CHAPTER Quadratic Functions Arches are used to support the weight of walls and ceilings in buildings. Arches were first used in architecture by the Mesopotamians over 4000 years ago. Later, the Romans incorporated arches into a wide range of structures, and they are commonly used in modern buildings. The arch in the image above is in the shape of a parabola, which is a graphical representation of a quadratic function. You will learn the properties and shapes of quadratic functions.. Lots and Projectiles Introduction to Quadratic Functions p. 3.2 Parabolas Properties of the Graphs of Quadratic Functions p. 37.3 Extremes Increase, Decrease, and Rates of Change p. 45.4 Solving Quadratic Equations Reviewing Roots and Zeros p. 5.5 Finding the Middle Determining the Vertex of a Quadratic Function p. 65.6 Other Forms of Quadratic Functions Vertex Form of a Quadratic Function p. 77.7 Graphing Quadratic Functions Basic Functions and Transformations p. 85 Chapter Quadratic Functions 29
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. Lots and Projectiles Introduction to Quadratic Functions Objectives In this lesson, you will: Write quadratic functions. Use quadratic functions to model area. Use quadratic functions to model vertical motion. Key Terms quadratic function vertical motion Problem Plots and Lots In a new housing development, every rectangular plot that is laid out must be six feet longer than it is wide to accommodate a sidewalk and a tree lawn (the area between the sidewalk and the road). Answer the following questions about this situation.. How long or wide would the plot be if the plot is a. 50 feet wide? b. 20 feet long? c. 75 feet long? 2. What would be the area of the plot if the plot is a. 60 feet wide? b. 80 feet long? c. 50 feet long? 3. Define a variable for the width of the plot. Lesson. Introduction to Quadratic Functions 3
4. Write an expression for the length of the plot. 5. Write an equation for the area of the plot. 6. Using this information, complete the following table. Then use the information in the table to graph the area of the plot versus the width of the plot. Quantity Name Unit Expression 7. Is the graph linear? Explain. 8. If you haven t done so already, use the distributive property to rewrite the equation for the area without parentheses. 32 Chapter Quadratic Functions
This equation is an example of a quadratic function. A quadratic function is defined as any equation of the form y ax 2 bx c where a, b, and c are real-number constants and a 0. Problem 2 Galileo s Discovery Galileo Galilei was a famous scientist who made many contributions in the areas of astronomy and physics. One of his most important discoveries was vertical motion when an object is dropped or falls, the distance it travels is a quadratic function of the time. Any object thrown, launched, or shot upward can be modeled by the following equation: s 2 at2 v 0 t s 0 where a is the acceleration from gravity, v 0 is the initial upward velocity, s 0 is the initial distance off the ground, and s is the height after t seconds.. For instance, a cannon ball is launched directly upward from the ground with an initial velocity of 320 feet per second. The acceleration due to gravity is 32 feet per second squared. The following equation models this situation. s 6t 2 320t 2. How high will the cannon ball be after a. 2 seconds? b. 0 seconds? c. 3. seconds? d. 20 seconds? Lesson. Introduction to Quadratic Functions 33
3. At what time(s) will the cannon ball be a. 304 feet above the ground? b. 576 feet above the ground? c. 2000 feet above the ground? 4. Using the information from Questions 2 and 3, complete the following table. Then use the information in the table to graph the height of the cannon ball versus the time. Quantity Name Unit Expression 34 Chapter Quadratic Functions
