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

Application of the Binomial Theorem

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

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

Binomial Theorem Expansion

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

Break-even Analysis via Synthetic Division

A company’s cost model is represented by the polynomial function $$C(x) = x^3 - 6*x^2 + 11*x - 6$$,

Characterizing End Behavior and Asymptotes

A rational function modeling a population is given by $$R(x)=\frac{3*x^2+2*x-1}{x^2-4}$$. Analyze th

Composite Function Analysis in Environmental Modeling

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

Concavity and Inflection Points of a Polynomial Function

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

Continuous Piecewise Function Modification

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

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 Domain and Range of a Transformed Rational Function

Consider the function $$g(x)= \frac{x^2 - 9}{x-3}$$. Answer the following:

Determining Function Behavior from a Data Table

A function $$f(x)$$ is represented by the table below: | x | f(x) | |-----|------| | -3 | 10 |

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

Discontinuity Analysis in a Rational Function with High Degree

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

End Behavior of a Quartic Polynomial

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

Engineering 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:

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

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.

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 Simplification and Graph Analysis

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

Graph Analysis and Identification of Discontinuities

A function is defined by $$r(x)=\frac{(x-1)(x+1)}{(x-1)(x+2)}$$ and is used to model a physical phen

Graphical Interpretation of Inverse Functions from a Data Table

A table below represents selected values of a polynomial function $$f(x)$$: | x | f(x) | |----|---

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 Quartic Polynomial Function

Consider the quartic function $$f(x)= (x-1)^4 + 2$$. Answer the following questions concerning its i

Inverse Analysis of a Shifted Cubic Function

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

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

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+

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

Manufacturing Efficiency Polynomial Model

A company's manufacturing efficiency is modeled by a polynomial function. The function, given by $$P

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

Piecewise Financial Growth Model

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

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

Polynomial Long Division and Slant Asymptote

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

Polynomial Long Division and Slant 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

Population Growth Modeling with a Polynomial Function

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

Predator-Prey Dynamics as a Rational Function

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

Product Revenue Rational Model

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

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 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 Graph and Asymptote Identification

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

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 Analysis

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

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

Real-World Inverse Function: Temperature Conversion

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

Regression Model Selection for Experimental Data

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

Return to a Rational Expression under Transformation

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

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 +

Slant Asymptote Determination for a Rational Function

Determine the slant (oblique) asymptote of the rational function $$r(x)= \frac{2*x^2 + 3*x - 5}{x -

Transformation in Composite Functions

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

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

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.

Zeros and Factorization Analysis

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

Acoustics and the Logarithmic Scale

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

Analyzing a Logarithmic Function from Data

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

Arithmetic Sequence Analysis

Consider an arithmetic sequence with initial term $$a_0 = 5$$ and constant difference $$d$$. Given t

Cellular Data Usage Trend

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

Comparing Arithmetic and Exponential Models in Population Growth

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

Comparing Linear and Exponential Growth Models

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

Competing Exponential Cooling Models

Two models are proposed for the cooling of an object. Model A is $$T_A(t) = T_env + 30·e^(-0.5*t)$$

Composite 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 and Their Inverses

For the functions $$f(x) = 2^x$$ and $$g(x) = \log_2(x)$$, analyze their composite functions.

Composite Sequences: Combining Geometric and Arithmetic Models in Production

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

Composition and Transformation Functions

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

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.

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

Determining an Exponential Model from Data

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

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

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)

Environmental Pollution Decay

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

Exploring Logarithmic Scales: pH and Hydrogen Ion Concentration

In chemistry, the pH of a solution is defined by the relation $$pH = -\log([H^+])$$, where $$[H^+]$$

Exploring the Properties of Exponential Functions

Analyze the exponential function $$f(x)= 4 * 2^x$$.

Exponential Decay and Log Function Inverses in Pharmacokinetics

In a pharmacokinetics study, the concentration of a drug in a patient’s bloodstream is observed to d

Exponential Decay in Pollution Reduction

The concentration of a pollutant in a lake decreases exponentially according to the model $$f(t)= a\

Exponential Function Transformation

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

Exponential Growth from Percentage Increase

A process increases by 8% per unit time. Write an exponential function that models this growth.

Fitting a Logarithmic Model to Sales Data

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

Fractal Pattern Growth

A fractal pattern is generated such that after its initial creation, each iteration adds an area tha

Geometric Investment Growth

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

Inverse and Domain of a Logarithmic Transformation

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

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 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

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

Log-Exponential Hybrid Function and Its Inverse

Consider the function $$f(x)=\log_3(8*3^(x)-5)$$. Analyze its domain, prove its one-to-one property,

Logarithmic Analysis of Earthquake Intensity

The magnitude of an earthquake on the Richter scale is determined using a logarithmic function. Cons

Logarithmic 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 Function with Scaling and Inverse

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

Modeling Bacterial Growth with Exponential Functions

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

Natural Logarithms in Continuous Growth

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

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 and Logarithmic Functions

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

Piecewise Exponential and Logarithmic Function Discontinuities

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

Population Growth of Bacteria

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

Profit Growth with Combined Models

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

Radioactive Decay and Exponential Functions

A sample of a radioactive substance is monitored over time. The decay in mass is recorded in the tab

Radioactive Decay and Half-Life Estimation Through Data

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

Radioactive Decay Model

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

Real Estate Price Appreciation

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

Savings Account Growth: Arithmetic vs Geometric Sequences

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

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) + \

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 Effects on Exponential Functions

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

Transformations of Exponential Functions

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

Transformations of Exponential Functions

Consider the exponential function $$f(x) = 3 \cdot 2^x$$. This function is transformed to produce $$

Transformed Exponential Equation

Solve the exponential equation $$5 \cdot (1.2)^{(x-3)} = 20$$.

