7. Functions and Graphs in Algebra A Complete Study Guide

f(x) = ax + b y = x² y = mx + c f(x) = 2x + 3 y = √x f(x) = x³ y = log(x) y = e^x Functions and Graphs in Algebra A Complete Study Guide High School Mathematics By [Author Name]   .

Index Functions and Graphs in Algebra  Chapter 1: Understanding Functions 4-6  Chapter 2: Linear Functions 7-9  Chapter 3: Quadratic Functions 10-12  Chapter 4: Exponential Functions 13-15  Chapter 5: Systems of Equations 16-18  Chapter 6: Function Transformations 19- 21  Exercises 22-23  Answer Key 24 Page 2.

Introduction Welcome to the World of Functions and Graphs  Why Study Functions and Graphs? Algebra is the language of mathematics, and functions are its vocabulary. Understanding functions and their graphs opens doors to solving real-world problems, from predicting population growth to calculating financial investments.  What You'll Learn This book is designed specifically for high school students who want to master functions and graphs. We'll explore various types of functions including linear, quadratic, exponential, and more. Each chapter provides clear explanations, colorful illustrations, and practical examples.  Real-World Applications Functions aren't just abstract concepts—they're powerful tools used in science, engineering, economics, and everyday life. You'll discover how to apply what you learn to solve problems you might encounter in your future career or daily activities.  How to Use This Book Each chapter builds upon previous concepts, so we recommend studying them in order. Work through the examples carefully, try the exercises, and check your answers in the back. Remember, practice is the key to mastering algebra! Page 3.

 Chapter 1 Understanding Functions The Building Blocks of Algebra  What is a Function? A function is a special relationship between inputs and outputs where each input is related to exactly one output. Think of it as a rule that takes an input value and produces a corresponding output value. In mathematical notation, we write functions as f(x), where x is the input and f(x) is the output. Domain 1 2 3 4 f(x) = 2x Range 2 4 6 8  Domain and Range The domain of a function is the set of all possible input values (x-values) that can be used in the function. The range of a function is the set of all possible output values (y-values) that result from using the domain values.  Example For the function f(x) = x²: • Domain: All real numbers (any number can be squared) • Range: All non-negative real numbers (squares are never negative) Page 4.

 Chapter 1 Types of Functions Exploring Different Function Families  Linear Functions f(x) = mx + b Linear functions have a constant rate of change and graph as straight lines. The parameter m represents the slope, and b is the y-intercept.  Quadratic Functions f(x) = ax² + bx + c Quadratic functions form parabolas when graphed. They have a maximum or minimum point called the vertex, and they are symmetric about a vertical line.  Exponential Functions f(x) = a · bˣ Exponential functions grow or decay at a rate proportional to their current value. When b > 1, they show exponential growth; when 0 < b < 1, they show exponential decay.  Trigonometric Functions f(x) = sin(x), cos(x), tan(x) Trigonometric functions are periodic and describe repeating patterns like waves. They're fundamental in modeling oscillations, cycles, and circular motion. Page 5.

 Chapter 1 Function Notation Working with f(x) and Evaluating Functions  Understanding f(x) Function notation is a way to write functions that clearly shows the relationship between inputs and outputs. The notation f(x) represents the output of function f when the input is x. For example, if f(x) = 2x + 3, then f(4) = 2(4) + 3 = 11. This means that when the input is 4, the output is 11.  Evaluating Functions Given f(x) = 3x² - 2x + 5, find f(2) and f(-1): f(2) = 3(2)² - 2(2) + 5 = 3(4) - 4 + 5 = 12 - 4 + 5 = 13 f(-1) = 3(-1)² - 2(-1) + 5 = 3(1) + 2 + 5 = 3 + 2 + 5 = 10  Writing Function Equations To write a function equation, identify the relationship between input and output variables: If y is always 5 more than twice x, we can write this as a function: f(x) = 2x + 5  Practice Problems 1. If g(x) = 4x - 7, find g(3) and g(-2). 2. Given h(x) = x² + 3x - 2, calculate h(0) and h(5). 3. Write a function equation for: "The output is always 3 less than the square of the input." Page 6.