5. Is this graph linear? Explain. 6. From the graph, can you tell the maximum height that the cannon ball attains? If so, what is this height and after how many seconds does the cannon ball reach it? 7. Does this graph make sense based on your own understanding of the path of a cannon ball? Explain. Be prepared to share your work with another pair, group, or the entire class. Lesson. Introduction to Quadratic Functions 35
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.2 Parabolas Properties of the Graphs of Quadratic Functions Objectives In this lesson, you will: Graph quadratic functions. Calculate the vertex, axis of symmetry, zeros, and intercepts of quadratic functions. Key Terms parabola vertex axis of symmetry zeros Problem Exploring Quadratic Functions. Complete the table of values for the quadratic function y x 2. Then use the table to construct a graph of the function. x y 0 2 3 2 3 2. Every quadratic function has a distinctive U-shape. Why? The graph of a quadratic function is called a parabola. The vertex of a parabola is the lowest or highest point on the curve. The axis of symmetry is the line that passes through the vertex and divides the parabola into two mirror images. For the parabolas we will be looking at in this chapter, the axis of symmetry is a vertical line. 3. Identify the vertex and the axis of symmetry for the graph of y x 2. Lesson.2 Properties of the Graphs of Quadratic Functions 37
One way to determine a parabola s vertex and axis of symmetry is graphically. On the graph, locate the coordinates of the highest or lowest point. This point is the vertex. For the parabolas we will be exploring, the axis of symmetry is the vertical line that passes through the vertex. The following table and graph are completed for the function y x 2 4x. The coordinates of the vertex and intercepts are shown on the graph. y x y 0 0 3 2 0 8 6 x = 2 y = x 2 4x 2 4 5 4 0 2 2 6 4 (0, 0) (4, 0) 2 6 2 4 (2, 4) 8 0 x 5 5 Vertex: (2, 4) x-intercepts: (0, 0) and (4, 0) y-intercept: (0, 0) Axis of symmetry: x 2 4. For each quadratic function, complete the table and sketch a graph. Then, determine the coordinates of the vertex, x-intercept(s), y-intercept, and the equation for the axis of symmetry. Label these key characteristics on the graph. a. y x 2 x 0 2 y 3 4 5 Vertex: x-intercept(s): y-intercept: Axis of symmetry: 38 Chapter Quadratic Functions
b. f(x) x 2 4x 3 x y Vertex: x-intercept(s): y-intercept: Axis of symmetry: c. f(x) x 2 4x x y Vertex: x-intercept(s): y-intercept: Axis of symmetry: Lesson.2 Properties of the Graphs of Quadratic Functions 39
d. y x 2 3x 2 x y Vertex: x-intercept(s): y-intercept: Axis of symmetry: e. y x 2 4x 3 x y Vertex: x-intercept(s): y-intercept: Axis of symmetry: 40 Chapter Quadratic Functions
5. The x-intercepts of a quadratic function are also called zeros. Why? 6. The standard form of the quadratic function is y ax 2 bx c. Use your graphs from Question 4 to answer the following questions. a. How does the sign of a affect the graph of a quadratic function? Explain. b. What does the value of c determine in the graph of a quadratic function? Explain. c. How is the x-value of the vertex related to the x-intercepts? Explain. 7. For each given axis of symmetry and point on a parabola, determine another point on the parabola. a. Axis of symmetry x 2; given point (0, 5): b. Axis of symmetry x 5; given point ( 7, 3): c. Axis of symmetry x ; given point ( 2, 7): 2 d. Axis of symmetry x 5; given point (0, 0): Lesson.2 Properties of the Graphs of Quadratic Functions 4
Problem 2 Key Characteristics of Quadratic Functions. As we have seen, the graph of a quadratic function y ax2 bx c is a parabola. Some characteristics of parabolas include the vertex, intercepts, and the axis of symmetry. For each question, use the information provided to determine the remaining characteristics, complete the table, and sketch a graph, if possible. a. The vertex of a parabola is ( 2, 4) and it passes through the point (0, 0). Axis of symmetry: x-intercept(s): x y 2 4 0 0 b. A parabola passes through the points (0, 4) and (8, 4). 42 Axis of symmetry: x-intercept(s): y-intercept: Vertex: x y 0 4 8 4 Chapter Quadratic Functions y-intercept:
