Student Resource Book Unit 2

Transcription

1 Student Resource Book Unit 2

2 ISBN Copyright 2013 J. Weston Walch, Publisher Portland, ME Printed in the United States of America WALCH EDUCATION

3 Table of Contents Introduction... v Unit 2: Quadratic Functions and Modeling Lesson 1: Analyzing Quadratic Functions... U2-1 Lesson 2: Interpreting Quadratic Functions... U2-35 Lesson 3: Building Functions... U2-69 Lesson 4: Graphing Other Functions... U2-91 Lesson 5: Analyzing Functions... U2-153 Lesson 6: Transforming Functions... U2-175 Lesson 7: Finding Inverse Functions... U2-204 Answer Key...AK-1 iii Table of Contents

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5 Introduction Welcome to the CCSS Integrated Pathway: Mathematics II Student Resource Book. This book will help you learn how to use algebra, geometry, data analysis, and probability to solve problems. Each lesson builds on what you have already learned. As you participate in classroom activities and use this book, you will master important concepts that will help to prepare you for mathematics assessments and other mathematics courses. This book is your resource as you work your way through the Math II course. It includes explanations of the concepts you will learn in class; math vocabulary and definitions; formulas and rules; and exercises so you can practice the math you are learning. Most of your assignments will come from your teacher, but this book will allow you to review what was covered in class, including terms, formulas, and procedures. In Unit 1: Extending the Number System, you will learn about rational exponents and the properties of rational and irrational numbers. This is followed by operating with polynomials. Finally, you will define an imaginary number and learn to operate with complex numbers. In Unit 2: Quadratic Functions and Modeling, you will begin by exploring and interpreting the the graphs of quadratic functions. Then you will learn how to build quadratic functions from a context and how to carry out operations with functions. This gives way to the exploration of other types of functions, including square root, cube root, absolute value, step, and piecewise functions. The unit progresses to analyzing exponential functions and comparing linear, quadratic, and exponential models given in different forms. The unit ends with transforming functions and finding the inverse of functions. In Unit 3: Expressions and Equations, you will reexamine the basic structures of expressions, but this time apply these structures to quadratic expressions. Then you will learn to solve quadratic equations using various methods, as well as how to apply structures of quadratic expressions in solving these equations. The structures of expressions theme continues into having you create quadratic equations of various forms; here, you will learn how to rearrange formulas to solve for a quadratic variable of interest. The unit builds on previous units by introducing the Fundamental Theorem of Algebra and showing you how complex numbers are solutions to quadratic equations. Then v Introduction

6 you will be introduced to rational functions. Again, you will learn to write exponentially structured expressions in equivalent forms. The unit ends with returning to a familiar topic solving systems of equations but now complex solutions can be determined. In Unit 4: Applications of Probability, you will start by defining events, applying the addition rule, and learning about independence. Then you will progress toward conditional probabilities and the multiplication rule. This builds into using combinatorics to count and calculate probabilities. Finally, you will learn to make and analyze decisions based on probability. In Unit 5: Simliarity, Right Triangle Trigonometry, and Proof, you will begin by learning about midpoints and other points of interest in a line segment. Then you will work with dilations and similarity. This builds into learning about and proving the various similarity statements. Then you will learn about special angles in intersecting lines and about relationships among the angles formed by a set of parallel lines intersected by a transversal. You will then return to working with triangles and proving theorems about them, including the Interior Angle Sum Theorem, theorems about isosceles triangles, midsegments, and centers of triangles. The unit ends with an introduction to trigonometric ratios and problem solving with those ratios and the Pythagorean theorem. In Unit 6: Circles With and Without Coordinates, you will study the properties of circles, including central and inscribed angles, chords of a circle, and tangents of a circle. Then you build on this to explore polygons circumscribed and inscribed in a circle. You will then learn about the properties and construction of tangent lines. The measurement units of radians are introduced, and you will use radians to measure the area of a sector and the circumference and area of a circle. You build from a 1- and 2-dimensional arena to a 3-dimensional one by exploring more deeply the volume formulas for cylinders, pyramids, cones, and spheres. Then you will study the links between algebra and geometry by deriving the equations for circles and parabolas. Finally, you will use coordinates to prove geometric theorems about circles and parabolas. Each lesson is made up of short sections that explain important concepts, including some completed examples. Each of these sections is followed by a few problems to help you practice what you have learned. The Words to Know section at the beginning of each lesson includes important terms introduced in that lesson. As you move through your Math II course, you will become a more confident and skilled mathematician. We hope this book will serve as a useful resource as you learn. vi Introduction

7 UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 1: Analyzing Quadratic Functions Common Core State Standards F IF.7 F IF.8 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Essential Questions 1. How is a quadratic equation similar to a linear equation? How is it different? 2. How is a quadratic equation similar to an exponential equation? How is it different? 3. In what situations is it appropriate to use a quadratic model? WORDS TO KNOW axis of symmetry of a parabola extrema factored form of a quadratic function intercept the line through the vertex of a parabola about which the parabola is symmetric. The equation of the axis of b symmetry is x =. 2a the minima or maxima of a function the intercept form of a quadratic equation, written as f(x) = a(x p)(x q), where p and q are the x-intercepts of the function; also known as intercept form of a quadratic function the point at which a line intercepts the x- or y-axis U2-1 Lesson 1: Analyzing Quadratic Functions