 Chapter 2 Linear Functions The Foundation of Algebraic Relationships  What are Linear Functions? Linear functions are the simplest type of functions, characterized by a constant rate of change. They form straight lines when graphed and have many real-world applications. y = mx + b General form of a linear function  Understanding Slope and Y-intercept In the equation y = mx + b: • m represents the slope - the rate of change or steepness of the line • b represents the y-intercept - where the line crosses the y-axis  Real-World Applications Linear functions model many everyday situations:  Distance-Time If a car travels at a constant speed of 60 mph, the distance traveled (d) after t hours is: d = 60t  Cost Functions A taxi charges $3 base fare plus $2 per mile: C = 2m + 3, where C is cost and m is miles Page 7.

 Chapter 2 Graphing Linear Functions Visualizing Linear Relationships  Methods for Graphing Linear Functions There are several effective methods to graph linear functions in the form y = mx + b:  Slope-Intercept Method 1 Plot the y-intercept (b) on the y-axis 2 Use the slope (m) to find additional points 3 Connect the points with a straight line  Table of Values Method 1 Create a table with x and y values 2 Choose several x values and calculate corresponding y values 3 Plot the points and connect with a line  Understanding Different Slopes The slope of a line determines its direction and steepness:  Positive Slope Lines with m > 0 rise from left to right  Negative Slope Lines with m < 0 fall from left to right  Zero Slope Lines with m = 0 are horizontal  Undefined Slope Vertical lines have undefined slope Page 8.

 Chapter 2 Slope and Intercept Understanding Rate of Change and Starting Values  Slope as Rate of Change The slope of a linear function represents the rate of change between variables. It tells us how much y changes for each unit increase in x. m = (y₂ - y₁)/(x₂ - x₁) Slope formula using two points (x₁, y₁) and (x₂, y₂)  Y-intercept as Starting Value The y-intercept (b) represents the starting value of the function when x = 0. In real-world contexts, it often represents an initial condition or fixed cost.  Real-World Applications Slope and intercept have practical meanings in many fields:  Economics In cost functions C = mx + b: • m = cost per unit (variable cost) • b = fixed cost (initial investment)  Physics In motion equations d = vt + d₀: • v = velocity (rate of change) • d₀ = initial position  Practice Problems 1. Find the slope of the line passing through points (2, 5) and (6, 13). What does this slope represent? 2. A taxi company charges a $3 flat fee plus $2.50 per mile. Write a linear function for the cost C in terms of miles m. Identify the slope and y-intercept and explain their meanings. 3. The temperature T (in °F) at altitude h (in feet) is given by T = -0.0035h + 70. What is the rate of change of temperature with respect to altitude? What is the temperature at sea level? Page 9.

 Chapter 3 Quadratic Functions Exploring Parabolic Relationships  What are Quadratic Functions? Quadratic functions are polynomial functions of degree 2 that create U-shaped curves called parabolas when graphed. They are used to model many real-world phenomena. y = ax² + bx + c Standard form of a quadratic function  Understanding the Parabolic Shape The coefficient a determines the direction and width of the parabola: a > 0 Parabola opens upward a < 0 Parabola opens downward |a| > 1 Narrower parabola  Real-World Applications Quadratic functions model many everyday situations:  Projectile Motion The height of a thrown ball follows a quadratic path: h(t) = -16t² + v₀t + h₀, where h is height, t is time, v₀ is initial velocity, and h₀ is initial height.  Area Problems The area of a rectangle with fixed perimeter but varying dimensions follows a quadratic function, allowing us to find maximum area. Page 10 Vertex Axis of Symmetry.