c. The vertex of a parabola is ( 6, 4) and it passes through the point ( 8, 0). Axis of symmetry: x-intercept(s): y-intercept: x y 6 4 8 0 Be prepared to share your work with another pair, group, or the entire class. Lesson.2 Properties of the Graphs of Quadratic Functions 43
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.3 Extremes Increase, Decrease, and Rates of Change Objectives In this lesson, you will: Define extreme points. Determine intervals of increase and decrease. Calculate rates of change. Key Terms second difference extreme points intervals open interval closed interval half-closed or half-open interval Problem Calculating Rates of Change Consider the table and graph for the quadratic function y x 2. x y 0 2 3 2 3 Earlier you learned that linear functions have a constant rate of change. This rate of change, or slope, is calculated by dividing the vertical change by the horizontal change: m y. x Lesson.3 Increase, Decrease, and Rates of Change 45
. Complete each table by calculating the unit rate of change, or slope, between each pair of points on the graph of y x 2. x y x 2 x 0 0 y m y x 2 3 x y x 2 x 0 0 y m y x 2 3 2. What can you conclude about the rate of change of y x 2? Explain. 3. Consider the table and graph for the quadratic function y x 2. x y 0 2 3 4 5 46 Chapter Quadratic Functions
4. Complete each table by calculating the unit rate of change, or slope, between each pair of points on the graph of y x 2. x y x 2 x 0 0 y m y x 2 3 x y x 2 x 0 0 y m y x 2 3 5. What can you conclude about the rate of change of y x 2? Explain. 6. The graphs of y x 2 and y x 2 both have a vertex at the point (0, 0). Describe the change in the rate of change from one side of the vertex to the other. Lesson.3 Increase, Decrease, and Rates of Change 47
7. The unit rate of change in y for a function is called a first difference. The unit rate of change in the first difference for a function is called a second difference. Complete each table by calculating the first and second differences for y x 2. x y x 2 y ( y) 0 0 2 4 3 2 3 9 5 x y x 2 y ( y) 0 0 2 4 3 2 3 9 5 8. Complete each table by calculating the first and second differences for y x 2. x y x 2 y ( y) 0 0 2 4 3 2 3 9 5 x y x 2 y ( y) 0 0 2 4 3 2 3 9 5 48 Chapter Quadratic Functions
9. What conclusion can you make about the second difference of a quadratic function? Explain. Problem 2 Extreme Points. Consider the graphs of quadratic functions of the form y ax2 bx c with positive a values. a. Describe how the vertex relates to all the other points on the parabola. b. Describe how the values of y change on each side of the vertex. 2. Consider the graphs of quadratic functions of the form y ax2 bx c with negative a values. a. Describe how the vertex relates to all the other points on the parabola. b. Describe how the values of y change on each side of the vertex. A maximum or a minimum point on a graph, such as the vertex of a parabola, is called an extreme point. To locate extreme points, it helps to use intervals. An interval is defined as the set of real numbers between two given numbers. The following notation is used for intervals: An open interval (a, b) is the set of all numbers between a and b, but not including a or b. A closed interval [a, b] is the set of all numbers between a and b, including a and b. A half-closed or half-open interval (a, b] is the set of all numbers between a and b, including b but not including a. A half-closed or half-open interval [a, b) is the set of all numbers between a and b, including a but not including b. Intervals that are unbounded can be written using the symbol for infinity,. For instance, the interval [a, ) means all numbers greater than or equal to a. Lesson.3 Increase, Decrease, and Rates of Change 49
