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 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 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$

Analyzing End Behavior of a Polynomial

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

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

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 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

Constructing a Function Model from Experimental Data

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

Constructing a Piecewise Function from Data

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

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

Data Analysis with Polynomial Interpolation

A scientist measures the decay of a radioactive substance at different times. The following table sh

Degree Determination from Finite Differences

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

Designing a Piecewise Function for a Temperature Model

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

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

End Behavior of a Quartic Polynomial

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

Engineering Application: Stress Analysis Model

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

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 Polynomial Function Behavior

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

Finding and Interpreting Inflection Points

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

Function Model Construction from Data Set

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

Function Simplification and Graph Analysis

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

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

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 Reciprocal Function

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

Inverse Analysis of an Even Function with Domain Restriction

Consider the function $$f(x)=x^2$$ defined on the restricted domain $$x \ge 0$$. Answer the followin

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

Investigation of Refund Policy via Piecewise Continuous Functions

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

Linear Function Inverse Analysis

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

Loan Payment Model using Rational Functions

A bank uses the rational function $$R(x) = \frac{2*x^2 - 3*x - 5}{x - 2}$$ to model the monthly inte

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

Manufacturing Efficiency Polynomial Model

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

Modeling with Inverse Variation: A Rational Function

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

Modeling with Rational Functions

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

Optimizing Production Using a Polynomial Model

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

Piecewise Financial Growth Model

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

Piecewise Function without a Calculator

Let the function $$f(x)=\begin{cases} x^2-1 & \text{for } x<2, \\ \frac{x^2-4}{x-2} & \text{for } x\

Polynomial 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 Interpolation and Curve Fitting

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

Polynomial Long Division and Slant Asymptote

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

Product Revenue Rational Model

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

Projectile Motion Analysis

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

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: Machine Efficiency Ratios

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

Revenue Function Transformations

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

Roller Coaster Curve Analysis

A roller coaster's vertical profile is modeled by the polynomial function $$f(x)= -0.05*x^3 + 1.2*x^

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 -

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.

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 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

Trigonometric Function Analysis and Identity Verification

Consider the trigonometric function $$g(x)= 2*\tan(3*x-\frac{\pi}{4})$$, where $$x$$ is measured in

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

Zero Finding and Sign Charts

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

Analyzing a Logarithmic Function

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

Analyzing Social Media Popularity with Logarithmic Growth

A social media analyst is studying the early-stage growth of a new account's followers. Initially, t

Arithmetic Savings Plan

A person decides to save money every month, starting with an initial deposit of $$50$$ dollars, with

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

In a laboratory experiment, a bacteria colony doubles every 3 hours. The initial count is $$500$$ ba

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(

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 Functions and Their Inverses

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

Composite Functions with Exponential and Logarithmic Elements

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

Composition of Exponential and Log Functions

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

Composition of Exponential and Logarithmic Functions

Consider the functions $$f(x)= \log_5\left(\frac{x}{2}\right)$$ and $$g(x)= 10\cdot 5^x$$. Answer th

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

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 Magnitude and Logarithms

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

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

Exponential Decay in Pollution Reduction

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

Exponential Function Transformations

Given the exponential function f(x) = 4ˣ, describe the transformation that produces the function g(x

Exponential Function with Compound Transformations and Its Inverse

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

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 $$g_7=256$$.

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

Inverse Functions in Exponential Contexts

Consider the function $$f(x)= 5^x + 3$$. Analyze its inverse function.

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

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 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

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:

Natural Logarithms in Continuous Growth

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

pH and Logarithmic Functions

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

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,

Population Growth Inversion

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

Population Growth Modeling with Exponential Functions

A small town has its population recorded every 5 years, as shown in the table below: | Year | Popul

Population Growth with an Immigration Factor

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

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 and Logarithmic Inversion

A radioactive substance decays such that its mass halves every 8 years. At time \(t=0\), the substan

Radioactive Decay Model

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

Radioactive Decay Modeling

A radioactive substance decays according to the model N(t) = N₀ · e^(-k*t), where t is measured in y

System of Exponential Equations

Solve the following system of equations: $$2\cdot 2^x + 3\cdot 3^y = 17$$ $$2^x - 3^y = 1$$.

Transformation of an Exponential Function

Consider the basic exponential function $$f(x)= 2^x$$. A transformed function is given by $$g(x)= 3\

Transformation of Exponential Functions

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

Transformations in Logarithmic Functions

Given \(f(x)=\log_{3}(x)\), consider the transformed function \(g(x)=-2\log_{3}(2x-6)+4\). (a) Dete

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 $$f(x)= 7 * e^{0.3x}$$. Investigate its transformations.

Transformed Exponential Equation

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

Translated Exponential Function and Its Inverse

Consider the function $$f(x)=5*2^(x+3)-8$$. Analyze its properties by confirming its one-to-one natu

Tumor Growth with Time Dilation Effects

A medical researcher is studying the growth of a tumor, which is modeled by the exponential function

Weekly Population Growth Analysis

A species exhibits exponential growth in its weekly population. If the initial population is $$2000$

Analysis of a Limacon

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

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 Phase Shifts in Sinusoidal Functions

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

Application of Trigonometric Sum Identities

Utilize trigonometric sum identities to simplify and solve expressions.