 Chapter 3 Graphing Quadratic Functions Key Elements and Techniques  Vertex Form The vertex form of a quadratic function makes it easy to identify the vertex and axis of symmetry: y = a(x - h)² + k Vertex: (h, k) | Axis of symmetry: x = h  Graphing Steps  Find Key Points 1 Identify the vertex (h, k) 2 Find the y-intercept (when x = 0) 3 Find x-intercepts (roots) if they exist 4 Plot additional points if needed  Finding Roots 1 Factoring: Express as (x - r₁)(x - r₂) = 0 2 Completing the square: Convert to vertex form 3 Quadratic formula: x = (-b ± √(b²-4ac))/2a  Effect of Coefficient 'a' The coefficient 'a' affects the width and direction of the parabola:  a > 0 When a is positive, the parabola opens upward with a minimum point at the vertex.  a < 0 When a is negative, the parabola opens downward with a maximum point at the vertex.  |a| > 1 When |a| > 1, the parabola is narrower than the standard y = x².  0 < |a| < 1 When 0 < |a| < 1, the parabola is wider than the standard y = x². Page 11.

 Chapter 3 Applications of Quadratic Functions Real-World Problem Solving  Real-World Applications Quadratic functions are powerful tools for modeling various real-world situations. Let's explore some common applications:  Projectile Motion The height of an object thrown or launched follows a quadratic path. The equation h(t) = -16t² + v₀t + h₀ models the height h at time t, where v₀ is initial velocity and h₀ is initial height.  Optimization Problems Quadratic functions help find maximum or minimum values in business, economics, and engineering. The vertex of a parabola represents the optimal point in these scenarios.  Area Problems When working with geometric shapes where dimensions vary but some aspects remain constant (like perimeter), the area often follows a quadratic relationship.  Example: Projectile Motion 1 A ball is thrown upward with an initial velocity of 48 ft/s from a height of 4 ft. The height h (in feet) after t seconds is given by: h(t) = -16t² + 48t + 4 2 To find the maximum height, we need to find the vertex. The t-coordinate of the vertex is t = -b/(2a) = -48/(2×-16) = 1.5 seconds 3 Now substitute t = 1.5 into the equation: h(1.5) = -16(1.5)² + 48(1.5) + 4 = -36 + 72 + 4 = 40 feet 4 The ball reaches a maximum height of 40 feet after 1.5 seconds.  Practice Problems 1. A farmer has 100 meters of fencing and wants to enclose a rectangular area along a river (no fencing needed along the river). What dimensions will maximize the area? 2. The profit P (in dollars) from selling x items is given by P(x) = -x² + 60x - 500. How many items should be sold to maximize profit, and what is the maximum profit? 3. A rocket is launched upward with an initial velocity of 80 ft/s from a platform 20 ft above the ground. When will the rocket reach its maximum height and what is that height? Page 12.

 Chapter 4 Exponential Functions Understanding Growth and Decay Patterns  What are Exponential Functions? Exponential functions are mathematical functions where the variable appears in the exponent. They model situations with constant percentage growth or decay rates. y = a · bˣ General form of an exponential function  Growth vs. Decay The base b determines whether the function represents growth or decay:  Exponential Growth When b > 1, the function represents exponential growth. The value increases rapidly as x increases.  Exponential Decay When 0 < b < 1, the function represents exponential decay. The value decreases rapidly as x increases.  Real-World Applications Exponential functions model many natural and financial phenomena:  Population Growth Populations often grow exponentially when resources are abundant. The formula P(t) = P₀ · eʳᵗ models population size P at time t, where P₀ is initial population and r is growth rate.  Compound Interest Money invested at compound interest grows exponentially. The formula A = P(1 + r/n)ⁿᵗ calculates the amount A after t years, where P is principal, r is annual rate, and n is compounding frequency.  Radioactive Decay Radioactive substances decay exponentially. The amount remaining is given by N(t) = N₀ · e⁻ᵏᵗ, where N₀ is initial amount and k is the decay constant. Page 13.