3. In Problem Question, the function y x 2 is decreasing over the interval (, 0) and increasing over the interval (0, ). Complete the table and graph and determine the intervals over which each quadratic function is increasing and over which it is decreasing. a. f(x) x 2 4x x y b. y x 2 3x 2 x y Be prepared to share your methods and solutions with the class. 50 Chapter Quadratic Functions
.4 Solving Quadratic Equations Reviewing Roots and Zeros Objectives In this lesson, you will: Solve quadratic equations using factoring. Solve quadratic equations by extracting square roots. Problem Solving Quadratic Equations by Factoring The solutions or roots of a quadratic equation ax 2 bx c 0 are the same as the x-intercepts or zeros of a quadratic function f(x) ax 2 bx c. One method for calculating the roots of a quadratic equation or the zeros of a quadratic function is by using factoring: If a function is given, set the function equal to zero. If an equation is given, perform transformations so that one side of the equation is equal to zero. Factor the quadratic expression on the other side of the equation. Set each factor equal to zero. Solve each resulting equation for the roots or zeros. For example, solve x 2 4x 3 using factoring by performing the following steps: x 2 4x 3 x 2 4x 3 3 3 x 2 4x 3 0 (x 3)(x ) 0 (x 3) 0 x 3 3 0 3 x 3 or or or (x ) 0 x 0 x Check: x 2 4x (3) 2 4(3) 9 2 3 x 2 4x () 2 4() 4 3 Lesson.4 Reviewing Roots and Zeros 5
. Calculate the roots of x 2 8x 7. 2. Calculate the roots of x 2 x 30. 3. Calculate the roots of x 2 5x 3x 8. 52 Chapter Quadratic Functions
4. Calculate the zeros of y x 2 2x 45. 5. Calculate the zeros of f(x) 5x 2 45x. 6. Calculate the roots of x 2 7x 60. Lesson.4 Reviewing Roots and Zeros 53
Problem 2 Multiplication of Binomials The functions and equations in Problem all had a coefficient of on the x 2 term. Factoring when the coefficient of the x 2 term is not is more complex. Before factoring quadratic expressions with a coefficient other than on the x 2 term, let s review multiplication of binomials. For example, you can multiply (2x 3)(5x 2) using the distributive property. (2x 3)(5x 2) 2x(5x 2) 3(5x 2) 0x2 4x 5x 6 0x2 9x 6 Alternatively, you can multiply using a multiplication table. 2x 3 5x 0x2 5x 2 4x 6 2x 3 5x 0x2 5x 2 4x 6. Perform each multiplication using the distributive property. a. ( 7x 4)( 3x 5) b. (8x 3) ( 5x ) 54 Chapter Quadratic Functions You can also multiply using an area model.
2. Perform each multiplication using a multiplication table. a. ( x 7)( 9x 0) b. (3x 2)( 5x 9) 3. Perform each multiplication using an area model. a. (2x 5)(3x 0) b. ( 3x 4)(3x 7) 4. Perform each multiplication using any method. a. ( 5x 7)(4x ) b. ( x 9)(3x ) c. (9x 4)(3x ) d. (3x 7)(3x 7) Lesson.4 Reviewing Roots and Zeros 55
Problem 3 Factoring When a In Problem 2, we saw that (2x 3)(5x 2) 0x 2 9x 6. 2x 3 5x 0x 2 5x 2 4x 6 Remember that the general form of a quadratic is ax 2 bx c with factors (cx d)(ex f). Answer the following questions.. Complete the multiplication table for the product (dx e)(fx g). 2. Use the general form of a quadratic equation, ax 2 bx c, to write an equivalent expression for each value. a. a b. c c. b Knowing the relationship between the coefficients of the quadratic equation and the coefficients of the binomial factors can help to factor. To factor 4x 2 23x 5 we know the following: The coefficient of the x 2 term, 4, is the product of d and f, the coefficients of the x terms of the binomials. The constant term, 5, is the product of e and g, the constant terms of the binomials. The product ac is equal to edfg, the product of all the coefficients in the factors. 3. The coefficient of the x term, b, is the sum of a pair of factors of ac. Why? 56 Chapter Quadratic Functions
