Quadratic Functions CHAPTER. 1.1 Lots and Projectiles Introduction to Quadratic Functions p. 31

Size: px
Start display at page:

Download "Quadratic Functions CHAPTER. 1.1 Lots and Projectiles Introduction to Quadratic Functions p. 31"

Transcription

1 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 Extremes Increase, Decrease, and Rates of Change p Solving Quadratic Equations Reviewing Roots and Zeros p. 5.5 Finding the Middle Determining the Vertex of a Quadratic Function p Other Forms of Quadratic Functions Vertex Form of a Quadratic Function p Graphing Quadratic Functions Basic Functions and Transformations p. 85 Chapter Quadratic Functions 29

2 30 Chapter Quadratic Functions

3 . 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 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

5 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

6 3. At what time(s) will the cannon ball be a. 304 feet above the ground? b. 576 feet above the ground? c 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

7 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

8 36 Chapter Quadratic Functions

9 .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 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

10 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 x = 2 y = x 2 4x (0, 0) (4, 0) (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 Vertex: x-intercept(s): y-intercept: Axis of symmetry: 38 Chapter Quadratic Functions

11 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

12 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

13 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

14 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 b. A parabola passes through the points (0, 4) and (8, 4). 42 Axis of symmetry: x-intercept(s): y-intercept: Vertex: x y Chapter Quadratic Functions y-intercept:

15 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 Be prepared to share your work with another pair, group, or the entire class. Lesson.2 Properties of the Graphs of Quadratic Functions 43

16 44 Chapter Quadratic Functions

17 .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 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

18 . 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 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 Chapter Quadratic Functions

19 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 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

20 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) x y x 2 y ( y) Complete each table by calculating the first and second differences for y x 2. x y x 2 y ( y) x y x 2 y ( y) Chapter Quadratic Functions

21 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

22 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

23 .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 x 2 4x 3 0 (x 3)(x ) 0 (x 3) 0 x x 3 or or or (x ) 0 x 0 x Check: x 2 4x (3) 2 4(3) x 2 4x () 2 4() 4 3 Lesson.4 Reviewing Roots and Zeros 5

24 . Calculate the roots of x 2 8x Calculate the roots of x 2 x Calculate the roots of x 2 5x 3x Chapter Quadratic Functions

25 4. Calculate the zeros of y x 2 2x 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

26 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.

27 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

28 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

29 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 Complete the multiplication table for 4x 2 23x 5. 4x 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

30 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

31 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

32 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

33 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 using perfect squares by performing the following steps: 5x x x x 2 9 x 2 9 x 3 Check: 5( 3) 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

34 b. x(x 5) 44 5x c. 3(x 3) x 62 Chapter Quadratic Functions

35 d. 5x 2 35 Be prepared to share your methods and solutions. Lesson.4 Reviewing Roots and Zeros 63

36 64 Chapter Quadratic Functions

37 .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

38 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 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

39 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 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

40 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 ) (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

41 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

42 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 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

43 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

44 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 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

45 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

46 c. y x 2 8x 5 y-intercept: Coordinates of the point symmetric to the y-intercept: 74 Chapter Quadratic Functions

47 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

48 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

49 .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

50 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

51 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 f(x) 2(x 3) 2 25 x y 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

52 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) 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?

53 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 x 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) Lesson.6 Vertex Form of a Quadratic Function 8

54 ( c. f(x) 4 x ) 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) vertex: (2, 7) f(x) 3(x 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

55 c. f(x) 5x 2 6x 3 Be prepared to share your methods and solutions. Lesson.6 Vertex Form of a Quadratic Function 83

56 84 Chapter Quadratic Functions

57 .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 What do you notice about these three graphs? Explain. Lesson.7 Basic Functions and Transformations 85

58 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

59 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 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

60 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

61 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

62 2. y x 2 5x 4 Vertex: 3. y x 2 5x 4 Vertex: 90 Chapter Quadratic Functions

63 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

64 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

65 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

66 c. y x 2 5x 4 Vertex: Graphical transformations: d. y 2x 2 2x 7 Vertex: Graphical transformations: 94 Chapter Quadratic Functions

67 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

68 96 Chapter Quadratic Functions

3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS

3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS 3.1 INTRODUCTION TO THE FAMILY OF QUADRATIC FUNCTIONS Finding the Zeros of a Quadratic Function Examples 1 and and more Find the zeros of f(x) = x x 6. Solution by Factoring f(x) = x x 6 = (x 3)(x + )

More information

UNIT 3 EXPRESSIONS AND EQUATIONS Lesson 3: Creating Quadratic Equations in Two or More Variables

UNIT 3 EXPRESSIONS AND EQUATIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Guided Practice Example 1 Find the y-intercept and vertex of the function f(x) = 2x 2 + x + 3. Determine whether the vertex is a minimum or maximum point on the graph. 1. Determine the y-intercept. The

