Graphing Quadratics using Transformations 5-1

Transcription

1 Graphing Quadratics using Transformations 5-1

2 5-1 Using Transformations to Graph Quadratic Functions Warm Up For each translation of the point ( 2, 5), give the coordinates of the translated point units down 2. 3 units right ( 2, 1) (1, 5) For each function, evaluate f( 2), f(0), and f(3). 3. f(x) = x 2 + 2x f(x) = 2x 2 5x + 1 6; 6; 21 19; 1; 4

3 5-1 Using Transformations to Graph Quadratic Functions Transform quadratic functions. Describe the effects of changes in the coefficients of y = a(x h) 2 + k. quadratic function parabola vertex of a parabola vertex form Objectives Vocabulary

4 5-1 Using Transformations to Graph Quadratic Functions In Chapters 2 and 3, you studied linear functions of the form f(x) = mx + b. A quadratic function is a function that can be written in the form of f(x) = a (x h) 2 + k (a 0). In a quadratic function, the variable is always squared. The table shows the linear and quadratic parent functions.

5 5-1 Using Transformations to Graph Quadratic Functions Notice that the graph of the parent function f(x) = x 2 is a U- shaped curve called a parabola. As with other functions, you can graph a quadratic function by plotting points with coordinates that make the equation true.

6 5-1 Using Transformations to Graph Quadratic Functions Example 1: Graphing Quadratic Functions Using a Table Graph f(x) = x 2 4x + 3 by using a table. Make a table. Plot enough ordered pairs to see both sides of the curve. x f(x)= x 2 4x + 3 (x, f(x)) 0 f(0)= (0) 2 4(0) + 3 (0, 3) 1 f(1)= (1) 2 4(1) + 3 (1, 0) 2 f(2)= (2) 2 4(2) + 3 (2, 1) 3 f(3)= (3) 2 4(3) + 3 (3, 0) 4 f(4)= (4) 2 4(4) + 3 (4, 3)

7 5-1 Using Transformations to Graph Quadratic Functions Example 1 Continued f(x) = x 2 4x + 3

8 5-1 Using Transformations to Graph Quadratic Functions Check It Out! Example 1 Graph g(x) = x 2 + 6x 8 by using a table. Make a table. Plot enough ordered pairs to see both sides of the curve. x g(x)= x 2 +6x 8 (x, g(x)) 1 g( 1)= ( 1) 2 + 6( 1) 8 ( 1, 15) 1 g(1)= (1) 2 + 6(1) 8 (1, 3) 3 g(3)= (3) 2 + 6(3) 8 (3, 1) 5 g(5)= (5) 2 + 6(5) 8 (5, 3) 7 g(7)= (7) 2 + 6(7) 8 (7, 15)

9 5-1 Using Transformations to Graph Quadratic Functions Check It Out! Example 1 Continued f(x) = x 2 + 6x 8

10 5-1 Using Transformations to Graph Quadratic Functions You can also graph quadratic functions by applying transformations to the parent function f(x) = x 2. Transforming quadratic functions is similar to transforming linear functions (Lesson 2-6).

11 5-1 Using Transformations to Graph Quadratic Functions Example 2A: Translating Quadratic Functions Use the graph of f(x) = x 2 as a guide, describe the transformations and then graph each function. g(x) = (x 2) Identify h and k. g(x) = (x 2) h k Because h = 2, the graph is translated 2 units right. Because k = 4, the graph is translated 4 units up. Therefore, g is f translated 2 units right and 4 units up.

12 5-1 Using Transformations to Graph Quadratic Functions Example 2B: Translating Quadratic Functions Use the graph of f(x) = x 2 as a guide, describe the transformations and then graph each function. g(x) = (x + 2) 2 3 Identify h and k. g(x) = (x ( 2)) 2 + ( 3) h k Because h = 2, the graph is translated 2 units left. Because k = 3, the graph is translated 3 units down. Therefore, g is f translated 2 units left and 4 units down.

13 5-1 Using Transformations to Graph Quadratic Functions Check It Out! Example 2a Using the graph of f(x) = x 2 as a guide, describe the transformations and then graph each function. g(x) = x 2 5 Identify h and k. g(x) = x 2 5 k Because h = 0, the graph is not translated horizontally. Because k = 5, the graph is translated 5 units down. Therefore, g is f is translated 5 units down.