4. Calculate ac for this expression and list all the possible factor pairs for this product. Remember to include negative factor pairs. ac Factor pairs: 5. Which of these factor pairs has a sum that is equal to b? 6. List all the factors of 4 and 5. Factors of 4: Factors of 5: 7. Identify the factors of 4 and the factors of 5 that, when multiplied, result in a product of 3 and 20. 8. Complete the multiplication table for 4x 2 23x 5. 4x 2 5 9. Use the process in Questions 4 through 8 to factor each quadratic expression. a. 5x 2 4x 8 ac Factor pairs of ac: Pair with sum of 4: Factors of 5: Factors of 8: Lesson.4 Reviewing Roots and Zeros 57
Factors of 5 and 8 that produce the desired products of and 40: 5x 2 8 5x 2 4x 8 b. 6x 2 x 0 ac Factor pairs of ac: Pair with sum of : Factors of 6: Factors of 0: Factors of 6 and 0 that produce the desired products of 4 and 5: 6x 2 0 6x 2 x 0 58 Chapter Quadratic Functions
0. Use the process in Questions 4 through 8 to solve each quadratic equation. a. 3x 2 22x 7 0 ac Factor pairs of ac: Pair with sum of 22: Factors of 3: Factors of 7: Factors of 3 and 7 that produce the desired products of and 2: Lesson.4 Reviewing Roots and Zeros 59
b. 8x 2 2x 2 0 ac Factor pairs of ac: Pair with sum of 2: Factors of 8: Factors of 2: Factors of 8 and 2 that produce the desired products of 2 and 4: 60 Chapter Quadratic Functions
Problem 4 Solving for a Perfect Square A second method for calculating the roots of a quadratic equation or the zeros of a quadratic function is by solving for a perfect square and then extracting the square roots. For example, solve 5x 2 45 0 using perfect squares by performing the following steps: 5x 2 45 0 5x 2 45 45 0 45 5x 2 5 45 5 x 2 9 x 2 9 x 3 Check: 5( 3) 2 45 45 45 0. Solve each equation by solving for a perfect square and then extracting the square roots. a. 3x 2 47 Lesson.4 Reviewing Roots and Zeros 6
b. x(x 5) 44 5x c. 3(x 3) 2 29 8x 62 Chapter Quadratic Functions
d. 5x 2 35 Be prepared to share your methods and solutions. Lesson.4 Reviewing Roots and Zeros 63
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.5 Finding the Middle Determining the Vertex of a Quadratic Function Objectives In this lesson, you will: Determine the vertex of a parabola given the equation of a quadratic function. Determine the vertex for the standard form of the quadratic function. Problem Exploring the Vertex In previous activities, we found the coordinates of the vertex from the graph of a quadratic function. We also explored the importance of the vertex as a maximum or minimum point and in determining intervals of increase and decrease.. Graph each of the following quadratic functions using a graphing calculator and sketch each on the grid. a. y x 2 b. y 2x 2 2 c. y 3x 2 4 Lesson.5 Determining the Vertex of a Quadratic Function 65
2. Graph each of the following quadratic functions using a graphing calculator and sketch each on the grid. a. y x 2 b. y 3x 2 2 c. y 2x 2 4 3. All of the functions in Questions and 2 are in the form y ax 2 c. What do you notice about the relationship between the vertex and axis of symmetry of the parabola and the equation of the function? 4. What are the coordinates of the vertex for a quadratic function in the form y ax 2 c with a 0? Is the vertex a maximum or a minimum? Explain. 5. What are the coordinates of the vertex for a quadratic function in the form y ax 2 c with a 0? Is the vertex a maximum or a minimum? Explain. 66 Chapter Quadratic Functions
Problem 2 The Vertex and Other Key Characteristics For a quadratic function in the form y ax2 bx c with b 0, determining the coordinates of the vertex and the equation of the line of symmetry is more difficult.. Consider the quadratic function y x2 4x 3. Complete the table for the function. Then, graph the quadratic function using a graphing calculator and sketch the graph on the grid. x y 0 2 3 4 6 a. What is the vertex of the function y x2 4x 3? Explain how you determined the coordinates. Lesson.5 Determining the Vertex of a Quadratic Function 67