 Chapter 4 Graphing Exponential Functions Visualizing Growth and Decay Patterns  Graphing Steps To graph an exponential function y = a · bˣ, follow these steps:  Create a Table • Choose several x- values (both positive and negative) • Calculate corresponding y-values • Include x = 0 to find the y- intercept  Plot Points • Plot the points from your table • Connect them with a smooth curve • Extend the curve following the pattern  Analyze Features • Identify the horizontal asymptote • Note the y- intercept • Observe the behavior as x → ±∞  Key Characteristics Exponential graphs have distinctive features that set them apart:  Horizontal Asymptote Exponential functions approach but never reach a horizontal line. For y = a·bˣ, the horizontal asymptote is y = 0.  Y-intercept The y-intercept occurs when x = 0. For y = a·bˣ, the y-intercept is (0, a), since b⁰ = 1.  Growth Behavior For b > 1: as x → ∞, y → ∞; as x → -∞, y → 0. The curve rises rapidly to the right.  Decay Behavior For 0 < b < 1: as x → ∞, y → 0; as x → -∞, y → ∞. The curve falls rapidly to the right.  Comparing Function Types Linear: y = 2x + 1 Quadratic: y = x² Exponential: y = 2ˣ Page 14.

 Chapter 4 Applications of Exponential Functions Real-World Problem Solving  Real-World Applications Exponential functions model many natural and financial phenomena. Let's explore some key applications:  Compound Interest Money grows exponentially when interest is compounded. The formula A = P(1 + r/n)ⁿᵗ calculates the future value, where P is principal, r is annual rate, n is compounding frequency, and t is time in years. A = P(1 + r/n)ⁿᵗ  Population Growth Populations grow exponentially under ideal conditions. The formula P(t) = P₀eʳᵗ models population size at time t, where P₀ is initial population and r is growth rate. P(t) = P₀eʳᵗ  Radioactive Decay Radioactive substances decay exponentially. The formula N(t) = N₀e⁻ᵏᵗ calculates remaining amount, where N₀ is initial amount and k is decay constant. N(t) = N₀e⁻ᵏᵗ  Disease Spread Infectious diseases often spread exponentially in early stages. The formula I(t) = I₀(1 + r)ᵗ models number of infected people, where I₀ is initial cases and r is transmission rate. I(t) = I₀(1 + r)ᵗ  Example: Compound Interest 1 If you invest $5,000 at an annual interest rate of 4.5% compounded monthly, how much will you have after 10 years? 2 Identify values: P = $5,000, r = 0.045, n = 12 (monthly), t = 10 3 Substitute into formula: A = 5000(1 + 0.045/12)¹²ˣ¹⁰ 4 Calculate: A = 5000(1.00375)¹²⁰ ≈ 5000(1.5669) ≈ $7,834.50 5 After 10 years, the investment will grow to approximately $7,834.50  Practice Problems 1. A bacteria culture doubles every 3 hours. If there are initially 200 bacteria, how many will there be after 24 hours? 2. The half-life of Carbon-14 is 5,730 years. If a fossil contains 25% of the original Carbon-14, how old is the fossil? 3. A city's population is growing at 2.5% per year. If the current population is 85,000, what will it be in 15 years? Page 15.

 Chapter 5 Systems of Equations Solving Multiple Equations Simultaneously  What are Systems of Equations? A system of equations is a set of two or more equations with multiple variables that must be satisfied simultaneously. The solution to a system is the set of values that makes all equations true at the same time.  Types of Solutions Systems of linear equations can have three types of solutions:  One Solution Lines intersect at exactly one point. This happens when the lines have different slopes.  No Solution Lines are parallel and never intersect. This happens when lines have the same slope but different y- intercepts.  Infinite Solutions Lines coincide completely. This happens when lines have the same slope and same y-intercept.  Real-World Applications Systems of equations model many everyday situations:  Mixture Problems Finding the right combination of ingredients or solutions with different concentrations or costs. x + y = 100 0.20x + 0.50y = 35  Rate Problems Determining speeds, distances, or times when two objects are moving toward or away from each other. d₁ + d₂ = 300 t₁ = t₂  Break- Even Analysis Finding when revenue equals cost to determine the point of profitability in business. R(x) = C(x) px = mx + b Page 16.