More information

Unit 2: Functions and Graphs

Unit 2: Functions and Graphs AMHS Precalculus - Unit 16 Unit : Functions and Graphs Functions A function is a rule that assigns each element in the domain to exactly one element in the range. The domain is the set of all possible

More information

Algebra II Quadratic Functions

Algebra II Quadratic Functions 1 Algebra II Quadratic Functions 2014-10-14 www.njctl.org 2 Ta b le o f C o n te n t Key Terms click on the topic to go to that section Explain Characteristics of Quadratic Functions Combining Transformations

More information

Section 9.3 Graphing Quadratic Functions

Section 9.3 Graphing Quadratic Functions Section 9.3 Graphing Quadratic Functions A Quadratic Function is an equation that can be written in the following Standard Form., where a 0. Every quadratic function has a U-shaped graph called a. If the

More information

MAFS Algebra 1. Quadratic Functions. Day 17 - Student Packet

MAFS Algebra 1. Quadratic Functions. Day 17 - Student Packet MAFS Algebra 1 Quadratic Functions Day 17 - Student Packet Day 17: Quadratic Functions MAFS.912.F-IF.3.7a, MAFS.912.F-IF.3.8a I CAN graph a quadratic function using key features identify and interpret

More information

Slide 2 / 222. Algebra II. Quadratic Functions

Slide 2 / 222. Algebra II. Quadratic Functions Slide 1 / 222 Slide 2 / 222 Algebra II Quadratic Functions 2014-10-14 www.njctl.org Slide 3 / 222 Table of Contents Key Terms Explain Characteristics of Quadratic Functions Combining Transformations (review)

More information

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved.

Polynomial and Rational Functions. Copyright Cengage Learning. All rights reserved. 2 Polynomial and Rational Functions Copyright Cengage Learning. All rights reserved. 2.1 Quadratic Functions Copyright Cengage Learning. All rights reserved. What You Should Learn Analyze graphs of quadratic

More information

1. a. After inspecting the equation for the path of the winning throw, which way do you expect the parabola to open? Explain.

1. a. After inspecting the equation for the path of the winning throw, which way do you expect the parabola to open? Explain. Name Period Date More Quadratic Functions Shot Put Activity 3 Parabolas are good models for a variety of situations that you encounter in everyday life. Example include the path of a golf ball after it

More information

MAC Rev.S Learning Objectives. Learning Objectives (Cont.) Module 4 Quadratic Functions and Equations

MAC Rev.S Learning Objectives. Learning Objectives (Cont.) Module 4 Quadratic Functions and Equations MAC 1140 Module 4 Quadratic Functions and Equations Learning Objectives Upon completing this module, you should be able to 1. understand basic concepts about quadratic functions and their graphs.. complete

More information

UNIT 8: SOLVING AND GRAPHING QUADRATICS. 8-1 Factoring to Solve Quadratic Equations. Solve each equation:

UNIT 8: SOLVING AND GRAPHING QUADRATICS. 8-1 Factoring to Solve Quadratic Equations. Solve each equation: UNIT 8: SOLVING AND GRAPHING QUADRATICS 8-1 Factoring to Solve Quadratic Equations Zero Product Property For all numbers a & b Solve each equation: If: ab 0, 1. (x + 3)(x 5) = 0 Then one of these is true:

More information

Final Exam Review Algebra Semester 1

Final Exam Review Algebra Semester 1 Final Exam Review Algebra 015-016 Semester 1 Name: Module 1 Find the inverse of each function. 1. f x 10 4x. g x 15x 10 Use compositions to check if the two functions are inverses. 3. s x 7 x and t(x)

More information

Unit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form

Unit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form Unit 3, Lesson 3.1 Creating and Graphing Equations Using Standard Form Imagine the path of a basketball as it leaves a player s hand and swooshes through the net. Or, imagine the path of an Olympic diver

More information

Section 7.2 Characteristics of Quadratic Functions

Section 7.2 Characteristics of Quadratic Functions Section 7. Characteristics of Quadratic Functions A QUADRATIC FUNCTION is a function of the form " # $ N# 1 & ;# & 0 Characteristics Include:! Three distinct terms each with its own coefficient:! An x

More information

Unit 6 Quadratic Functions

Unit 6 Quadratic Functions Unit 6 Quadratic Functions 12.1 & 12.2 Introduction to Quadratic Functions What is A Quadratic Function? How do I tell if a Function is Quadratic? From a Graph The shape of a quadratic function is called

More information

More Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a

More Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a More Ways to Solve & Graph Quadratics The Square Root Property If x 2 = a and a R, then x = ± a Example: Solve using the square root property. a) x 2 144 = 0 b) x 2 + 144 = 0 c) (x + 1) 2 = 12 Completing