b. What is the y-intercept of this function? c. What are the x-intercepts of this function? d. What is the equation of the axis of symmetry of this function? The axis of symmetry divides the parabola into two halves that are mirror images of each other. Every point of the parabola on one side of the axis of symmetry has a symmetric point on the other side of the axis of symmetry. These symmetric points are equidistant from the axis of symmetry. e. The points (, 8) and ( 6, 5) are points on the parabola. Determine the point on the parabola that is symmetric to each. f. Explain how you determined these symmetric points. g. Verify that the points you identified in Question (e) are on the parabola by substituting the coordinates of each into the function. Take Note h. What do you notice about the y-coordinates of each pair of symmetric points? The midpoint of a segment with endpoints (x, y ) and ( x x 2, y y 2 2 2 ) (x 2, y 2 ) is. i. If you draw a line segment connecting each pair of symmetric points, the midpoint of these segments lies on the axis of symmetry. Why? 68 Chapter Quadratic Functions
j. Use the midpoint formula to calculate the midpoint of the line segments connecting each pair of symmetric points. k. What is the equation for the axis of symmetry? How do you know? 2. Explain how to calculate the equation for the axis of symmetry if you know the coordinates of two symmetric points on the parabola. 3. Calculate the equation for the axis of symmetry using each pair of symmetric points. a. (4, 8) and (0, 8) b. ( 5, 6) and ( 2, 6) c. (d, e) and (f, e) Lesson.5 Determining the Vertex of a Quadratic Function 69
4. Consider the quadratic function y 2x 2 7x 3. Complete the table for the function. Then, graph the quadratic function using a graphing calculator and sketch the graph on the grid. x y 0 2 3 4 a. Use the graph to estimate the coordinates of the vertex. b. Based on the estimate for the vertex, what is the equation for the axis of symmetry? c. What are the x-intercepts of this function? d. Calculate the equation for the axis of symmetry of y 2x 2 7x 3? 70 Chapter Quadratic Functions
e. How does the calculated value compare to the estimated value using the graph? f. Explain why it is difficult to use a graph to determine exact values for the vertex and axis of symmetry. Problem 3 Calculating the Vertex Coordinates. Consider the function y 2x2 7x 3. a. What is the y-intercept? b. The y-intercept has a symmetric point on the parabola that has the same y-coordinate. Substitute the y-coordinate of the y-intercept into the equation and determine the coordinates of the point symmetric to the y-intercept. Lesson.5 Determining the Vertex of a Quadratic Function 7
c. Use these symmetric points to calculate the equation of the axis of symmetry for this function. d. The vertex lies on the axis of symmetry, so the x-coordinate of the vertex is the x-coordinate of the axis of symmetry. You can calculate the y-coordinate of the vertex by evaluating the function for this x-coordinate. Calculate the vertex coordinates for y 2x 2 7x 3. 2. Calculate the vertex for each of the following quadratic functions. a. y x 2 4x 3 y-intercept: Coordinates of the point symmetric to the y-intercept: 72 Chapter Quadratic Functions
b. y x 2 7x 6 y-intercept: Coordinates of the point symmetric to the y-intercept: Lesson.5 Determining the Vertex of a Quadratic Function 73
c. y x 2 8x 5 y-intercept: Coordinates of the point symmetric to the y-intercept: 74 Chapter Quadratic Functions
d. y 3x 2 2x 5 y-intercept: Coordinates of the point symmetric to the y-intercept: Lesson.5 Determining the Vertex of a Quadratic Function 75
e. y ax 2 bx c y-intercept: Coordinates of the point symmetric to the y-intercept: Be prepared to share your solutions and methods. 76 Chapter Quadratic Functions