8 intercept form maximum minimum parabola the factored form of a quadratic equation, written as f(x) = a(x p)(x q), where p and q are the x-intercepts of the function the largest y-value of a quadratic equation the smallest y-value of a quadratic equation the U-shaped graph of a quadratic equation quadratic function a function that can be written in the form f(x) = ax 2 + bx + c, where a 0. The graph of any quadratic function is a parabola. standard form of a quadratic function vertex form vertex of a parabola x-intercept y-intercept a quadratic function written as f(x) = ax 2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term a quadratic function written as f(x) = a(x h) 2 + k, where the vertex of the parabola is the point (h, k); the form of a quadratic equation where the vertex can be read directly from the equation the point on a parabola that is the maximum or minimum the point at which the graph crosses the x-axis; written as (x, 0) the point at which the graph crosses the y-axis; written as (0, y) U2-2 Unit 2: Quadratic Functions and Modeling

9 Recommended Resources IXL Learning. Solve an Equation Using the Zero Product Property. This interactive website gives a series of problems and scores them immediately. If the user submits a wrong answer, a description and process for arriving at the correct answer are provided. Users solve quadratic equations by setting factors equal to 0. This activity is meant as a review. PhET Interactive Simulations. Equation Grapher This website allows users to compare the graphs of various self-created equations. West Texas A&M University Virtual Math Lab. Graphs of Quadratic Functions. This tutorial offers a review and worked examples for writing and graphing quadratic functions in different forms, as well as practice problems with worked solutions for reference. U2-3 Lesson 1: Analyzing Quadratic Functions

10 Lesson 2.1.1: Graphing Quadratic Functions Introduction You may recall that a line is the graph of a linear function and that all linear functions can be written in the form f(x) = mx + b, in which m is the slope and b is the y-intercept. The solutions to a linear function are the infinite set of points on the line. In this lesson, you will learn about a second type of function known as a quadratic function. Key Concepts A quadratic function is a function that can be written in the form f(x) = ax 2 + bx + c, where x is the variable, a, b, and c are constants, and a 0. This form is also known as the standard form of a quadratic function, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. Quadratic functions can be graphed on a coordinate plane. One method of graphing a quadratic function is to create a table of at least five x-values and calculate the corresponding y-values. Once graphed, all quadratic functions will have a U-shape called a parabola. Distinguishing characteristics can be used to describe, draw, and compare quadratic functions. These characteristics include the y-intercept, x-intercepts, the maximum or minimum of the function, and the axis of symmetry. The intercept of a graph is the point at which a line intercepts the x- or y-axis. The x-intercept is the point at which a graph crosses the x-axis. It is written as (x, 0). The x-intercepts of a quadratic function occur when the parabola intersects the x-axis at (x, 0). U2-4 Unit 2: Quadratic Functions and Modeling

11 The following graph of a quadratic function, f(x) = x 2 2x 3, shows the location of the parabola s x-intercepts x-intercepts ( 1, 0) 1 (3, 0) Note that the x-intercepts of this function are ( 1, 0) and (3, 0). The equation of the x-axis is y = 0; therefore, the x-intercepts can also be found in a table by identifying which values of x have a corresponding y-value that is 0. The table of values below corresponds to the function f(x) = x 2 2x 3. Notice that the same x-intercepts noted in the graph can be found where the table shows y is equal to 0. x y U2-5 Lesson 1: Analyzing Quadratic Functions

12 The y-intercept of a quadratic function is the point at which the graph intersects the y-axis. It is written as (0, y). The y-intercept of a quadratic is the c value of the quadratic equation when written in standard form. The following graph of a quadratic function, f(x) = x 2 2x 3, shows the location of the parabola s y-intercept y-intercept (0, 3) Note that the y-intercept of this equation is (0, 3). The c value of the function is also 3. The axis of symmetry of a parabola is the line through the parabola about which the parabola is symmetric. b The equation of the axis of symmetry is x =. 2a U2-6 Unit 2: Quadratic Functions and Modeling

13 The equation of the axis of symmetry for the function f(x) = x 2 2x 3 is x = 1 because the vertical line through 1 is the line that cuts the parabola in half Axis of symmetry The vertex of a parabola is the point on a parabola that is the maximum or minimum of the function. The maximum is the largest y-value of a quadratic equation and the minimum is the smallest y-value. The extrema of a graph are the minima or maxima of a function. In other words, an extremum is the function value that achieves either a minimum or maximum. The vertex of a quadratic lies on the axis of symmetry. The vertex is often written as (h, k). The formula b x = is also used to find the x-coordinate of the vertex. 2a To find the y-coordinate, substitute the value of x into the original function, b ( hk, ) 2 a, f b = 2a. U2-7 Lesson 1: Analyzing Quadratic Functions

14 The graph that follows shows the relationship between the vertex and the axis of symmetry of a parabola Vertex (1, 4) -5-6 Axis of symmetry Notice that the vertex of the function f(x) = x 2 2x 3 is (1, 4). If you know the x-intercepts of the graph, or any two points on the graph with the same y-value, the x-coordinate of the vertex is the point halfway between the values of the x-coordinates. For x-intercepts (r, 0) and (s, 0), the x-coordinate of the vertex is r + s. 2 From the equation of a function in standard form, you can determine if the function has a maximum or a minimum based on the sign of the coefficient of the quadratic term, a. If a > 0, then the parabola opens up and therefore has a minimum value. If a < 0, the parabola opens down and therefore has a maximum value. The value of a of the function f(x) = x 2 2x 3 is 1; therefore, the vertex is a minimum. U2-8 Unit 2: Quadratic Functions and Modeling