More information

QUADRATIC FUNCTIONS: MINIMUM/MAXIMUM POINTS, USE OF SYMMETRY. 7.1 Minimum/Maximum, Recall: Completing the square

QUADRATIC FUNCTIONS: MINIMUM/MAXIMUM POINTS, USE OF SYMMETRY. 7.1 Minimum/Maximum, Recall: Completing the square CHAPTER 7 QUADRATIC FUNCTIONS: MINIMUM/MAXIMUM POINTS, USE OF SYMMETRY 7.1 Minimum/Maximum, Recall: Completing the square The completing the square method uses the formula x + y) = x + xy + y and forces

More information

EXERCISE SET 10.2 MATD 0390 DUE DATE: INSTRUCTOR

EXERCISE SET 10.2 MATD 0390 DUE DATE: INSTRUCTOR EXERCISE SET 10. STUDENT MATD 090 DUE DATE: INSTRUCTOR You have studied the method known as "completing the square" to solve quadratic equations. Another use for this method is in transforming the equation

More information

y 1 ) 2 Mathematically, we write {(x, y)/! y = 1 } is the graph of a parabola with 4c x2 focus F(0, C) and directrix with equation y = c.

y 1 ) 2 Mathematically, we write {(x, y)/! y = 1 } is the graph of a parabola with 4c x2 focus F(0, C) and directrix with equation y = c. Ch. 10 Graphing Parabola Parabolas A parabola is a set of points P whose distance from a fixed point, called the focus, is equal to the perpendicular distance from P to a line, called the directrix. Since

More information

4.3 Quadratic functions and their properties

4.3 Quadratic functions and their properties 4.3 Quadratic functions and their properties A quadratic function is a function defined as f(x) = ax + x + c, a 0 Domain: the set of all real numers x-intercepts: Solutions of ax + x + c = 0 y-intercept:

More information

CHAPTER 6 Quadratic Functions

CHAPTER 6 Quadratic Functions CHAPTER 6 Quadratic Functions Math 1201: Linear Functions is the linear term 3 is the leading coefficient 4 is the constant term Math 2201: Quadratic Functions Math 3201: Cubic, Quartic, Quintic Functions

More information

+ bx + c = 0, you can solve for x by using The Quadratic Formula. x

+ bx + c = 0, you can solve for x by using The Quadratic Formula. x Math 33B Intermediate Algebra Fall 01 Name Study Guide for Exam 4 The exam will be on Friday, November 9 th. You are allowed to use one 3" by 5" index card on the exam as well as a scientific calculator.

More information

Exploring Quadratic Graphs

Exploring Quadratic Graphs Exploring Quadratic Graphs The general quadratic function is y=ax 2 +bx+c It has one of two basic graphs shapes, as shown below: It is a symmetrical "U"-shape or "hump"-shape, depending on the sign of

More information

6.4 Vertex Form of a Quadratic Function

6.4 Vertex Form of a Quadratic Function 6.4 Vertex Form of a Quadratic Function Recall from 6.1 and 6.2: Standard Form The standard form of a quadratic is: f(x) = ax 2 + bx + c or y = ax 2 + bx + c where a, b, and c are real numbers and a 0.

More information

Review for Quarter 3 Cumulative Test

Review for Quarter 3 Cumulative Test Review for Quarter 3 Cumulative Test I. Solving quadratic equations (LT 4.2, 4.3, 4.4) Key Facts To factor a polynomial, first factor out any common factors, then use the box method to factor the quadratic.

More information

Graphing Absolute Value Functions

Graphing Absolute Value Functions Graphing Absolute Value Functions To graph an absolute value equation, make an x/y table and plot the points. Graph y = x (Parent graph) x y -2 2-1 1 0 0 1 1 2 2 Do we see a pattern? Desmos activity: 1.

More information

Algebra II Quadratic Functions and Equations - Extrema Unit 05b

Algebra II Quadratic Functions and Equations - Extrema Unit 05b Big Idea: Quadratic Functions can be used to find the maximum or minimum that relates to real world application such as determining the maximum height of a ball thrown into the air or solving problems

More information

But a vertex has two coordinates, an x and a y coordinate. So how would you find the corresponding y-value?

But a vertex has two coordinates, an x and a y coordinate. So how would you find the corresponding y-value? We will work with the vertex, orientation, and x- and y-intercepts of these functions. Intermediate algebra Class notes More Graphs of Quadratic Functions (section 11.6) In the previous section, we investigated

More information

Lesson 8 Introduction to Quadratic Functions

Lesson 8 Introduction to Quadratic Functions Lesson 8 Introduction to Quadratic Functions We are leaving exponential and logarithmic functions behind and entering an entirely different world. As you work through this lesson, you will learn to identify

More information

Chapter 3 Practice Test

Chapter 3 Practice Test 1. Complete parts a c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry, and the x-coordinate of the vertex. b. Make a table of values that includes the vertex.