Applying Sine and Cosine Sum Identities in Modeling

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

Cardioid Polar Graphs

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

Comparing Sinusoidal Function Models

Two models for daily illumination intensity are given by: $$I_1(t)=6*\sin\left(\frac{\pi}{12}(t-4)\r

Composite Function Analysis with Polar and Trigonometric Elements

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

Conversion between Rectangular and Polar Coordinates

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

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 the Unit Circle

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

Coterminal Angles and Trigonometric Evaluations

Consider the angle $$750^\circ$$. Using properties of coterminal angles and the unit circle, answer

Coterminal Angles and Unit Circle Analysis

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

Damped Oscillations: Combining Sinusoidal Functions and Geometric Sequences

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

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 a Limacon

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

Exploring Rates of Change in Polar Functions

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

Extracting Sinusoidal Parameters from Data

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

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

Graphing the Tangent Function with Asymptotes

The tangent function, $$f(\theta) = \tan(\theta)$$, exhibits vertical asymptotes where it is undefin

Graphing the Tangent Function with Asymptotes

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

Identity Verification

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

Interpreting a Sinusoidal Graph

The graph provided displays a function of the form $$g(\theta)=a\sin[b(\theta-c)]+d$$. Use the graph

Interpreting Trigonometric Data Models

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

Inverse Trigonometric Analysis

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

Inverse Trigonometric Function Analysis

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

Inverse Trigonometric Functions in Navigation

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

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

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

Modeling Tidal Patterns with Sinusoidal Functions

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

Period Detection and Frequency Analysis

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

Periodic Temperature Variation Model

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

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 Coordinates: Converting and Graphing

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

Polar Rose Analysis

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

Polar to Cartesian Conversion for a Circle

Consider the polar equation $$r=6\cos(\theta)$$.

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

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.

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}{

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 Transformations

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

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 Trigonometric Equations

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

Special Triangles and Unit Circle Coordinates

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

Tidal Patterns and Sinusoidal Modeling

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

Transformations of Inverse Trigonometric Functions

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

Transformations of Sinusoidal Functions

Consider the function $$y = 3*\sin(2*(x - \pi/4)) - 1$$. Answer the following:

Trigonometric Identities and Sum Formulas

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

Verification and Application of Trigonometric Identities

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

Vibration Analysis

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

Acceleration in a Vector-Valued Function

Given the particle's position vector $$\mathbf{r}(t) = \langle t^2, t^3 - 3*t \rangle$$, answer the

Analysis of Vector Directions and Transformations

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

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 Parametrization

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

Complex Parametric and Matrix Analysis in Planar Motion

A particle moves in the plane according to the parametric equations $$x(t)=3\cos(t)+2*t$$ and $$y(t)

Converting an Explicit Function to Parametric Form

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

Determinant and Inverse Calculation

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

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^

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

Dot Product, Projection, and Angle Calculation

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

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 }

Evaluating Limits in a Parametrically Defined Motion Scenario

A particle’s motion is given by the parametric equations: $$x(t)=\begin{cases} \frac{t^2-9}{t-3} & \

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

FRQ 1: Parametric Path and Motion Analysis

Consider the parametric function $$f(t)=(x(t),y(t))$$ defined by $$x(t)=t^2-4*t+3$$ and $$y(t)=2*t-1

FRQ 6: Implicit Function to Parametric Representation

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

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 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

Graphical and Algebraic Analysis of a Function with a Removable Discontinuity

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

Implicit Function Analysis

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

Implicitly Defined Circle

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

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 Function and Transformation Mapping

Given the function $$f(x)=\frac{x+2}{3}$$, analyze its invertibility and the relationship between th

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 Transformations in the Plane

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

Matrices as Representations of Rotation

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

Matrix Modeling in Population Dynamics

A biologist is studying a species with two age classes: juveniles and adults. The population dynamic

Matrix Multiplication and Linear Transformations

Consider the matrices $$A= \begin{pmatrix} 1 & 3 \\ 2 & -1 \end{pmatrix}$$ and $$B= \begin{pmatrix}

Matrix Representation of Linear Transformations

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

Matrix Transformation in Graphics

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

Matrix Transformation of a Vector

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

Modeling Particle Trajectory with Parametric Equations

A particle’s motion is described by the parametric equations $$x(t)=3*t+1$$ and $$y(t)=-2*t^2+8*t-1$

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

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)

Parabolic and Elliptical Parametric Representations

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

Parametric Equations and Inverses

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

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 Ellipse

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

Parametric Representation on the Unit Circle and Special Angles

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

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 $$

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

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

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

Position and Velocity Vectors

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

Projectile Motion: Parabolic Path

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

Properties of a Parametric Curve

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

Rate of Change Analysis in Parametric Motion

A particle’s movement is described by the parametric equations $$x(t)=t^3-6*t+4$$ and $$y(t)=2*t^2-t

Transformation Matrices in Computer Graphics

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

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 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.

Trigonometric Function Analysis

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

Vector Addition and Scalar Multiplication

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

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