15 To graph a function using a graphing calculator, follow these general steps for your calculator model. On a TI-83/84: Step 1: Press the [Y=] button. Step 2: Type the function into Y1, or any available equation. Use the [X, T, θ, n] button for the variable x. Use the [x 2 ] button for a square. Step 3: Press [WINDOW]. Enter values for Xmin, Xmax, Ymin, and Ymax. The Xscl and Yscl are arbitrary. Leave Xres = 1. Step 4: Press [GRAPH]. On a TI-Nspire: Step 1: Press the [home] key. Step 2: Arrow over to the graphing icon and press [enter]. Step 3: Type the function next to f1(x), or any available equation, and press [enter]. Use the [X] button for the variable x. Use the [x 2 ] button for a square. Step 4: To change the viewing window, press [menu]. Select 4: Window/ Zoom and select A: Zoom Fit. U2-9 Lesson 1: Analyzing Quadratic Functions

16 Guided Practice Example 1 Given the function f(x) = x 2, identify the key features of the graph: the extremum, vertex, and y-intercept. Then sketch the graph. 1. Determine the extremum of the graph. The extreme value is a minimum when a > 0. It is a maximum when a < 0. Because a = 1, the graph opens upward and the quadratic has a minimum. 2. Determine the vertex of the graph. The minimum value occurs at the vertex. The vertex is of the form b 2 a, f b 2a. Use the original function f(x) = x 2 to find the values of a and b in order to find the x-value of the vertex. b x = Formula to find the x-coordinate of 2a the vertex of a quadratic (0) x = Substitute 1 for a and 0 for b. 2(1) x = 0 Simplify. The x-coordinate of the vertex is 0. Substitute 0 into the original equation to find the y-coordinate. f(x) = x 2 Original equation f(0) = (0) 2 Substitute 0 for x. f(0) = 0 Simplify. The y-coordinate of the vertex is 0. The vertex is located at (0, 0). U2-10 Unit 2: Quadratic Functions and Modeling

17 3. Determine the y-intercept of the graph. The y-intercept occurs when x = 0. The y-intercept of the function f(x) = x 2 is the same as the vertex, (0, 0). When the equation is written in standard form, the y-intercept is c. 4. Graph the function. Create a table of values and axis of symmetry to identify points on the graph. The axis of symmetry goes through the vertex, so the axis of symmetry is x = 0. For each point to the left of the axis of symmetry, there is another point the same distance on the right side of the axis and vice versa. Choose at least two values of x that are to the right and left of 0. Let s start with x = 2. f(x) = x 2 Original equation f(2) = (2) 2 Substitute 2 for x. f(2) = 4 An additional point is (2, 4). Simplify. (2, 4) is 2 units to the right of the vertex. The point ( 2, 4) is 2 units to the left of the vertex, so ( 2, 4) is also on the graph. To find another set of points on the graph, let s evaluate the original equation for x = 3. f(x) = x 2 Original equation f(3) = (3) 2 Substitute 3 for x. f(3) = 9 An additional point is (3, 9). Simplify. (3, 9) is 3 units to the right of the vertex. The point ( 3, 9) is 3 units to the left of the vertex, so ( 3, 9) is also on the graph. (continued) U2-11 Lesson 1: Analyzing Quadratic Functions

18 Plot the points and join them with a smooth curve. ( 3, 9) ( 2, 4) (2, 4) (3, 9) f(x) = x (0, 0) Example 2 Given the function f(x) = 2x x 30, identify the key features of the graph: the extremum, vertex, and y-intercept. Then sketch the graph. 1. Determine the extremum of the graph. The extreme value is a minimum when a > 0. It is a maximum when a < 0. Because a = 2, the graph opens downward and the quadratic has a maximum. U2-12 Unit 2: Quadratic Functions and Modeling

19 2. Determine the vertex of the graph. The maximum value occurs at the vertex. b The vertex is of the form a f b 2, 2a. Use the original equation f (x) = 2x x 30 to find the values of a and b in order to find the x-value of the vertex. b x = Formula to find the x-coordinate of 2a the vertex of a quadratic ( 16) x = 2( Substitute 2 for a and 16 for b. 2) x = 4 Simplify. The x-coordinate of the vertex is 4. Substitute 4 into the original equation to find the y-coordinate. f(x) = 2x x 30 Original equation f(4) = 2(4) (4) 30 Substitute 4 for x. f(4) = 2 Simplify. The y-coordinate of the vertex is 2. The vertex is located at (4, 2). 3. Determine the y-intercept of the graph. The y-intercept occurs when x = 0. Substitute 0 for x in the original equation. f(x) = 2x x 30 Original equation f(0) = 2(0) (0) 30 Substitute 0 for x. f(0) = 30 The y-intercept is (0, 30). Simplify. When the quadratic equation is written in standard form, the y-intercept is c. U2-13 Lesson 1: Analyzing Quadratic Functions

20 4. Graph the function. Use symmetry to identify additional points on the graph. The axis of symmetry goes through the vertex, so the axis of symmetry is x = 4. For each point to the left of the axis of symmetry, there is another point the same distance on the right side of the axis and vice versa. The point (0, 30) is on the graph, and 0 is 4 units to the left of the axis of symmetry. The point that is 4 units to the right of the axis is 8, so the point (8, 30) is also on the graph. Determine two additional points on the graph. Choose an x-value to the left or right of the vertex and find the corresponding y-value. f(x) = 2x x 30 Original equation f(1) = 2(1) (1) 30 Substitute 1 for x. f(1) = 16 Simplify. An additional point is (1, 16). (1, 16) is 3 units to the left of the axis of symmetry. The point that is 3 units to the right of the axis is 7, so the point (7, 16) is also on the graph. Plot the points and join them with a smooth curve. 5 0 f(x) = 2x x 30 (4, 2) (3, 0) (5, 0) (1, 16) (7, 16) (0, 30) (8, 30) U2-14 Unit 2: Quadratic Functions and Modeling