More information

Warm-Up Exercises. Find the x-intercept and y-intercept 1. 3x 5y = 15 ANSWER 5; y = 2x + 7 ANSWER ; 7

Warm-Up Exercises. Find the x-intercept and y-intercept 1. 3x 5y = 15 ANSWER 5; y = 2x + 7 ANSWER ; 7 Warm-Up Exercises Find the x-intercept and y-intercept 1. 3x 5y = 15 ANSWER 5; 3 2. y = 2x + 7 7 2 ANSWER ; 7 Chapter 1.1 Graph Quadratic Functions in Standard Form A quadratic function is a function that

More information

WHAT YOU SHOULD LEARN

WHAT YOU SHOULD LEARN GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of

More information

Y. Butterworth Lehmann & 9.2 Page 1 of 11

Y. Butterworth Lehmann & 9.2 Page 1 of 11 Pre Chapter 9 Coverage Quadratic (2 nd Degree) Form a type of graph called a parabola Form of equation we'll be dealing with in this chapter: y = ax 2 + c Sign of a determines opens up or down "+" opens

More information

OpenStax-CNX module: m Quadratic Functions. OpenStax OpenStax Precalculus. Abstract

OpenStax-CNX module: m Quadratic Functions. OpenStax OpenStax Precalculus. Abstract OpenStax-CNX module: m49337 1 Quadratic Functions OpenStax OpenStax Precalculus This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you

More information

Properties of Quadratic functions

Properties of Quadratic functions Name Today s Learning Goals: #1 How do we determine the axis of symmetry and vertex of a quadratic function? Properties of Quadratic functions Date 5-1 Properties of a Quadratic Function A quadratic equation

More information

A I only B II only C II and IV D I and III B. 5 C. -8

A I only B II only C II and IV D I and III B. 5 C. -8 1. (7A) Points (3, 2) and (7, 2) are on the graphs of both quadratic functions f and g. The graph of f opens downward, and the graph of g opens upward. Which of these statements are true? I. The graphs

More information

MATHS METHODS QUADRATICS REVIEW. A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation

MATHS METHODS QUADRATICS REVIEW. A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation MATHS METHODS QUADRATICS REVIEW LAWS OF EXPANSION A reminder of some of the laws of expansion, which in reverse are a quick reference for rules of factorisation a) b) c) d) e) FACTORISING Exercise 4A Q6ace,7acegi

More information

Chapter 2. Polynomial and Rational Functions. 2.2 Quadratic Functions

Chapter 2. Polynomial and Rational Functions. 2.2 Quadratic Functions Chapter 2 Polynomial and Rational Functions 2.2 Quadratic Functions 1 /27 Chapter 2 Homework 2.2 p298 1, 5, 17, 31, 37, 41, 43, 45, 47, 49, 53, 55 2 /27 Chapter 2 Objectives Recognize characteristics of

More information

Chapter 2: Polynomial and Rational Functions Power Standard #7

Chapter 2: Polynomial and Rational Functions Power Standard #7 Chapter 2: Polynomial and Rational s Power Standard #7 2.1 Quadratic s Lets glance at the finals. Learning Objective: In this lesson you learned how to sketch and analyze graphs of quadratic functions.

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 6: Analyzing Quadratic Functions Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 6: Analyzing Quadratic Functions Instruction Prerequisite Skills This lesson requires the use of the following skills: factoring quadratic expressions finding the vertex of a quadratic function Introduction We have studied the key features of the

More information

Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola. Day #1

Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola. Day #1 Algebra I Unit #3: Quadratic Functions Lesson #13: The Almighty Parabola Name Period Date Day #1 There are some important features about the graphs of quadratic functions we are going to explore over the

More information

Student Exploration: Quadratics in Polynomial Form

Student Exploration: Quadratics in Polynomial Form Name: Date: Student Exploration: Quadratics in Polynomial Form Vocabulary: axis of symmetry, parabola, quadratic function, vertex of a parabola Prior Knowledge Questions (Do these BEFORE using the Gizmo.)

More information

2. The diagram shows part of the graph of y = a (x h) 2 + k. The graph has its vertex at P, and passes through the point A with coordinates (1, 0).