14 5-1 Using Transformations to Graph Quadratic Functions Check It Out! Example 2b Use the graph of f(x) =x 2 as a guide, describe the transformations and then graph each function. g(x) = (x + 3) 2 2 Identify h and k. g(x) = (x ( 3)) 2 + ( 2) h k Because h = 3, the graph is translated 3 units left. Because k = 2, the graph is translated 2 units down. Therefore, g is f translated 3 units left and 2 units down.

15 5-1 Using Transformations to Graph Quadratic Functions Recall that functions can also be reflected, stretched, or compressed.

16 5-1 Using Transformations to Graph Quadratic Functions

17 5-1 Using Transformations to Graph Quadratic Functions Example 3A: Reflecting, Stretching, and Compressing Quadratic Functions Using the graph of f(x) = x 2 as a guide, describe the transformations and then graph each function. g ( x)= 1 4 x 2 Because a is negative, g is a reflection of f across the x- axis. Because a =, g is a vertical compression of f by a factor of.

18 5-1 Using Transformations to Graph Quadratic Functions Example 3B: Reflecting, Stretching, and Compressing Quadratic Functions Using the graph of f(x) = x 2 as a guide, describe the transformations and then graph each function. g(x) =(3x) 2 Because b =, g is a horizontal compression of f by a factor of.

19 5-1 Using Transformations to Graph Quadratic Functions Using the graph of f(x) = x 2 as a guide, describe the transformations and then graph each function. g(x) =(2x) 2 Because b =, g is a horizontal compression of f by a factor of. Check It Out! Example 3a

20 5-1 Using Transformations to Graph Quadratic Functions Using the graph of f(x) = x 2 as a guide, describe the transformations and then graph each function. g(x) = x 2 Because a is negative, g is a reflection of f across the x-axis. Because a =, g is a vertical compression of f by a factor of. Check It Out! Example 3b

21 5-1 Using Transformations to Graph Quadratic Functions If a parabola opens upward, it has a lowest point. If a parabola opens downward, it has a highest point. This lowest or highest point is the vertex of the parabola. The parent function f(x) = x 2 has its vertex at the origin. You can identify the vertex of other quadratic functions by analyzing the function in vertex form. The vertex form of a quadratic function is f(x) = a(x h) 2 + k, where a, h, and k are constants.

22 5-1 Using Transformations to Graph Quadratic Functions Because the vertex is translated h horizontal units and k vertical from the origin, the vertex of the parabola is at (h, k). Helpful Hint When the quadratic parent function f(x) = x 2 is written in vertex form, y = a(x h) 2 + k, a = 1, h = 0, and k = 0.

23 5-1 Using Transformations to Graph Quadratic Functions Example 4: Writing Transformed Quadratic Functions Use the description to write the quadratic function in vertex form. The parent function f(x) = x 2 is vertically 4 stretched by a factor of and then translated 2 units 3 left and 5 units down to create g. Step 1 Identify how each transformation affects the constant in vertex form. Vertical stretch by : a = Translation 2 units left: h = 2 Translation 5 units down: k =

24 5-1 Using Transformations to Graph Quadratic Functions Example 4: Writing Transformed Quadratic Functions Step 2 Write the transformed function. g(x) = a(x h) 2 + k 4 = (x ( 2)) 2 + ( 5) 3 4 = (x + 2) Vertex form of a quadratic function 4 Substitute for a, 2 for h, and 5 for k. 3 Simplify. g(x) = (x + 2)

25 5-1 Using Transformations to Graph Quadratic Functions Check Graph both functions on a graphing calculator. Enter f as Y 1, and g as Y 2. The graph indicates the identified transformations. f g

26 5-1 Using Transformations to Graph Quadratic Functions Check It Out! Example 4a Use the description to write the quadratic function in vertex form. The parent function f(x) = x 2 is vertically compressed by a factor of and then translated 2 units right and 4 units down to create g. Step 1 Identify how each transformation affects the constant in vertex form. Vertical compression by : a = Translation 2 units right: h = 2 Translation 4 units down: k = 4