.6 Other Forms of Quadratic Functions Vertex Form of a Quadratic Function Objective In this lesson, you will: Write quadratic functions in vertex form. Key Term vertex form of a quadratic function Problem Vertex and Axis of Symmetry For each quadratic function: a. Calculate the equation for the axis of symmetry. b. Calculate the coordinates of the vertex. c. Complete the table by including the x-coordinate of the vertex and the x-coordinates one and two units to the left and right of the vertex. d. Sketch a graph of the function.. f(x) x 2 8x + 9 a. Axis of symmetry: b. Vertex: c. Table: Vertex: x y Lesson.6 Vertex Form of a Quadratic Function 77
d. Graph: 2. f(x) 2x 2 2x + 7 a. Axis of symmetry: b. Vertex: c. Table: x y Vertex: d. Graph: 78 Chapter Quadratic Functions
Problem 2 Different Forms of Quadratic Functions For each quadratic function, complete the table and sketch a graph.. f(x) (x 4) 2 7 x y 2 3 4 5 6 2. f(x) 2(x 3) 2 25 x y 2 0 2 4 6 8 3. Compare the tables and graphs from Problem Question and Problem 2 Question. What do you notice? 4. Simplify the expression (x 4) 2 7. Lesson.6 Vertex Form of a Quadratic Function 79
5. What can you conclude about the functions from Problem Question and Problem 2 Question? 6. Compare the tables and graphs from Problem Question 2 and Problem 2 Question 2. What do you notice? 7. Simplify the expression 2(x 3)2 + 25. 8. What can you conclude about the functions from Problem Question 2 and Problem 2 Question 2? Problem 3 Working with Standard Form In Problem, the quadratic functions are written in standard form, f(x) ax2 bx + c.. Consider the function f(x) x2 8x 9. a. Identify the values of the constants a, b, and c. c. What information does the value of b provide about the graph of the function? d. What information does the value of c provide about the graph of the function? In Problem 2 the quadratic functions are written in the form f(x) a(x h)2 + k. 2. Consider the function f(x) (x 4)2 7. a. Identify the values of the constants a, h, and k. 80 Chapter Quadratic Functions b. What information does the value of a provide about the graph of the function?
b. What information does the value of a provide about the graph of the function? c. What information does the value of h provide about the graph of the function? d. What information does the value of k provide about the graph of the function? The form f(x) a(x h) 2 + k is called the vertex form of a quadratic function. Problem 4 Converting between Vertex and Standard Forms To convert from vertex form to standard form, simplify the quadratic expression in the function definition. For example, convert f(x) 5(x 4) 2 5 to standard form: f(x) 5(x 4) 2 5 5(x 2 8x 6) 5 5x 2 40x 80 5 5x 2 40x 65. Convert each quadratic function from vertex form to standard form. a. f(x) 2(x 8) 2 + b. f(x) 4(x 3) 2 + 7 Lesson.6 Vertex Form of a Quadratic Function 8
( c. f(x) 4 x 3 7 2 ) 2 To convert from standard form to vertex form, calculate the coordinates of the vertex. Use the vertex to rewrite the function. For example, to convert f(x) 3x 2 2x 5 to vertex form: x b 2a 2 2(3) 2 f(2) 3(2) 2 2(2) 5 2 24 5 7 vertex: (2, 7) f(x) 3(x 2) 2 7 2. Convert each quadratic function from standard form to vertex form. a. f(x) x 2 8x 6 b. f(x) 2x 2 2x 3 82 Chapter Quadratic Functions
c. f(x) 5x 2 6x 3 Be prepared to share your methods and solutions. Lesson.6 Vertex Form of a Quadratic Function 83
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.7 Graphing Quadratic Functions Basic Functions and Transformations Objectives In this lesson, you will: Graph the basic quadratic function. Transform the graph of the quadratic basic function. Dilate the graph of the basic quadratic function. Key Term basic quadratic function Problem Basic Function. The basic quadratic function is y x 2. Graph this function. 2. On the same grid, graph the following functions. a. y x 2 4 b. y x 2 3 3. What do you notice about these three graphs? Explain. Lesson.7 Basic Functions and Transformations 85