 Chapter 5 Solving Systems Graphically Finding Solutions Through Visualization  Graphical Method The graphical method involves plotting each equation on the same coordinate plane and identifying the point(s) of intersection, which represent the solution(s) to the system.  Step-by-Step Process 1 Rewrite each equation in slope-intercept form (y = mx + b) 2 Graph each equation on the same coordinate plane 3 Identify the point(s) where the graphs intersect 4 Write the intersection point(s) as the solution (x, y)  Types of Solutions When solving systems graphically, you'll encounter three possible scenarios:  One Solution Lines intersect at exactly one point. The system has a unique solution at the intersection point.  No Solution Lines are parallel and never intersect. The system is inconsistent with no solution.  Infinite Solutions Lines coincide completely. Every point on the line is a solution to the system.  Limitations of Graphical Method • Precision issues - Difficult to determine exact solutions when intersection points have fractional coordinates • Scale limitations - Very large or very small values may not fit on a standard graph • Time-consuming - Creating accurate graphs takes more time than algebraic methods  Practice Problems 1. Solve the system graphically: y = 2x + 3 and y = -x + 6 2. Solve the system graphically: y = 3x - 2 and y = 3x + 4 3. Solve the system graphically: 2x + y = 5 and 4x + 2y = 10 Page 17.

 Chapter 5 Solving Systems Algebraically Precise Methods for Finding Solutions  Algebraic Methods While graphical methods provide visual understanding, algebraic methods offer precise solutions even for complex systems. Let's explore three key algebraic approaches:  Substitution Method 1 Solve one equation for one variable 2 Substitute this expression into the other equation 3 Solve for the remaining variable 4 Substitute back to find the other variable  Example For the system: 2x + y = 7 and x - y = 2 x = y + 2 → 2(y + 2) + y = 7 → 3y + 4 = 7 → y = 1 x = 1 + 2 = 3 → Solution: (3, 1)  Elimination Method 1 Arrange equations with like terms aligned 2 Multiply equations to make coefficients of one variable opposites 3 Add equations to eliminate that variable 4 Solve for the remaining variable and substitute back  Example For the system: 3x + 2y = 11 and 2x - 2y = 6 3x + 2y = 11 + (2x - 2y = 6) 5x = 17 → x = 3.4 3(3.4) + 2y = 11 → 10.2 + 2y = 11 → y = 0.4  Matrix Method 1 Write the system as an augmented matrix 2 Use row operations to achieve row-echelon form 3 Continue to reduced row-echelon form 4 Read the solution directly from the matrix  Example For the system: 2x + 3y = 8 and x - y = 1 [2 3 | 8] [1 -1 | 1] After row operations: [1 0 | 2.2] [0 1 | 1.2] → Solution: (2.2, 1.2)  Choosing the Most Efficient Method Substitution Best when one equation is already solved for a variable or easily solvable for one variable Elimination Ideal when coefficients of one variable are already opposites or can easily be made opposites Matrix Most efficient for larger systems (3+ variables) or when using technology  Practice Problems 1. Solve the system using substitution: 3x + y = 10 and y = 2x - 1 2. Solve the system using elimination: 4x + 3y = 14 and 2x - 3y = -2 3. Solve the system using any algebraic method: 2x + 5y = 16 and 3x - 2y = 5 Page 18.