21 Example 3 Given the function f(x) = x 2 + 6x + 9, identify the key features of its graph: the extremum, vertex, and y-intercept. Then sketch the graph. 1. Determine the extremum of the graph. The extreme value is a minimum when a > 0. It is a or maximum when a < 0. Because a = 1, the graph opens upward and the quadratic has a minimum. 2. Determine the vertex of the graph. The minimum value occurs at the vertex. b The vertex is of the form 2 a, f b 2a. Use the original function f(x) = x 2 + 6x + 9 to find the values of a and b in order to find the x-value of the vertex. b Formula to find the x-coordinate of x = 2a the vertex of a quadratic (6) x = Substitute 1 for a and 6 for b. 2(1) x = 3 Simplify. The x-coordinate of the vertex is 3. Substitute 3 into the original equation to find the y-coordinate. f(x) = x 2 + 6x + 9 Original equation f( 3) = ( 3) 2 + 6( 3) + 9 Substitute 3 for x. f( 3) = 0 Simplify. The y-coordinate of the vertex is 0. The vertex is located at ( 3, 0). U2-15 Lesson 1: Analyzing Quadratic Functions

22 3. Determine the y-intercept of the graph. The y-intercept occurs when x = 0. Substitute 0 for x in the original equation. f(x) = x 2 + 6x + 9 Original equation f(0) = (0) 2 + 6(0) + 9 Substitute 0 for x. f(0) = 9 Simplify. The y-intercept is (0, 9). 4. Graph the function. Use symmetry to identify an additional point on the graph. The axis of symmetry goes through the vertex, so the axis of symmetry is x = 3. For each point to the left of the axis of symmetry, there is another point the same distance on the right side of the axis and vice versa. The point (0, 9) is on the graph, and 0 is 3 units to the right of the axis of symmetry. The point that is 3 units to the left of the axis is 6, so the point ( 6, 9) is also on the graph. Determine two additional points on the graph. Choose an x-value to the left or right of the vertex and find the corresponding y-value. f(x) = x 2 + 6x + 9 Original equation f( 1) = ( 1) 2 + 6( 1) + 9 Substitute 1 for x. f( 1) = 4 An additional point is ( 1, 4). Simplify. ( 1, 4) is 2 units to right of the axis of symmetry. The point that is 2 units to the left of the axis is 5, so the point ( 5, 4) is also on the graph. (continued) U2-16 Unit 2: Quadratic Functions and Modeling

23 Plot the points and join them with a smooth curve ( 6, 9) (0, 9) 8 f(x) = x 2 + 6x ( 5, 4) ( 1, 4) ( 3, 0) Example 4 Given the function f(x) = 2x 2 12x 10, identify the key features of its graph: the extremum, vertex, and y-intercept. Then sketch the graph. 1. Determine the extremum of the graph. The extreme value is either a minimum, when a > 0, or a maximum, when a < 0. Because a = 2, the graph opens down and the quadratic has a maximum. U2-17 Lesson 1: Analyzing Quadratic Functions

24 2. Determine the vertex of the graph. b The vertex is of the form 2 a, f b 2a. Use the original function f(x) = 2x 2 12x 10 to find the values of a and b in order to find the x-value of the vertex. b Formula to find the x-coordinate of x = 2a the vertex of a quadratic ( 12) x = 2( 2) x = 3 Substitute 2 for a and 12 for b. Simplify. The x-coordinate of the vertex is 3. Substitute 3 into the original equation to find the y-coordinate. f(x) = 2x 2 12x 10 Original equation f( 3) = 2( 3) 2 12( 3) 10 Substitute 3 for x. f( 3) = 8 The y-coordinate of the vertex is 8. The vertex is ( 3, 8). Simplify. 3. Determine the y-intercept of the graph. The y-intercept occurs when x = 0. Substitute 0 for x in the original equation. f(x) = 2x 2 12x 10 Original equation f(0) = 2(0) 2 12(0) 10 Substitute 0 for x. f(0) = 10 The y-intercept is (0, 10). U2-18 Unit 2: Quadratic Functions and Modeling

25 4. Graph the function. Use symmetry to identify another point on the graph. Because 0 is 3 units to the right of the axis of symmetry, the point 3 units to the left of the axis will have the same value, so ( 6, 10) is also on the graph. Determine two additional points on the graph. Choose an x-value to the left or right of the vertex and find the corresponding y-value. f(x) = 2x 2 12x 10 Original equation f(0) = 2( 2) 2 12( 2) 10 Substitute 2 for x. f( 2) = 6 An additional point is ( 2, 6). Simplify. ( 2, 6) is 1 unit to right of the axis of symmetry. The point that is 1 unit to the left of the axis is 4, so the point ( 4, 6) is also on the graph. Plot the points and join them with a smooth curve. f(x) = 2x 2 12x 10 2 ( 5, 0) ( 1, 0) ( 6, 10) ( 3, 8) ( 4, 6) ( 2, 6) (0, 10) U2-19 Lesson 1: Analyzing Quadratic Functions