4. For each part of Question 2, describe how the graph and the equation have been transformed from the basic function. 5. The basic quadratic function is y x 2. Graph this function. 6. On the same grid, graph the following functions. a. y x 2 4x 4 b. y x 2 4x 4 7. What do you notice about these three graphs? Explain. 8. For each part of Question 6, describe how the graph and the equation have been transformed from the basic function. 9. Factor: y x 2 4x 4 y x 2 4x 4 a. Calculate the following values of f(x) x 2 and g(x) (x 2) 2. i. f(0) iv. g(0) ii. g( 2) v. f( 3) iii. f(2) vi. g( 5) 86 Chapter Quadratic Functions
b. Calculate the following values of f(x) x 2 and h(x) (x 2) 2. i. f(0) iv. h(4) ii. h(2) v. f( 3) iii. f(2) vi. h( ) c. Describe how the factors (x 2) and (x 2) are related to the transformations you described in Question 8. 0. The basic quadratic function is y x 2. Graph this function.. On the same grid, graph the following functions. a. y 2x 2 b. y 2 x2 2. What do you notice about these three graphs? Explain. 3. For each part of Question, describe how the graph and the equation have been transformed from the basic function. Lesson.7 Basic Functions and Transformations 87
4. The basic quadratic function is y x 2. Graph this function. 5. On the same grid, graph the following functions. a. y x 2 2x b. y 2x 2 8x 8 6. What do you notice about these three graphs? Explain. 7. For each part of Question 5, describe how the graph and the equation have been transformed from the basic function. In each of the previous graphs of quadratic functions, one or more of the four following transformations were performed on the quadratic basic function: A. Vertical shift B. Horizontal shift C. Reflection D. Dilation 8. Which one of these transformations changed the shape of the parabola? Explain. 88 Chapter Quadratic Functions
9. Given that only one transformation changed the shape of the parabola, which of the coefficients of the standard quadratic function, y ax2 bx c, determines the shape of the parabola? Why? Remember that the zeros of a quadratic function can be determined by using the quadratic formula and that the x-value of the vertex of the function is the average of the zeros. Problem 2 For each of the following, determine the vertex. First determine the x-value by calculating the average of the zeros. Then use substitution to determine the y-value. Vertex:. y x2 0x 24 Lesson.7 Basic Functions and Transformations 89
2. y x 2 5x 4 Vertex: 3. y x 2 5x 4 Vertex: 90 Chapter Quadratic Functions
4. y 2x 2 2x 7 Vertex: 5. Use the vertices you calculated in Questions to 4 and your knowledge of the shape determined by the value of a to graph each of the following functions. a. y x 2 0x 24 Vertex: Lesson.7 Basic Functions and Transformations 9
b. y x 2 5x 4 Vertex: c. y x 2 5x 4 Vertex: d. y 2x 2 2x 7 Vertex: 92 Chapter Quadratic Functions
6. Using the vertices you have already calculated in Questions 4, graph the following functions. Then describe the graphical transformations that can be used to transform the basic function to each function. a. y x 2 0x 24 Vertex: Graphical transformations: b. y x 2 5x 4 Vertex: Graphical transformations: Lesson.7 Basic Functions and Transformations 93
c. y x 2 5x 4 Vertex: Graphical transformations: d. y 2x 2 2x 7 Vertex: Graphical transformations: 94 Chapter Quadratic Functions
7. Using the Quadratic Formula, derive a general formula for the average of the zeros in terms of a, b, and c. 8. Using your result from Question 7, what is the x-coordinate of the vertex of y ax 2 bx c? Be prepared to share your work with another pair, group, or the entire class. Lesson.7 Basic Functions and Transformations 95
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