Traveling Sales Discount Sequence

A traveling salesman offers discounts on his products following a geometric sequence. The initial pr

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 Cotangent Function

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

Analysis of Rose Curves

A polar curve is given by the equation $$r=4*\cos(3*θ)$$ which represents a rose curve. Analyze the

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

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

Combining Logarithmic and Trigonometric Equations

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

Comparing Sinusoidal Functions

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

Composite Function Analysis with Polar and Trigonometric Elements

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

Conversion between Rectangular and Polar Coordinates

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

Conversion Between Rectangular and Polar Coordinates

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

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

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:

Graph Transformations: Sine and Cosine Functions

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

Graphing a Rose Curve

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

Graphing and 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.

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

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 Seasonal Temperature Data with Sinusoidal Functions

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

Modeling Tidal Patterns with Sinusoidal Functions

A coastal scientist studies tide levels at a beach that vary periodically. Using collected tide data

Multiple Angle Equation

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

Periodic Phenomena: Seasonal Daylight Variation

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

Phase Shift Analysis in Sinusoidal Functions

A sinusoidal function describing a physical process is given by $$f(\theta)=5*\sin(\theta-\phi)+2$$.

Phase Shift and Frequency Analysis

Analyze the function $$f(x)=\cos\Bigl(4\bigl(x-\frac{\pi}{8}\bigr)\Bigr)$$.

Polar Coordinates and Graphing a Circle

Answer the following questions on polar coordinates:

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

Probability and Trigonometry: Dartboard Game

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

Rate of Change in Polar Functions

For the polar function $$r(\theta)=4+\cos(\theta)$$, investigate its rate of change.

Roses and Limacons in Polar Graphs

Consider the polar curves described below and 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}{

Secant Function and Its Transformations

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

Sinusoidal Combination

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

Sinusoidal Data Analysis

An experimental setup records data that follows a sinusoidal pattern. The table below gives the disp

Sinusoidal Function and Its Inverse

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

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 a Trigonometric Equation

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

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

Tangent Function and Asymptotes

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

Transformation and Reflection of a Cosine Function

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

Trigonometric Identities and Sum Formulas

Trigonometric identities are important for simplifying expressions that arise in wave interference a

Understanding Coterminal Angles Through Art Installation

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

Using Trigonometric Sum and Difference Identities

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

Verification and Application of Trigonometric Identities

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

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

Advanced Matrix Modeling in Economic Transitions

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

Analysis of a Function with Trigonometric Components and Discontinuities

Examine the function $$f(\theta)=\begin{cases} \frac{1-\cos(\theta)}{\theta} & \text{if } \theta \ne

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$

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 a Piecewise Function Representing a Linear Transformation

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

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

Composite Transformations in the Plane

Consider two linear transformations in $$\mathbb{R}^2$$: a rotation by 90° counterclockwise and a re

Composition of Linear Transformations

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

Composition of Linear Transformations

Let $$L_1: \mathbb{R}^2 \to \mathbb{R}^2$$ be defined by $$L_1(x,y)=(x+y,\,2x-y)$$ and $$L_2: \mathb

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

Discontinuity Analysis in a Function Modeling Particle Motion

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

Discontinuity Analysis in an Implicitly Defined Function

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

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

Exponential Decay Modeled by Matrices

Consider a system where decay over time is modeled by the matrix $$M(t)= e^{-k*t}I$$, where I is the

FRQ 3: Linear Parametric Motion - Car Journey

A car travels along a linear path described by the parametric equations $$x(t)=3+2*t$$ and $$y(t)=4-

FRQ 6: Implicit Function to Parametric Representation

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

FRQ 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 10: Unit Vectors and Direction

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

FRQ 11: Matrix Inversion and Determinants

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

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 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

FRQ 18: Dynamic Systems and Transition Matrices

Consider a transition matrix modeling state changes given by $$M=\begin{bmatrix}0.7 & 0.3\\0.4 & 0.6

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

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

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 Matrix with a Parameter

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

Linear Transformation and Area Scaling

Consider the linear transformation L on \(\mathbb{R}^2\) defined by the matrix $$A= \begin{pmatrix}

Linear Transformation Evaluation

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

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

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 Multiplication and Non-Commutativity

Let the matrices be defined as $$A=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$$ and $$B=\begin{pma

Matrix Multiplication Exploration

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

Modified Circular Motion: Transformation Effects

Consider the parametric equations $$x(t)=2+4\cos(t)$$ and $$y(t)=-3+4\sin(t)$$ which describe a curv

Parabolic and Elliptical Parametric Representations

A parabola is given by the equation $$y=x^2-4*x+3$$.

Parametric Curve with Logarithmic and Exponential Components

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

Parametric Equations of an Ellipse

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

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

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 an Ellipse

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

Parametrizing a Linear Path: Car Motion

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

Parametrizing a Parabola

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

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

Properties of a Parametric Curve

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

Resolving Discontinuities in an Elliptical Parameterization

An ellipse is parameterized by the following equations: $$x(\theta)=\begin{cases} 5\cos(\theta) & \t

Transformation Matrices in Computer Graphics

A transformation matrix $$A = \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$$ is applied to points in

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 \

Transition Matrix in Markov Chains

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

Uniform Circular Motion

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

Vector Components and Magnitude

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

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$