AP Precalculus FRQ Room

Analysis of a Quartic Function

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

Analysis of a Rational Function with Factorable Denominator

A function is given by $$f(x)=\frac{x^2-5*x+6}{x^2-4}$$. Examine its domain and discontinuities.

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

Analysis of Removable Discontinuities in an Experiment

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

Analyzing a Rational Function with Asymptotes

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

Analyzing End Behavior of a Polynomial

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

Average Rate of Change and Tangent Lines

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

Average Rate of Change of a Rational Function

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

Binomial Theorem Expansion

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

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|)

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 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 Function Inverse Analysis

Consider the cubic function $$f(x) = x^3 - 6*x^2 + 9*x$$. Answer the following questions related to

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 Degree from Discrete Data

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

Determining Polynomial Degree from Finite Differences

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

Discontinuity Analysis in a Rational Function with High Degree

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

Engineering Curve Analysis: Concavity and Inflection

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

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:

Expanding a Binomial: Application of the Binomial Theorem

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

Exploring End Behavior and Leading Coefficients

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

Exploring Symmetry in Polynomial Functions

Let $$f(x)= x^4-5*x^2+4$$.

Exploring the Effect of Multiplicities on Graph Behavior

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

Factoring and Zero Multiplicity

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

Graph Interpretation and Log Transformation

An experiment records the reaction time R (in seconds) of an enzyme as a power function of substrate

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

Inverse Analysis of a Reciprocal Function

Consider the function $$f(x)= \frac{1}{x+2} + 3$$. 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 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

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 Equation Solving in a Financial Model

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

Logarithmic Linearization in Exponential Growth

An ecologist is studying the growth of a bacterial population in a laboratory experiment. The popula

Model Interpretation: End Behavior and Asymptotic Analysis

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

Modeling with Rational Functions

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

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

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 Construction for Utility Rates

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

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\

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 Division in Limit Evaluation

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

Polynomial End Behavior and Zeros Analysis

A polynomial function is given by $$f(x)= 2*x^4 - 3*x^3 - 12*x^2$$. This function models a physical

Polynomial Long Division and Slant Asymptote

Consider the rational function $$F(x)= \frac{x^3 + 2*x^2 - 5*x + 1}{x - 2}$$. Answer the following p

Polynomial Long Division and Slant Asymptotes

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

Projectile Motion Analysis

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

Quadratic Function Inverse Analysis with Domain Restriction

Consider the function $$f(x) = x^2 - 4*x + 5$$. Assume that the domain of $$f$$ is restricted so tha

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 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:

Real-World Inverse Function: Temperature Conversion

The function $$f(x)= \frac{9}{5}*x + 32$$ converts a temperature in degrees Celsius to degrees Fahre

Real-World Modeling: Population Estimation

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

Regression Model Selection for Experimental Data

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

Revenue Function Transformations

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

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 a Polynomial Inequality

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

Solving a System of Equations: Polynomial vs. Rational

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

Temperature Rate of Change Analysis

In a manufacturing process, the temperature in a reactor is recorded over time. Using the table prov

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 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

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 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

Arithmetic Sequence Derived from Logarithms

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

Arithmetic Sequence in Savings

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

Bacterial Growth Model and Inverse Function

A bacterial culture grows according to the function $$f(x)=500*2^(x/3)$$, where $$x$$ is time in hou

Bacterial Growth Modeling

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

Bacterial Growth: Arithmetic vs Exponential Models

A laboratory study records the growth of a bacterial culture at regular one‐hour intervals. The data

Base Transformation and End Behavior

Consider the functions \(f(x)=2^{x}\) and \(g(x)=5\cdot2^{(x+3)}-7\). (a) Express the function \(f(

Cellular Data Usage Trend

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

Comparing Linear and Exponential Revenue Models

A company is forecasting its revenue growth using two models based on different assumptions. Initial

Composite Function Analysis: Identity and Inverses

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

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 Financial Growth

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

Connecting Exponential Functions with Geometric Sequences

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

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 Restrictions in Logarithmic Functions

Consider the logarithmic function $$f(x) = \log_4(x^2 - 9)$$.

Economic Inflation Model Analysis

An economist proposes a model for the inflation rate given by R(t) = A · ln(B*t + C) + D, where R(t)

Experimental Data Modeling Using Semi-Log Plots

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

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 Growth in a Bacterial Culture

A bacterial culture grows according to the model $$P(t) = P₀ · 2^(t/3)$$, where t (in hours) is the

Exponential Inequality Solution

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

Finding Terms in a Geometric Sequence

A geometric sequence is known to satisfy $$g_3=16$$ and $$g_7=256$$.

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}(

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 Construction

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

Geometric Sequence in Compound Interest

An investment grows according to a geometric sequence. The initial investment is $$1000$$ dollars an

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

Inverse Functions of Exponential and Logarithmic Forms

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

Inverse of a Composite Function

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

Inverse Relationships in Exponential and Logarithmic Functions

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

Investment Growth: Compound Interest

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

Investment Scenario Convergence

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

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 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

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

pH and Logarithmic Functions

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

pH Measurement and Inversion

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

Piecewise Exponential and Logarithmic Function Discontinuities

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

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 Demographics Model

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

Population Growth of Bacteria

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

Population Growth with an Immigration Factor

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

Profit Growth with Combined Models

A company's profit is modeled by a function that combines an arithmetic increase with exponential gr

Radioactive Decay 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 Logarithmic Equations and Checking Domain

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

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

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

Using Exponential Product Property in Function Analysis

Consider the function $$f(x)= 3^x * 2^{2x}.$$

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$

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 Sinusoidal Function Rate of Change

A sound wave is modeled by the function $$f(t)=4*\sin(\frac{\pi}{2}*(t-1))+5$$, where t is measured

Analyzing the Tangent Function

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

Application of Trigonometric Sum Identities

Utilize trigonometric sum identities to simplify and solve expressions.