26 PRACTICE UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 1: Analyzing Quadratic Functions Practice 2.1.1: Graphing Quadratic Functions For each function that follows, identify the intercepts, vertex, and maximum or minimum. Then, sketch the graph of the function. 1. y = x 2 + 6x 7 2. y = x 2 8x y = x 2 + 4x y= x 2x 2 5. y = x x y = 3x 2 + 6x y = 2x 2 12x 16 For each problem that follows, determine whether the function has a minimum or maximum, identify the maximum or minimum, and identify the intercepts. 8. A golfer s ball lands in a sand trap 4 feet below the playing green. The path of the ball on her next shot is given by the equation y = 16x x 4, where y represents the height of the ball after x seconds. 9. The revenue, R(x), generated by an increase in price of x dollars for an item is represented by the equation R(x) = 10x x The flight of a rubber band follows the quadratic equation H(x) = x 2 + 6x + 7, where H(x) represents the height of the rubber band in inches and x is the horizontal distance the rubber band travels in inches after launch. U2-20 Unit 2: Quadratic Functions and Modeling

27 Lesson 2.1.2: Interpreting Various Forms of Quadratic Functions Introduction Quadratic equations can be written in several forms, including standard form, vertex form, and factored form. While each form is equivalent, certain forms easily reveal different features of the graph of the quadratic function. In this lesson, you will learn to use the various forms of quadratic functions to show the key features of the graph and determine how these key features relate to the characteristics of a real-world situation. Key Concepts Standard Form Recall that the standard form, or general form, of a quadratic function is written as f(x) = ax 2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. When a function is written in standard form, the y-intercept is the value of c. The vertex of the function can be found by first determining the value of x, b x = b, and then finding the corresponding y-value, y= f 2a 2a. b The vertex is often written as 2 a, f b 2a. If a > 0, the function has a minimum and the graph opens up. If a < 0, the function has a maximum and the graph opens down. Vertex Form The vertex form of a quadratic function is written as f(x) = a(x h) 2 + k. In vertex form, the maximum or minimum of the function is identified using the vertex of the parabola, the point (h, k). If a > 0, the function has a minimum, where k is the y-coordinate of the minimum and h is the x-coordinate of the minimum. If a < 0, the function has a maximum, where k is the y-coordinate of the maximum and h is the x-coordinate of the maximum. U2-21 Lesson 1: Analyzing Quadratic Functions

28 Because the axis of symmetry goes through the vertex, the axis of symmetry can be identified from vertex form as x = h. The graph of a quadratic function is symmetric about the axis of symmetry. Factored Form The factored form, or intercept form, of a quadratic function is written as f(x) = a(x p)(x q). Recall that the x-intercepts of a function are the x-values where the function is 0. In factored form, the x-intercepts of the function are identified as p and q. Recall that the y-intercept of a function is the point at which the function intersects the y-axis. To determine the y-intercept, substitute 0 for x and simplify. The axis of symmetry can be identified from the factored form since it occurs at the midpoint between the x-intercepts. Therefore, the axis of symmetry is p q x = +. 2 To determine the vertex of the function, calculate the y-value that corresponds to the x-value of the axis of symmetry. If a > 0, the function has a minimum and the graph opens up. If a < 0, the function has a maximum and the graph opens down. U2-22 Unit 2: Quadratic Functions and Modeling

29 Guided Practice Example 1 Suppose that the flight of a launched bottle rocket can be modeled by the function f(x) = (x 1)(x 6), where f(x) measures the height above the ground in meters and x represents the horizontal distance in meters from the launching spot at x = 1. How far does the bottle rocket travel in the horizontal direction from launch to landing? What is the maximum height the bottle rocket reaches? How far has the bottle rocket traveled horizontally when it reaches its maximum height? Graph the function. 1. Identify the x-intercepts of the function. In the function, f(x) represents the height of the bottle rocket. At launch and landing, the height of the bottle rocket is 0. The function f(x) = (x 1)(x 6) is of the form f(x) = a(x p)(x q), where p and q are the x-intercepts. The x-intercepts of the function are at x = 1 and x = 6. Find the distance between the two points to determine how far the bottle rocket traveled in the horizontal direction. 6 1 = 5 The bottle rocket traveled 5 meters in the horizontal direction from launch to landing. 2. Determine the maximum height of the bottle rocket. The maximum height occurs at the vertex. p q Find the axis of symmetry using the formula x = +. 2 p q x = + Formula to determine the axis of symmetry x = + Substitute 6 for p and 1 for q. 2 x = 3.5 Simplify. The axis of symmetry is x = 3.5. (continued) U2-23 Lesson 1: Analyzing Quadratic Functions

30 Use the axis of symmetry to determine the vertex of the function. f(x) = (x 1)(x 6) Original function f(3.5) = [(3.5) 1][(3.5) 6] Substitute 3.5 for x. f(3.5) = (2.5)( 2.5) Simplify. f(3.5) = 6.25 The y-coordinate of the vertex is The maximum height reached by the bottle rocket is 6.25 meters. 3. Determine the horizontal distance from the launch point to the maximum height of the bottle rocket. We know that the bottle rocket is launched from the point (1, 0) and reaches a maximum height at (3.5, 6.25). Subtract the x-value of the two points to find the distance traveled horizontally = 2.5 Another method is to take the total distance traveled horizontally from launch to landing and divide by 2 to find the same answer. This is because the maximum value occurs halfway between the x-intercepts of the function = The bottle rocket travels 2.5 meters horizontally when it reaches its maximum. U2-24 Unit 2: Quadratic Functions and Modeling