2. The diagram shows part of the graph of y = a (x h) 2 + k. The graph has its vertex at P, and passes through the point A with coordinates (1, 0). Quadratics Vertex Form 1. Part of the graph of the function y = d (x m) + p is given in the diagram below. The x-intercepts are (1, 0) and (5, 0). The vertex is V(m, ). (a) Write down the value of (i)

More information

NO CALCULATOR ON ANYTHING EXCEPT WHERE NOTED

NO CALCULATOR ON ANYTHING EXCEPT WHERE NOTED Algebra II (Wilsen) Midterm Review NO CALCULATOR ON ANYTHING EXCEPT WHERE NOTED Remember: Though the problems in this packet are a good representation of many of the topics that will be on the exam, this

More information

QUESTIONS 1 10 MAY BE DONE WITH A CALCULATOR QUESTIONS ARE TO BE DONE WITHOUT A CALCULATOR. Name

QUESTIONS 1 10 MAY BE DONE WITH A CALCULATOR QUESTIONS ARE TO BE DONE WITHOUT A CALCULATOR. Name QUESTIONS 1 10 MAY BE DONE WITH A CALCULATOR QUESTIONS 11 5 ARE TO BE DONE WITHOUT A CALCULATOR Name 2 CALCULATOR MAY BE USED FOR 1-10 ONLY Use the table to find the following. x -2 2 5-0 7 2 y 12 15 18

More information

Quadratic Functions. *These are all examples of polynomial functions.

Quadratic Functions. *These are all examples of polynomial functions. Look at: f(x) = 4x-7 f(x) = 3 f(x) = x 2 + 4 Quadratic Functions *These are all examples of polynomial functions. Definition: Let n be a nonnegative integer and let a n, a n 1,..., a 2, a 1, a 0 be real

More information

x 2 + 8x - 12 = 0 Aim: To review for Quadratic Function Exam #1 Homework: Study Review Materials

x 2 + 8x - 12 = 0 Aim: To review for Quadratic Function Exam #1 Homework: Study Review Materials Aim: To review for Quadratic Function Exam #1 Homework: Study Review Materials Do Now - Solve using any strategy. If irrational, express in simplest radical form x 2 + 8x - 12 = 0 Review Topic Index 1.

More information

Lecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal

Lecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal Distance; Circles; Equations of the form Lecture 5 y = ax + bx + c In this lecture we shall derive a formula for the distance between two points in a coordinate plane, and we shall use that formula to

More information

Section 6.1: Quadratic Functions and their Characteristics Vertical Intercept Vertex Axis of Symmetry Domain and Range Horizontal Intercepts

Section 6.1: Quadratic Functions and their Characteristics Vertical Intercept Vertex Axis of Symmetry Domain and Range Horizontal Intercepts Lesson 6 Quadratic Functions and Equations Lesson 6 Quadratic Functions and Equations We are leaving exponential functions behind and entering an entirely different world. As you work through this lesson,

More information

8-4 Transforming Quadratic Functions

8-4 Transforming Quadratic Functions 8-4 Transforming Quadratic Functions Warm Up Lesson Presentation Lesson Quiz Algebra 1 Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward

More information

Sketching graphs of polynomials

Sketching graphs of polynomials Sketching graphs of polynomials We want to draw the graphs of polynomial functions y = f(x). The degree of a polynomial in one variable x is the highest power of x that remains after terms have been collected.

More information

1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check

1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check Thinkwell s Placement Test 5 Answer Key If you answered 7 or more Test 5 questions correctly, we recommend Thinkwell's Algebra. If you answered fewer than 7 Test 5 questions correctly, we recommend Thinkwell's

More information

3.1 Investigating Quadratic Functions in Vertex Form

3.1 Investigating Quadratic Functions in Vertex Form Math 2200 Date: 3.1 Investigating Quadratic Functions in Vertex Form Degree of a Function - refers to the highest exponent on the variable in an expression or equation. In Math 1201, you learned about

More information

MAC Learning Objectives. Module 4. Quadratic Functions and Equations. - Quadratic Functions - Solving Quadratic Equations

MAC Learning Objectives. Module 4. Quadratic Functions and Equations. - Quadratic Functions - Solving Quadratic Equations MAC 1105 Module 4 Quadratic Functions and Equations Learning Objectives Upon completing this module, you should be able to: 1. Understand basic concepts about quadratic functions and their graphs. 2. Complete

More information

Objective. 9-4 Transforming Quadratic Functions. Graph and transform quadratic functions.

Objective. 9-4 Transforming Quadratic Functions. Graph and transform quadratic functions. Warm Up Lesson Presentation Lesson Quiz Warm Up For each quadratic function, find the axis of symmetry and vertex, and state whether the function opens upward or downward. 1. y = x 2 + 3 2. y = 2x 2 x

More information

Algebra I. Slide 1 / 137. Slide 2 / 137. Slide 3 / 137. Quadratic & Non-Linear Functions. Table of Contents

Algebra I. Slide 1 / 137. Slide 2 / 137. Slide 3 / 137. Quadratic & Non-Linear Functions. Table of Contents Slide 1 / 137 Slide 2 / 137 Algebra I Quadratic & Non-Linear Functions 2015-11-04 www.njctl.org Table of Contents Slide 3 / 137 Click on the topic to go to that section Key Terms Explain Characteristics

More information

II. Functions. 61. Find a way to graph the line from the problem 59 on your calculator. Sketch the calculator graph here, including the window values:

II. Functions. 61. Find a way to graph the line from the problem 59 on your calculator. Sketch the calculator graph here, including the window values: II Functions Week 4 Functions: graphs, tables and formulas Problem of the Week: The Farmer s Fence A field bounded on one side by a river is to be fenced on three sides so as to form a rectangular enclosure

More information

Quadratic Functions (Section 2-1)

Quadratic Functions (Section 2-1) Quadratic Functions (Section 2-1) Section 2.1, Definition of Polynomial Function f(x) = a is the constant function f(x) = mx + b where m 0 is a linear function f(x) = ax 2 + bx + c with a 0 is a quadratic

More information

Properties of Graphs of Quadratic Functions

Properties of Graphs of Quadratic Functions H e i g h t (f t ) Lesson 2 Goal: Properties of Graphs of Quadratic Functions Identify the characteristics of graphs of quadratic functions: Vertex Intercepts Domain and Range Axis of Symmetry and use

More information

9.1: GRAPHING QUADRATICS ALGEBRA 1

9.1: GRAPHING QUADRATICS ALGEBRA 1 9.1: GRAPHING QUADRATICS ALGEBRA 1 OBJECTIVES I will be able to graph quadratics: Given in Standard Form Given in Vertex Form Given in Intercept Form What does the graph of a quadratic look like? https://www.desmos.com/calculator

More information

Section 4.4: Parabolas

Section 4.4: Parabolas Objective: Graph parabolas using the vertex, x-intercepts, and y-intercept. Just as the graph of a linear equation y mx b can be drawn, the graph of a quadratic equation y ax bx c can be drawn. The graph

More information

This is called the vertex form of the quadratic equation. To graph the equation

This is called the vertex form of the quadratic equation. To graph the equation Name Period Date: Topic: 7-5 Graphing ( ) Essential Question: What is the vertex of a parabola, and what is its axis of symmetry? Standard: F-IF.7a Objective: Graph linear and quadratic functions and show

More information

Section 5: Quadratics

Section 5: Quadratics Chapter Review Applied Calculus 46 Section 5: Quadratics Quadratics Quadratics are transformations of the f ( x) x function. Quadratics commonly arise from problems involving area and projectile motion,

More information

Test 3 review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Test 3 review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Test 3 review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Approximate the coordinates of each turning point by graphing f(x) in the standard viewing

More information

Lesson 3.1 Vertices and Intercepts. Important Features of Parabolas

Lesson 3.1 Vertices and Intercepts. Important Features of Parabolas Lesson 3.1 Vertices and Intercepts Name: _ Learning Objective: Students will be able to identify the vertex and intercepts of a parabola from its equation. CCSS.MATH.CONTENT.HSF.IF.C.7.A Graph linear and

More information

Algebra II Chapter 4: Quadratic Functions and Factoring Part 1

Algebra II Chapter 4: Quadratic Functions and Factoring Part 1 Algebra II Chapter 4: Quadratic Functions and Factoring Part 1 Chapter 4 Lesson 1 Graph Quadratic Functions in Standard Form Vocabulary 1 Example 1: Graph a Function of the Form y = ax 2 Steps: 1. Make

More information

1.1 - Functions, Domain, and Range

1.1 - Functions, Domain, and Range 1.1 - Functions, Domain, and Range Lesson Outline Section 1: Difference between relations and functions Section 2: Use the vertical line test to check if it is a relation or a function Section 3: Domain

More information

Quadratics Functions: Review

Quadratics Functions: Review Quadratics Functions: Review Name Per Review outline Quadratic function general form: Quadratic function tables and graphs (parabolas) Important places on the parabola graph [see chart below] vertex (minimum

More information

Lesson 1: Analyzing Quadratic Functions

Lesson 1: Analyzing Quadratic Functions UNIT QUADRATIC FUNCTIONS AND MODELING Lesson 1: Analyzing Quadratic Functions Common Core State Standards F IF.7 F IF.8 Essential Questions Graph functions expressed symbolically and show key features

More information

) 2 + (y 2. x 1. y c x2 = y

) 2 + (y 2. x 1. y c x2 = y Graphing Parabola Parabolas A parabola is a set of points P whose distance from a fixed point, called the focus, is equal to the perpendicular distance from P to a line, called the directrix. Since this

More information

Quadratic Equations. Learning Objectives. Quadratic Function 2. where a, b, and c are real numbers and a 0

Quadratic Equations. Learning Objectives. Quadratic Function 2. where a, b, and c are real numbers and a 0 Quadratic Equations Learning Objectives 1. Graph a quadratic function using transformations. Identify the vertex and axis of symmetry of a quadratic function 3. Graph a quadratic function using its vertex,

More information

171S3.3p Analyzing Graphs of Quadratic Functions. October 04, Vertex of a Parabola. The vertex of the graph of f (x) = ax 2 + bx + c is