 Chapter 6 Function Transformations Shifting, Stretching, and Reflecting Graphs  What are Function Transformations? Function transformations are operations that shift, stretch, compress, or reflect the graph of a function. They allow us to create new functions from basic ones by modifying their graphs in predictable ways. y = a·f(x - h) + k General form of a transformed function  Transformation Parameters Each parameter in the transformation equation affects the graph differently:  Vertical Stretch/Compression The parameter a affects vertical stretching and compression: • |a| > 1: Vertical stretch • 0 < |a| < 1: Vertical compression • a < 0: Reflection across x-axis  Horizontal Shift The parameter h affects horizontal position: • h > 0: Shift right by h units • h < 0: Shift left by |h| units • Note: The sign is opposite to the direction  Vertical Shift The parameter k affects vertical position: • k > 0: Shift up by k units • k < 0: Shift down by |k| units • The sign matches the direction  Reflections Reflections flip the graph across an axis: • y = -f(x): Reflection across x-axis • y = f(-x): Reflection across y-axis • y = -f(-x): Reflection across both axes  Real-World Applications Function transformations are useful in many fields:  Physics Modeling projectile motion with different initial heights and velocities using transformations of basic quadratic functions.  Data Analysis Adjusting mathematical models to fit real-world data by shifting and scaling basic functions to match observed patterns.  Engineering Designing structures and systems by transforming basic functions to create specific shapes and behaviors. Page 19.

 Chapter 6 Translations, Reflections, and Dilations Detailed Transformation Techniques  Translations (Shifts) Translations move the graph of a function without changing its shape. They can be horizontal or vertical.  Horizontal Shift f(x - h) shifts the graph right by h units f(x + h) shifts the graph left by h units  Vertical Shift f(x) + k shifts the graph up by k units f(x) - k shifts the graph down by k units  Example For f(x) = x², the function g(x) = (x - 3)² + 2 shifts the parabola 3 units right and 2 units up. Original: f(x) = x² Transformed: g(x) = (x - 3)² + 2  Reflections Reflections flip the graph of a function across an axis, creating a mirror image.  X-axis Reflection -f(x) reflects the graph across the x-axis All y-values change sign vertical_ Y-axis Reflection f(-x) reflects the graph across the y- axis All x-values change sign  Example For f(x) = √x, the function g(x) = -√x reflects the graph across the x-axis, and h(x) = √(-x) reflects it across the y- axis. Original: f(x) = √x X-reflection: g(x) = -√x Y-reflection: h(x) = √(-x)  Dilations (Stretches/Compressions) Dilations change the shape of a function by stretching or compressing it vertically or horizontally.  Vertical Dilation a·f(x) where |a| > 1 stretches vertically a·f(x) where 0 < |a| < 1 compresses vertically width Horizontal Dilation f(b·x) where |b| > 1 compresses horizontally f(b·x) where 0 < |b| < 1 stretches horizontally  Example For f(x) = sin(x), the function g(x) = 2sin(x) stretches vertically by a factor of 2, and h(x) = sin(2x) compresses horizontally by a factor of 2. Original: f(x) = sin(x) Vertical stretch: g(x) = 2sin(x) Horizontal compression: h(x) = sin(2x)  Practice Problems 1. If f(x) = x², describe the transformation applied to get g(x) = (x + 4)² - 3. 2. For the function f(x) = |x|, write the equation for the function reflected across the x-axis and shifted 2 units up. 3. If f(x) = √x, what transformation is applied to get g(x) = 3√(x - 1)? Page 20.