Average Rate of Change in a Polar Function

Given the polar function $$r(\theta) = 5*\sin(2*\theta) + 7$$ over the interval $$\theta \in [0, \fr

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 $$r=1+\cos(\theta)$$.

Combining Logarithmic and Trigonometric Equations

Consider a model where the amplitude of a cosine function is modulated by an exponential decay. The

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

Comparing Sinusoidal Functions

Consider the functions $$f(x)=\sin(x)$$ and $$g(x)=\cos\Bigl(x-\frac{\pi}{2}\Bigr)$$.

Conversion Between Rectangular and Polar Coordinates

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

Conversion Between Rectangular and Polar Coordinates

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

Converting and Graphing Polar Equations

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

Converting Complex Numbers to Polar Form

Convert the complex number $$3-3*\text{i}$$ to polar form and use this representation to compute the

Coterminal Angles and Unit Circle Analysis

Identify coterminal angles and determine the corresponding coordinates on the unit circle.

Daily Temperature Fluctuations

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

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 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:

Graphical Analysis of a Periodic Function

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

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 $$r=3*\cos(2\theta)$$.

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:

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 $$f(x) = \arctan(\tan(x))$$.

Inverse Trigonometric Function Analysis

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

Inverse Trigonometric Function Analysis

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

Inverse Trigonometric Functions in Navigation

A navigation system uses inverse trigonometric functions to determine heading angles. Answer the fol

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 Seasonal Temperature Data with Sinusoidal Functions

A sinusoidal pattern is observed in average monthly temperatures. Refer to the provided temperature

Pendulum Motion and Periodic Phenomena

A pendulum's angular displacement from the vertical is observed to follow a periodic pattern. Refer

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 Circle Graph

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

Polar Coordinates: Converting and Graphing

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

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 $$r=4*\cos(\theta)+3$$.

Rate of Change in Polar Functions

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

Reciprocal Trigonometric Functions

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

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}{

Secant Function and Its Transformations

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

Sine and Cosine Graph Transformations

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

Sinusoidal Function Transformation Analysis

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

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 $$2*\cos(\theta) - 1 = 0$$ for $$\theta$$ in the interval $$[0, 2\p

Solving a Trigonometric Equation with Sum and Difference Identities

Solve the equation $$\sin\left(x+\frac{\pi}{6}\right)=\cos(x)$$ for $$0\le x<2\pi$$.

Solving a Trigonometric Inequality

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

Solving Trigonometric Equations

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

Solving Trigonometric Equations in a Specified Interval

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

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

Tidal Patterns and Sinusoidal Modeling

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

Transformation and Reflection of a Cosine Function

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

Transformations of Inverse Trigonometric Functions

Analyze the inverse trigonometric function $$g(x)=\arccos(x)$$ and its transformation into $$h(x)=2-

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

Verification and Application of Trigonometric Identities

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

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 Involving Absolute Value and Removability

Consider the function $$F(x)=\begin{cases} \frac{|x-2|(x+1) - (x-2)(x+1)}{x-2} & \text{if } x \neq 2

Analyzing the Composition of Two Matrix Transformations

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

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 Transformations and Inverses

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

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 Area of a Parallelogram

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

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

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

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

Ferris Wheel Motion

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

Finding Angle Between Vectors

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

FRQ 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 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 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 19: Parametric Functions and Matrix Transformation

A particle's motion is given by the parametric equations $$f(t)=(t, t^2)$$ for $$t\in[0,2]$$. A line

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

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:

Implicitly Defined Circle

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

Inverse Matrix with a Parameter

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

Inverses and Solving a Matrix Equation

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

Investigating a Rational Piecewise Function with a Jump Discontinuity

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

Linear Transformation Evaluation

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

Matrices as Models for Population Dynamics

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

Matrix Applications in State Transitions

In a system representing transitions between two states, the following transition matrix is used: $

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 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 Linear Motion Using Parametric Equations

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

Parametric Curve with Logarithmic and Exponential Components

A curve is described by the parametric equations $$x(t)= t + \ln(t)$$ and $$y(t)= e^{t} - 3$$ for t

Parametric 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 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

A parabola is given by the equation $$y=x^2-2*x+1$$. A parametric representation for this parabola i

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

Parametrization of an Ellipse

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

Particle Motion from Parametric Equations

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

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$

Population Transition Matrix Analysis

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

Position and Velocity in Vector-Valued Functions

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

Projectile Motion: Parabolic Path

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

Transition Matrices in Dynamic Models

A system with two states is modeled by the transition matrix $$T=\begin{bmatrix}0.8 & 0.3\\ 0.2 & 0.

Transition Matrix and State Changes

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

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 Operations

Given the vectors $$u=\langle 3, -2 \rangle$$ and $$v=\langle -1, 4 \rangle$$, (a) Compute the magn

Vector Operations and Dot Product

Let $$\mathbf{u}=\langle 3,-1 \rangle$$ and $$\mathbf{v}=\langle -2,4 \rangle$$. Use these vectors t

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 Operations in the Plane

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