31 4. Graph the function. Use a graphing calculator or complete a table of values. Use the x-intercepts and vertex as three of the known points. Choose x-values on either side of the vertex for two additional x-values. x y To determine the y-coordinates of the additional points, substitute each x-value into the original function and solve. f(x) = (x 1)(x 6) Original function f(2) = [(2) 1][(2) 6] Substitute 2 for x. f(2) = (1)( 4) Simplify. f(2) = 4 f(x) = (x 1)(x 6) Original function f(5) = [(5) 1][(5) 6] Substitute 5 for x. f(5) = (4)( 1) Simplify. f(5) = 4 Fill in the missing table values. x y (continued) U2-25 Lesson 1: Analyzing Quadratic Functions

32 Notice that the points (2, 4) and (5, 4) are the same number of units from the vertex. Plot the points on a coordinate plane and connect them using a smooth curve. Since the function models the flight of a bottle rocket, it is important to only show the portion of the graph where both time and height are positive. 8 7 (3.5, 6.25) 6 Height 5 4 (2, 4) (5, 4) (1, 0) (6, 0) Time U2-26 Unit 2: Quadratic Functions and Modeling

33 Example 2 Reducing the cost of an item can result in a greater number of sales. The revenue function that predicts the revenue in dollars, R(x), for each $1 change in price, x, for a particular item is R(x) = 100(x 7) ,900. What is the maximum value of the function? What does the maximum value mean in the context of the problem? What price increase maximizes the revenue and what does it mean in the context of the problem? Graph the function. 1. Determine the maximum value of the function. The function R(x) = 100(x 7) ,900 is written in vertex form, f(x) = a(x h) 2 + k, where (h, k) is the vertex. The vertex of the function is (7, 28,900); therefore, the maximum value is 28, Determine what the maximum value means in the context of the problem. The maximum value of 28,900 means that the maximum revenue resulting from increasing the price by x dollars is $28, Determine the price increase that will maximize the revenue and what it means in the context of the problem. The maximum value occurs at the vertex (7, 28,900). This means an increase in price of $7 will result in the maximum revenue. U2-27 Lesson 1: Analyzing Quadratic Functions

34 4. Graph the function. Use a graphing calculator or complete a table of coordinates. Use the vertex as one known point. Choose x-values on either side of the vertex to have four additional x-values. x y , To determine the y-coordinates of the additional points, substitute each x-value into the original function and solve. R(x) = 100(x 7) ,900 Original function R(0) = 100[(0) 7] ,900 Substitute 0 for x. R(0) = 24,000 Simplify. R(x) = 100(x 7) ,900 Original function R(5) = 100[(5) 7] ,900 Substitute 5 for x. R(5) = 28,500 Simplify. R(x) = 100(x 7) ,900 Original function R(9) = 100[(9) 7] ,900 Substitute 9 for x. R(9) = 28,500 Simplify. R(x) = 100(x 7) ,900 Original function R(14) = 100[(14) 7] ,900 Substitute 14 for x. R(14) = 24,000 Simplify. (continued) U2-28 Unit 2: Quadratic Functions and Modeling

35 Fill in the missing table values. x y 0 24, , , , ,000 Notice that the points (0, 24,000) and (14, 24,000) are the same number of units from the vertex. The same is true for (5, 28,500) and (9, 28,500). Plot the points on a coordinate plane and connect using a smooth curve. Since the function models revenue, it is important to only graph the portion of the graph where both the x- and y-values are positive. Revenue 29,000 28,000 27,000 26,000 25,000 24,000 23,000 22,000 21,000 20,000 19,000 18,000 17,000 16,000 15,000 14,000 13,000 12,000 11,000 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 (5, 28,500) (0, 24,000) (7, 28,900) (9, 28,500) (14, 24,000) $1 change in price U2-29 Lesson 1: Analyzing Quadratic Functions

36 Example 3 A football is kicked and follows a path given by f(x) = 0.03x x, where f(x) represents the height of the ball in feet and x represents the horizontal distance in feet. What is the maximum height the ball reaches? What horizontal distance maximizes the height? Graph the function. 1. Determine the maximum height of the ball. The function f(x) = 0.03x x is written in standard form, f(x) = ax 2 + bx + c, where a = 0.03, b = 1.8, and c = 0. b The maximum occurs at the vertex, 2 a, f b 2a. Determine the x-value of the vertex. b x = Formula to find the x-coordinate for the 2a vertex of a quadratic equation (1.8) x = Substitute values for a and b. 2( 0.03) x = 30 Simplify. Determine the y-value of the vertex. f(x) = 0.03x x Original function f(30) = 0.03(30) (30) Substitute 30 for x. f(30) = 27 Simplify. The vertex is (30, 27) and the maximum value is 27 feet. The maximum height the ball reaches is 27 feet. 2. Determine the horizontal distance of the ball when it reaches its maximum height. The x-coordinate of the vertex maximizes the quadratic. The vertex is (30, 27). The ball will have traveled 30 feet in the horizontal direction when it reaches its maximum height. U2-30 Unit 2: Quadratic Functions and Modeling

37 3. Graph the function. Use a graphing calculator or complete a table of coordinates. Use the vertex as one known point. Choose x-values on either side of the vertex to have four additional x-values. x y To determine the y-coordinates of the additional points, substitute each x-value into the original function and solve. f(x) = 0.03x x Original function f(5) = 0.03(5) (5) Substitute 5 for x. f(5) = 8.25 Simplify. f(x) = 0.03x x Original function f(20) = 0.03(20) (20) Substitute 20 for x. f(20) = 24 Simplify. f(x) = 0.03x x Original function f(40) = 0.03(40) (40) Substitute 40 for x. f(40) = 24 Simplify. f(x) = 0.03x x Original function f(55) = 0.03(55) (55) Substitute 55 for x. f(55) = 8.25 Simplify. (continued) U2-31 Lesson 1: Analyzing Quadratic Functions