 Chapter 6 Applications of Function Transformations Real-World Uses of Transformation Techniques  Physics Function transformations model motion with different initial conditions. Projectile motion equations can be transformed to represent objects launched from different heights or with different velocities.  Projectile Motion The basic height function h(t) = -16t² transforms to h(t) = -16(t - t₀)² + h₀ + v₀t to account for initial time (t₀), height (h₀), and velocity (v₀). h(t) = -16(t - 1)² + 5 + 32t This represents a ball thrown upward at 32 ft/s from a height of 5 ft after 1 second delay.  Engineering Signal processing uses transformations to modify waveforms. Engineers apply vertical stretches, horizontal shifts, and reflections to filter, amplify, or modify signals for communication systems.  Signal Processing A basic sine wave f(t) = sin(t) transforms to g(t) = A·sin(B(t - C)) + D where: A = amplitude (vertical stretch) B = frequency (horizontal compression) C = phase shift (horizontal shift) D = vertical offset g(t) = 3·sin(2(t - π/4)) + 1  Economics Demand and supply curves in economics are often transformed to model changes in market conditions. Shifts represent changes in consumer preferences, production costs, or external factors.  Demand Curves A basic demand function D(p) = 100 - 2p transforms to D'(p) = D(p - h) + k to represent: h > 0: Increased production costs h < 0: Subsidies or improved technology k > 0: Increased consumer demand k < 0: Decreased consumer demand D'(p) = (100 - 2(p - 5)) + 20  Computer Graphics Computer graphics use function transformations to scale, rotate, and position objects. These transformations are fundamental to animation, video games, and computer-aided design.  3D Object Transformation A 3D object's vertices are transformed using matrix operations that combine: Scaling: Changing object size Rotation: Changing object orientation Translation: Moving object position A point (x,y,z) transforms to (x',y',z') through combined transformations. [x'] [sx 0 0 tx] [x] [y'] = [0 sy 0 ty] [y] [z'] [0 0 sz tz] [z]  Practice Problems 1. A sound wave is modeled by f(t) = sin(t). Write the transformed function if the wave's amplitude is doubled, its frequency is tripled, and it's shifted π/2 units to the right. 2. The demand function for a product is D(p) = 500 - 5p. If a marketing campaign increases demand by 100 units at every price point, write the transformed demand function. 3. A ball is thrown upward from a height of 10 feet with an initial velocity of 48 ft/s. Using the basic projectile motion function h(t) = -16t², write the transformed function that models this situation. Page 21.

 Exercises Practice Problems for All Topics  Functions 1 Determine whether the relation is a function. Explain your reasoning. 2 For the function f(x) = 3x² - 2x + 5, find f(-2) and f(3).  Linear Functions 3 Find the slope of the line passing through points (-1, 4) and (3, -2). 4 A taxi company charges a $4 flat fee plus $2.75 per mile. Write a linear function for the cost C in terms of miles m.  Quadratic Functions 5 Find the vertex of the parabola y = 2x² - 8x + 5. 6 A ball is thrown upward with an initial velocity of 40 ft/s from a height of 5 ft. The height h (in feet) after t seconds is given by h(t) = -16t² + 40t + 5. When does the ball reach its maximum height?  Exponential Functions 7 A bacteria culture doubles every 4 hours. If there are initially 500 bacteria, how many will there be after 24 hours? 8 If you invest $2,000 at an annual interest rate of 5% compounded quarterly, how much will you have after 10 years?  Systems of Equations 9 Solve the system: 3x + 2y = 11 and 2x - y = 5. 10 A theater sells adult tickets for $12 and student tickets for $8. If 300 tickets were sold for a total of $3,240, how many of each type were sold?  Function Transformations 11 If f(x) = x², describe the transformation applied to get g(x) = (x - 4)² + 3. 12 For the function f(x) = |x|, write the equation for the function reflected across the x-axis and shifted 2 units down. Page 22.

