.

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

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

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 Inequalities

Solve the inequality $$\log_{2}(x-1) > 3$$.

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

Model Validation and Error Analysis in Exponential Trends

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

Parameter Sensitivity in Exponential Functions

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

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 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 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 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

Solving Exponential Equations Using Logarithms

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

Solving Exponential Equations Using Logarithms

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

Solving Logarithmic Equations 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 Exponential Functions

Consider the exponential function $$f(x)= 3 * 5^x$$. A new function $$g(x)$$ is defined by applying

Transformations of Exponential Functions

Consider the exponential function \(f(x)=3\cdot2^{x}\). (a) Determine the equation of the transform

Transformations of Exponential Functions

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

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 $$b^{-3}$$.

Wildlife Population Decline

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

Analysis of a Rose Curve

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

Analyzing a Rose Curve

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

Application of Trigonometric Sum Identities

Utilize trigonometric sum identities to simplify and solve expressions.

Calculating the Area Enclosed by a Polar Curve

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

Cardioid Polar Graphs

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

Conversion Between Rectangular and Polar Coordinates

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

Coterminal Angles and the Unit Circle

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

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

Evaluating Inverse Trigonometric Functions

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

Evaluating Sine and Cosine Values Using Special Triangles

Using the properties of special triangles, answer the following:

Exploring Coterminal Angles and Periodicity

Analyze the concept of coterminal angles.

Exploring Limacons in Polar Coordinates

Consider the polar function $$r=2+3*\cos(θ)$$ which represents a limacon. Evaluate its key features

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

Graphical Analysis of a Periodic Function

A periodic function is depicted in the graph provided. Analyze the function’s key features based on

Graphical Reflection of Trigonometric Functions and Their Inverses

Consider the sine function and its inverse. The graph of an inverse function is the reflection of th

Graphing 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 Analyzing a Transformed Sine Function

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

Graphing Polar Circles and Roses

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

Identity Verification

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

Inverse Function Analysis

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

Inverse Tangent Composition and Domain

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

Limacon Analysis

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

Limacons and Cardioids

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

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 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 vary seasonally and are modeled by $$D(t)=11+1.5\sin\left(\frac{2\pi}{365}(t

Modeling Tidal Motion with a Sinusoidal Function

A coastal town uses the model $$h(t)=4*\sin\left(\frac{\pi}{6}*(t-2)\right)+10$$ (with $$t$$ in hour

Multiple Angle Equation

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

Period Detection and Frequency Analysis

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

Periodic Phenomena in Weather Patterns

A city's average daily temperature over the course of a year is modeled by a sinusoidal function. Th

Phase Shifts and Reflections of Sine Functions

Analyze the relationship between the functions $$f(\theta)=\sin(\theta)$$ and $$g(\theta)=\sin(\thet

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 Graphs: Conversion and Analysis

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

Polar Rate of Change

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

Polar Rose Analysis

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

Rate of Change in Polar Functions

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

Reciprocal and Pythagorean Identities

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

Reciprocal Trigonometric Functions

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

Rewriting and Graphing a Composite Trigonometric Function

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

Roses and Limacons in Polar Graphs

Consider the polar curves described below and answer the following:

Roulette Wheel Outcomes and Angle Analysis

A casino roulette wheel is divided into 12 equal sectors. Answer the following:

Seasonal Demand Modeling

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

Seasonal Temperature Modeling

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

Secant, Cosecant, and Cotangent Functions Analysis

Consider the reciprocal trigonometric functions. Answer the following:

Sinusoidal Combination

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

Sinusoidal Function Transformation Analysis

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

Sinusoidal Transformation and Logarithmic Manipulation

An electronic signal is modeled by $$S(t)=5*\sin(3*(t-2))$$ and its decay is described by $$D(t)=\ln

Solving a Basic Trigonometric Equation

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

Solving Trigonometric Equations in a Survey

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

Special Triangles and Unit Circle Coordinates

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

Tangent Function and Asymptotes

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

Tidal Patterns and Sinusoidal Modeling

A coastal engineer models tide heights (in meters) as a function of time (in hours) using the sinuso

Tide Height Model: Using Sine Functions

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

Transformation and Reflection of a Cosine Function

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

Trigonometric Inequality Solution

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

Unit Circle and Special Triangle Values

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

Using Trigonometric Sum and Difference Identities

Prove the identity $$\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$$ and apply it.

Verification and Application of Trigonometric Identities

Consider the sine addition identity $$\sin(\alpha+\beta)=\sin(\alpha)\cos(\beta)+\cos(\alpha)\sin(\b

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

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

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

Average Rate of Change in Parametric Motion

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

Circular Motion and Transformation

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

Circular Motion Parametrization

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

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

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

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 Applications in Area Computation

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

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)=

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

Ferris Wheel Motion

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

FRQ 5: Parametrically Defined Ellipse

An ellipse is described parametrically by $$x(t)=3*\cos(t)$$ and $$y(t)=2*\sin(t)$$ for $$t\in[0,2\p

FRQ 12: Matrix Multiplication in Transformation

Let matrices $$A=\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$$ and $$B=\begin{bmatrix}0 & 1\\1 & 0\end{

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 15: Composition of Linear Transformations

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

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

Graphical Analysis of Parametric Motion

A particle moves in the plane with its position defined by the functions $$x(t)= t^2 - 2*t$$ and $$y

Graphical and Algebraic Analysis of a Function with a Removable Discontinuity

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

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

Hyperbola Parametrization Using Trigonometric Functions

Consider the hyperbola defined by $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$. 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 Matrix

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

Inverse Matrix and Transformation of the Unit Square

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

Linear Parametric Motion Modeling

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

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

Linear Transformations in the Plane

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

Linear Transformations via Matrices

A linear transformation \(L\) in \(\mathbb{R}^2\) is defined by $$L(x,y)=(3*x- y, 2*x+4*y)$$. This t

Logarithmic and Exponential Parametric Functions

A particle’s position is defined by the parametric equations $$x(t)= \ln(1+t)$$ and $$y(t)= e^{1-t}$

Matrices as Models for Population Dynamics

A population of two species is modeled by the transition matrix $$P=\begin{pmatrix} 0.8 & 0.1 \\ 0.2

Matrix Modeling of Department Transitions

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

Matrix Representation of Linear Transformations

Consider the linear transformation defined by $$L(x,y)=(3*x-2*y, 4*x+y)$$.

Matrix Transformation of a Vector

Let the transformation matrix be $$A=\begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix},$$ and let the

Modeling Linear Motion Using Parametric Equations

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

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 Equations and Inverses

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

Parametric Function and Its Inverse: Parabolic Function

Consider the function $$f(x)= (x-1)^2 + 2$$ for x \(\ge\) 1. (a) Provide a parametrization for the

Parametric 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 Parabola

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

Parametric Representation of an Ellipse

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

Parametric Representation of an Implicit Curve

The equation $$x^2+y^2-6*x+8*y+9=0$$ defines a curve in the plane. Analyze this curve.

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 $$B=(7,8)$$.

Particle Motion Through Position and Velocity Vectors

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

Particle Motion with Quadratic Parametric Functions

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

Piecewise Function and Discontinuities

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

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

Reflection Transformation Using Matrices

A reflection over the line \(y=x\) in the plane can be represented by the matrix $$R=\begin{pmatrix}

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 Operations

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

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$