171S3.3p Analyzing Graphs of Quadratic Functions. October 04, Vertex of a Parabola. The vertex of the graph of f (x) = ax 2 + bx + c is MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and

More information

Section 1.5 Transformation of Functions

Section 1.5 Transformation of Functions 6 Chapter 1 Section 1.5 Transformation of Functions Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs and equations in order to explain or

More information

Sample: Do Not Reproduce QUAD4 STUDENT PAGES. QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 4: Quadratic Functions and Applications

Sample: Do Not Reproduce QUAD4 STUDENT PAGES. QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 4: Quadratic Functions and Applications Name Period Date QUADRATIC FUNCTIONS AND EQUATIONS Student Pages for Packet 4: Quadratic Functions and Applications QUAD 4.1 Vertex Form of a Quadratic Function 1 Explore how changing the values of h and

More information

3.1 Quadratic Functions and Models

3.1 Quadratic Functions and Models 3.1 Quadratic Functions and Models Objectives: 1. Identify the vertex & axis of symmetry of a quadratic function. 2. Graph a quadratic function using its vertex, axis and intercepts. 3. Use the maximum

More information

UNIT 5 QUADRATIC FUNCTIONS Lesson 7: Building Functions Instruction

UNIT 5 QUADRATIC FUNCTIONS Lesson 7: Building Functions Instruction Prerequisite Skills This lesson requires the use of the following skills: multiplying linear expressions factoring quadratic equations finding the value of a in the vertex form of a quadratic equation

More information

Writing Equivalent Forms of Quadratic Functions Adapted from Walch Education

Writing Equivalent Forms of Quadratic Functions Adapted from Walch Education Writing Equivalent Forms of Quadratic Functions Adapted from Walch Education Recall The standard form, or general form, of a quadratic function is written as f(x) = ax 2 + bx + c, where a is the coefficient

More information

Solving Simple Quadratics 1.0 Topic: Solving Quadratics

Solving Simple Quadratics 1.0 Topic: Solving Quadratics Ns Solving Simple Quadratics 1.0 Topic: Solving Quadratics Date: Objectives: SWBAT (Solving Simple Quadratics and Application dealing with Quadratics) Main Ideas: Assignment: Square Root Property If x

More information

Section 1.5 Transformation of Functions

Section 1.5 Transformation of Functions Section 1.5 Transformation of Functions 61 Section 1.5 Transformation of Functions Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs and equations

More information

2. From General Form: y = ax 2 + bx + c # of x-intercepts determined by the, D =

2. From General Form: y = ax 2 + bx + c # of x-intercepts determined by the, D = Alg2H 5-3 Using the Discriminant, x-intercepts, and the Quadratic Formula WK#6 Lesson / Homework --Complete without calculator Read p.181-p.186. Textbook required for reference as well as to check some

More information

1.1 Functions. Cartesian Coordinate System

1.1 Functions. Cartesian Coordinate System 1.1 Functions This section deals with the topic of functions, one of the most important topics in all of mathematics. Let s discuss the idea of the Cartesian coordinate system first. Cartesian Coordinate

More information

Lesson 6 - Practice Problems

Lesson 6 - Practice Problems Lesson 6 - Practice Problems Section 6.1: Characteristics of Quadratic Functions 1. For each of the following quadratic functions: Identify the coefficients a, b and c. Determine if the parabola opens

More information

Unit 1 Quadratic Functions

Unit 1 Quadratic Functions Unit 1 Quadratic Functions This unit extends the study of quadratic functions to include in-depth analysis of general quadratic functions in both the standard form f ( x) = ax + bx + c and in the vertex

More information

Graph Quadratic Functions Using Properties *

Graph Quadratic Functions Using Properties * OpenStax-CNX module: m63466 1 Graph Quadratic Functions Using Properties * OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 By the end of this

More information

Quadratics. March 18, Quadratics.notebook. Groups of 4:

Quadratics. March 18, Quadratics.notebook. Groups of 4: Quadratics Groups of 4: For your equations: a) make a table of values b) plot the graph c) identify and label the: i) vertex ii) Axis of symmetry iii) x- and y-intercepts Group 1: Group 2 Group 3 1 What

More information

Section 6.2: Properties of Graphs of Quadratic Functions. Vertex:

Section 6.2: Properties of Graphs of Quadratic Functions. Vertex: Section 6.2: Properties of Graphs of Quadratic Functions determine the vertex of a quadratic in standard form sketch the graph determine the y intercept, x intercept(s), the equation of the axis of symmetry,

More information

Mid Term Pre Calc Review

Mid Term Pre Calc Review Mid Term 2015-13 Pre Calc Review I. Quadratic Functions a. Solve by quadratic formula, completing the square, or factoring b. Find the vertex c. Find the axis of symmetry d. Graph the quadratic function

More information

Name: Chapter 7 Review: Graphing Quadratic Functions

Name: Chapter 7 Review: Graphing Quadratic Functions Name: Chapter Review: Graphing Quadratic Functions A. Intro to Graphs of Quadratic Equations: = ax + bx+ c A is a function that can be written in the form = ax + bx+ c where a, b, and c are real numbers

More information

Math Analysis Chapter 1 Notes: Functions and Graphs

Math Analysis Chapter 1 Notes: Functions and Graphs Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 1-1 Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian coordinate system) Practice: Label each on the

More information

Step 2: Find the coordinates of the vertex (h, k) Step 5: State the zeros and interpret what they mean. Step 6: Make sure you answered all questions.