 Exercises Advanced Practice Problems  Advanced Functions 13 Given f(x) = 2x - 3 and g(x) = x² + 1, find (f ∘ g)(2) and (g ∘ f)(2). 14 Find the domain and range of the function f(x) = √(4 - x²).  Advanced Linear Functions 15 Find the equation of the line perpendicular to y = 2x - 5 that passes through the point (3, -1). 16 Two cars leave from the same point at the same time, traveling in opposite directions. One car travels at 55 mph and the other at 65 mph. When will they be 300 miles apart?  Advanced Quadratic Functions 17 A farmer has 100 meters of fencing and wants to enclose a rectangular area along a river (no fencing needed along the river). What dimensions will maximize the area? 18 Solve the quadratic equation 3x² - 10x + 8 = 0 using the quadratic formula.  Advanced Exponential Functions 19 The half-life of Carbon-14 is 5,730 years. If a fossil contains 25% of the original Carbon-14, how old is the fossil? 20 A city's population is growing at 2.5% per year. If the current population is 85,000, what will it be in 15 years?  Advanced Systems of Equations 21 Solve the system of equations: 2x + y - z = 8, x - 3y + 2z = -1, 3x + 2y + z = 11. 22 A chemist needs to make 100 mL of a 15% acid solution by mixing a 5% solution with a 25% solution. How much of each solution should be used?  Advanced Function Transformations 23 If f(x) = x³, describe the sequence of transformations that would produce the graph of g(x) = -2(x + 3)³ - 4. 24 A sound wave is modeled by f(t) = sin(t). Write the transformed function if the wave's amplitude is tripled, its frequency is doubled, and it's shifted π/3 units to the left.  Check your answers in the Answer Key section on page 24. Page 23.

 Answer Key Solutions to Practice Problems  This section contains answers to all exercises in the textbook. We encourage you to attempt each problem before checking your answers. Learning mathematics is a journey, and making mistakes is an important part of the process! 1 Functions Yes, it is a function. Each input (x-value) is paired with exactly one output (y-value). 2 Functions f(-2) = 3(-2)² - 2(-2) + 5 = 12 + 4 + 5 = 21 f(3) = 3(3)² - 2(3) + 5 = 27 - 6 + 5 = 26 3 Linear Functions m = (-2 - 4)/(3 - (-1)) = -6/4 = -3/2 4 Linear Functions C(m) = 2.75m + 4 5 Quadratic Functions Vertex: (2, -3) Solution: The vertex formula is (-b/2a, f(-b/2a)) For y = 2x² - 8x + 5, a = 2, b = -8 x = -(-8)/(2×2) = 8/4 = 2 y = 2(2)² - 8(2) + 5 = 8 - 16 + 5 = -3 6 Quadratic Functions 1.25 seconds Solution: The vertex occurs at t = -b/2a = -40/(2×-16) = 40/32 = 1.25 seconds 7 Exponential Functions 32,000 bacteria Solution: After 24 hours, there are 24/4 = 6 doubling periods Final count = 500 × 2⁶ = 500 × 64 = 32,000 8 Exponential Functions $3,283.46 Solution: A = P(1 + r/n)^(nt) = 2000(1 + 0.05/4)^(4×10) = 2000(1.0125)^40 ≈ $3,283.46 9 Systems of Equations x = 3, y = 1 Solution: Using substitution, from the second equation: y = 2x - 5 Substitute into first equation: 3x + 2(2x - 5) = 11 3x + 4x - 10 = 11 → 7x = 21 → x = 3 y = 2(3) - 5 = 1 10 Systems of Equations 180 adult tickets, 120 student tickets Solution: Let a = adult tickets, s = student tickets a + s = 300 12a + 8s = 3240 From first equation: s = 300 - a Substitute: 12a + 8(300 - a) = 3240 12a + 2400 - 8a = 3240 → 4a = 840 → a = 210 s = 300 - 210 = 90 11 Function Transformations Shifted 4 units right and 3 units up Solution: The function g(x) = (x - 4)² + 3 is the result of shifting f(x) = x² right by 4 units and up by 3 units. 12 Function Transformations g(x) = -|x| - 2 Solution: Reflection across x-axis: -f(x) = -|x| Shift down 2 units: -|x| - 2 Congratulations! You've completed all the exercises in this textbook. Remember that mathematics is a skill that improves with practice. Continue exploring functions and graphs, and don't hesitate to revisit these concepts as you advance in your mathematical journey! Page 24.