38 Fill in the missing table values. x y Notice that the points (5, 8.25) and (55, 8.25) are the same number of units from the vertex. The same is true for (20, 24) and (40, 24). Plot the points on a coordinate plane and connect them using a smooth curve. Since the function models the path of a kicked football, it is important to only show the portion of the graph where both height and horizontal distance are positive (20, 24) (30, 27) (40, 24) Height (5, 8.25) (55, 8.25) Horizontal distance U2-32 Unit 2: Quadratic Functions and Modeling

39 PRACTICE UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 1: Analyzing Quadratic Functions Practice 2.1.2: Interpreting Various Forms of Quadratic Functions Use the given functions to complete all parts of problems f(x) = x 2 8x + 12 a. Identify the y-intercept. b. Identify the vertex. c. Identify whether the function has a maximum or minimum. 2. f(x) = 2(x 3)(x + 5) a. Identify the x-intercepts. b. Determine the y-intercept. c. Determine the axis of symmetry. d. Determine the vertex. 3. f(x) = 16(x 3) 2 a. Identify the vertex. b. Identify whether the function has a maximum or minimum. Use the given information in each scenario that follows to complete the remaining problems. 4. A butterfly descends toward the ground and then flies back up. The butterfly s descent can be modeled by the equation h(t) = t 2 10t + 26, where h(t) is the butterfly s height above the ground in feet and t is the time in seconds since you saw the butterfly. Graph the function and identify the vertex. What is the meaning of the vertex in the context of the problem? 5. A cliff diver jumps upward from the edge of a cliff then begins to descend, so that his path follows a parabola. The diver s height, h(t), above the water in feet is given by h(t) = 2(t 1) , where t represents the time in seconds. Graph the function. What is the vertex and what does it represent in the context of the problem? How many seconds after the start of the dive does the diver reach the initial height? continued U2-33 Lesson 1: Analyzing Quadratic Functions

40 PRACTICE UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 1: Analyzing Quadratic Functions 6. The revenue of producing and selling widgets is given by the function R(w) = 8(w 50)(w + 2), where w is the number of widgets produced and R(w) is the amount of revenue in dollars. Graph the function. What are the x-intercepts and what do they represent in the context of the problem? What number of widgets maximizes the revenue? 7. A football is kicked and follows a path given by y = 0.03x x, where y represents the height of the ball in feet and x represents the horizontal distance in feet. Graph the function. What is the vertex and what does it mean in the context of the problem? How far does the ball travel in the horizontal direction? 8. A frog hops from the bank of a creek onto a lily pad. The path of the jump 1 can be modeled by the equation ( ) 2 ( 2) 2 hx= x + 4, where h(x) is the frog s height in feet above the water and x is the number of seconds since the frog jumped. Graph the function. What does the vertex represent in the context of the problem? What is the axis of symmetry? After how many seconds does the height of the frog reach the initial height? 9. The revenue, R(x), generated by an increase in price of x dollars for an item is represented by the equation R(x) = 2x x Graph the function and identify the vertex. What does the vertex represent in the context of the problem? What is the axis of symmetry? What increase in price results in the same revenue as not increasing the price at all? 10. Decreasing the cost of an item can result in a greater number of sales. The revenue function that predicts the revenue in dollars, R(x), for each $1 decrease in price, x, for a certain item is R(x) = (x 26)(x + 10). Graph the function. Identify the x-intercepts. What do the x-intercepts represent in the context of the problem? What is the axis of symmetry? What increase in price results in the same revenue as not increasing the price at all? U2-34 Unit 2: Quadratic Functions and Modeling

41 UNIT 2 QUADRATIC FUNCTIONS AND MODELING Lesson 2: Interpreting Quadratic Functions Common Core State Standards F IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F IF.5 F IF.6 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Essential Questions 1. What information can be gathered by analyzing the key features of a quadratic function? 2. What properties must be true for a function to be identified as odd, even, or neither? 3. What are the applications of the domain of a quadratic function? 4. Why is it important to find the average rate of change when calculating the slope of a quadratic function? WORDS TO KNOW average rate of change the ratio of the difference of output values to the difference of the corresponding input values: f( b) f( a) ; a measure of how a quantity changes b a over some interval U2-35 Lesson 2: Interpreting Quadratic Functions

42 concave down concave up concavity decreasing domain end behavior even function extrema increasing inflection point U2-36 Unit 2: Quadratic Functions and Modeling a graph of a curve that is bent downward, such as a quadratic function with a maximum value a graph of a curve that is bent upward, such as a quadratic function with a minimum value with respect to a curve, the property of being arched upward or downward. A quadratic with positive concavity will increase on either side of the vertex, meaning that the vertex is the minimum or lowest point of the curve. A quadratic with negative concavity will decrease on either side of the vertex, meaning that the vertex is the maximum or highest point of the curve. the interval of a function for which the output values are becoming smaller as the input values are becoming larger the set of all input values (x-values) that satisfy the given function without restriction the behavior of the graph as x becomes larger or smaller a function that, when evaluated for x, results in a function that is the same as the original function; f( x) = f(x) the minima or maxima of a function the interval of a function for which the output values are becoming larger as the input values are becoming larger a point on a curve at which the sign of the curvature (i.e., the concavity) changes key features of a quadratic the x-intercepts, y-intercept, where the function is increasing and decreasing, where the function is positive and negative, relative minimums and maximums, symmetries, and end behavior of the function used to describe, draw, and compare quadratic functions neither odd function describes a function that, when evaluated for x, does not result in the opposite of the original function (odd) or the original function (even) a function that, when evaluated for x, results in a function that is the opposite of the original function; f( x) = f(x)