Step 2: Find the coordinates of the vertex (h, k) Step 5: State the zeros and interpret what they mean. Step 6: Make sure you answered all questions. Chapter 4 No Problem Word Problems! Name: Algebra 2 Period: 1 2 3 4 5 6 A. Solving from Standard Form 1. A ball is thrown so its height, h, in feet, is given by the equation h = 16t! + 10t where t is the

More information

Let s review some things we learned earlier about the information we can gather from the graph of a quadratic.

Let s review some things we learned earlier about the information we can gather from the graph of a quadratic. Section 6: Quadratic Equations and Functions Part 2 Section 6 Topic 1 Observations from a Graph of a Quadratic Function Let s review some things we learned earlier about the information we can gather from

More information

WK # Given: f(x) = ax2 + bx + c

WK # Given: f(x) = ax2 + bx + c Alg2H Chapter 5 Review 1. Given: f(x) = ax2 + bx + c Date or y = ax2 + bx + c Related Formulas: y-intercept: ( 0, ) Equation of Axis of Symmetry: x = Vertex: (x,y) = (, ) Discriminant = x-intercepts: When

More information

Algebra 1 Notes Quarter

Algebra 1 Notes Quarter Algebra 1 Notes Quarter 3 2014 2015 Name: ~ 1 ~ Table of Contents Unit 9 Exponent Rules Exponent Rules for Multiplication page 6 Negative and Zero Exponents page 10 Exponent Rules Involving Quotients page

More information

ALGEBRA 1 NOTES. Quarter 3. Name: Block

ALGEBRA 1 NOTES. Quarter 3. Name: Block 2016-2017 ALGEBRA 1 NOTES Quarter 3 Name: Block Table of Contents Unit 8 Exponent Rules Exponent Rules for Multiplication page 4 Negative and Zero Exponents page 8 Exponent Rules Involving Quotients page

More information

February 8 th February 12 th. Unit 6: Polynomials & Introduction to Quadratics

February 8 th February 12 th. Unit 6: Polynomials & Introduction to Quadratics Algebra I February 8 th February 12 th Unit 6: Polynomials & Introduction to Quadratics Jump Start 1) Use the elimination method to solve the system of equations below. x + y = 2 3x + y = 8 2) Solve: 13

More information

WHAT ARE THE PARTS OF A QUADRATIC?

WHAT ARE THE PARTS OF A QUADRATIC? 4.1 Introduction to Quadratics and their Graphs Standard Form of a Quadratic: y ax bx c or f x ax bx c. ex. y x. Every function/graph in the Quadratic family originates from the parent function: While

More information

College Pre Calculus A Period. Weekly Review Sheet # 1 Assigned: Monday, 9/9/2013 Due: Friday, 9/13/2013

College Pre Calculus A Period. Weekly Review Sheet # 1 Assigned: Monday, 9/9/2013 Due: Friday, 9/13/2013 College Pre Calculus A Name Period Weekly Review Sheet # 1 Assigned: Monday, 9/9/013 Due: Friday, 9/13/013 YOU MUST SHOW ALL WORK FOR EVERY QUESTION IN THE BOX BELOW AND THEN RECORD YOUR ANSWERS ON THE

More information

GSE Algebra 1 Name Date Block. Unit 3b Remediation Ticket

GSE Algebra 1 Name Date Block. Unit 3b Remediation Ticket Unit 3b Remediation Ticket Question: Which function increases faster, f(x) or g(x)? f(x) = 5x + 8; two points from g(x): (-2, 4) and (3, 10) Answer: In order to compare the rate of change (roc), you must

More information

Assignment Assignment for Lesson 9.1

Assignment Assignment for Lesson 9.1 Assignment Assignment for Lesson.1 Name Date Shifting Away Vertical and Horizontal Translations 1. Graph each cubic function on the grid. a. y x 3 b. y x 3 3 c. y x 3 3 2. Graph each square root function

More information

Math Analysis Chapter 1 Notes: Functions and Graphs

Math Analysis Chapter 1 Notes: Functions and Graphs Math Analysis Chapter 1 Notes: Functions and Graphs Day 6: Section 1-1 Graphs; Section 1- Basics of Functions and Their Graphs Points and Ordered Pairs The Rectangular Coordinate System (aka: The Cartesian

More information