43 slope the measure of the rate of change of one variable with y2 y1 y rise respect to another variable; slope = = = x2 x1 x run the slope in the equation y = mx + b is m. ; Recommended Resources ChiliMath. Finding the Domain and Range of a Function. This website provides a summary of finding the domain and range for various types of functions as well as practice problems. Illustrated examples walk through finding the domain and range for different situations. JamesRahn.com. Rate of Change. This website provides a summary, practice, and an answer key for problems related to average rate of change. The intended audience is precalculus and calculus students; however, the summary is written so students of all levels feel comfortable exploring the concept. MathIsFun.com. Even and Odd Functions. This website provides a summary and practice problems for even functions, odd functions, and functions that are neither odd nor even. The site also illustrates the differences in behavior for each type of function. U2-37 Lesson 2: Interpreting Quadratic Functions

44 Lesson 2.2.1: Interpreting Key Features of Quadratic Functions Introduction The tourism industry thrives on being able to provide travelers with an amazing travel experience. Specifically, in areas known for having tropical weather, tour planners want to maximize profit each month by identifying the warmest and coolest months, and then plan tours accordingly. Tour planners might use quadratic models to determine when profits are increasing or decreasing, when they maximized, and/or how profits change in the earlier months versus the later months by looking at the key features of the quadratic functions. In this lesson, you will review the definitions of key features of a quadratic function and how to use graphs, tables, and verbal descriptions to identify and apply the key features. Key Concepts The key features of a quadratic function are distinguishing characteristics used to describe, draw, and compare quadratic functions. Key features include the x-intercepts, y-intercept, minimums and maximums, and symmetries, as well as where the function is increasing and decreasing, where the function is positive and negative, and the end behavior of the function. Recall each of the forms of quadratic functions, outlined as follows. Standard Form U2-38 Unit 2: Quadratic Functions and Modeling The standard form, or general form, of a quadratic function is written as f(x) = ax 2 + bx + c, where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant term. The y-intercept is the value of c. The vertex of the function can be found by first determining the value of x, and then finding the corresponding y-value. Vertex Form The vertex form of a quadratic function is written as f(x) = a(x h) 2 + k. The vertex is (h, k). The axis of symmetry is identified from vertex form as x = h. Factored Form The factored form, or intercept form, of a quadratic function is written as f(x) = a(x p)(x q). The x-intercepts of the function are p and q.

45 The x-intercepts of a quadratic function occur when the parabola intersects the x-axis. In the graph that follows, the x-intercepts occur when x = 2 and when x = 2. y x-intercepts x 2 4 The equation of the x-axis is y = 0; therefore, the x-intercepts can also be found in a table by identifying when the y-value is 0. The table of values below corresponds to the parabola illustrated above. Notice that the same x-intercepts can be found where the table shows y is equal to 0. x y The ordered pair that corresponds to an x-intercept is always of the form (x, 0). U2-39 Lesson 2: Interpreting Quadratic Functions

46 The x-intercepts are also the solutions of a quadratic function. The y-intercepts of a quadratic function occur when the parabola intersects the y-axis. In the next graph, the y-intercept occurs when y = 4. y x 2 4 y-intercept The equation of the y-axis is x = 0; therefore, the y-intercept can also be found in a table by identifying when the x-value is 0. Notice in the table of values that corresponds to the parabola above, the same y-intercept can be found where x is 0. x y The ordered pair that corresponds to a y-intercept is always of the form (0, y). Recall that the vertex is the maximum or minimum of the function. The vertex is also the point where the parabola changes from increasing to decreasing. U2-40 Unit 2: Quadratic Functions and Modeling

47 Increasing refers to the interval of a function for which the output values are becoming larger as the input values are becoming larger. Decreasing refers to the interval of a function for which the output values are becoming smaller as the input values are becoming larger. Recall that parabolas are symmetric to a line that extends through the vertex, called the axis of symmetry. Any point to the right or left of the parabola is equidistant to another point on the other side of the parabola. A parabola only increases or decreases as x becomes larger or smaller. Read the graph from left to right to determine when the function is increasing or decreasing. Trace the path of the graph with a pencil tip. If your pencil tip goes down as you move toward increasing values of x, then f(x) is decreasing. If your pencil tip goes up as you move toward increasing values of x, then f(x) is increasing. For a quadratic, if the graph has a minimum value, then the quadratic will start by decreasing toward the vertex, and then it will increase. If the graph has a maximum value, then the quadratic will start by increasing toward the vertex, and then it will decrease. The vertex is called an extremum. Extrema are the maxima or minima of a function. The concavity of a parabola is the property of being arched upward or downward. A quadratic with positive concavity will increase on either side of the vertex, meaning that the vertex is the minimum or lowest point of the curve. A quadratic with negative concavity will decrease on either side of the vertex, meaning that the vertex is the maximum or highest point of the curve. A quadratic that has a minimum value is concave up because the graph of the function is bent upward. A quadratic that has a maximum value is concave down because the graph of the function is bent downward. The graphs that follow demonstrate examples of parabolas as they decrease and then increase, and vice versa. Trace the path of each parabola from left to right with your pencil to see the difference. U2-41 Lesson 2: Interpreting Quadratic Functions