27 5-1 Using Transformations to Graph Quadratic Functions Check It Out! Example 4a Continued Step 2 Write the transformed function. g(x) = a(x h) 2 + k = (x 2) 2 + ( 4) Vertex form of a quadratic function Substitute for a, 2 for h, and 4 for k. = (x 2) 2 4 Simplify. g(x) = (x 2) 2 4

28 5-1 Using Transformations to Graph Quadratic Functions Check It Out! Example 4a Continued Check Graph both functions on a graphing calculator. Enter f as Y 1, and g as Y 2. The graph indicates the identified transformations. f g

29 5-1 Using Transformations to Graph Quadratic Functions Check It Out! Example 4b Use the description to write the quadratic function in vertex form. The parent function f(x) = x 2 is reflected across the x-axis and translated 5 units left and 1 unit up to create g. Step 1 Identify how each transformation affects the constant in vertex form. Reflected across the x-axis: a is negative Translation 5 units left: h = 5 Translation 1 unit up: k = 1

30 5-1 Using Transformations to Graph Quadratic Functions Check It Out! Example 4b Continued Step 2 Write the transformed function. g(x) = a(x h) 2 + k = (x ( 5) 2 + (1) = (x +5) Vertex form of a quadratic function Substitute 1 for a, 5 for h, and 1 for k. Simplify. g(x) = (x +5) 2 + 1

31 5-1 Using Transformations to Graph Quadratic Functions Check It Out! Example 4b Continued Check Graph both functions on a graphing calculator. Enter f as Y 1, and g as Y 2. The graph indicates the identified transformations. f g

32 5-1 Using Transformations to Graph Quadratic Functions Lesson Quiz: Part I 1. Graph f(x) = x 2 + 3x 1 by using a table.

33 5-1 Using Transformations to Graph Quadratic Functions Lesson Quiz: Part II 2. Using the graph of f(x) = x 2 as a guide, describe the transformations, and then graph g(x) = (x + 1) 2. g is f reflected across x-axis, vertically compressed by a factor of, and translated 1 unit left.

34 5-1 Using Transformations to Graph Quadratic Functions Lesson Quiz: Part III 3. The parent function f(x) = x 2 is vertically stretched by a factor of 3 and translated 4 units right and 2 units up to create g. Write g in vertex form. g(x) = 3(x 4) 2 + 2

35 Quadratic Equations 5-2

36 5-2 Properties of Quadratic Functions in Standard Form Objectives Define, identify, and graph quadratic functions. Identify and use maximums and minimums of quadratic functions to solve problems.

37 5-2 Properties of Quadratic Functions in Standard Form axis of symmetry standard form minimum value maximum value Vocabulary

38 5-2 Properties of Quadratic Functions in Standard Form When you transformed quadratic functions in the previous lesson, you saw that reflecting the parent function across the y-axis results in the same function.

39 5-2 Properties of Quadratic Functions in Standard Form This shows that parabolas are symmetric curves. The axis of symmetry is the line through the vertex of a parabola that divides the parabola into two congruent halves.

40 5-2 Properties of Quadratic Functions in Standard Form Example 1: Identifying the Axis of Symmetry Identify the axis of symmetry for the graph of. Rewrite the function to find the value of h. Because h = 5, the axis of symmetry is the vertical line x = 5.

41 5-2 Properties of Quadratic Functions in Standard Form Check It Out! Example1 Identify the axis of symmetry for the graph of ( x ) f ( x ) = Rewrite the function to find the value of h. f(x) = [x - (3)] Because h = 3, the axis of symmetry is the vertical line x = 3.

42 5-2 Properties of Quadratic Functions in Standard Form Another useful form of writing quadratic functions is the standard form. The standard form of a quadratic function is f(x)= ax 2 + bx + c, where a 0. The coefficients a, b, and c can show properties of the graph of the function. You can determine these properties by expanding the vertex form. f(x)= a(x h) 2 + k f(x)= a(x 2 2xh +h 2 ) + k f(x)= a(x 2 ) a(2hx) + a(h 2 ) + k f(x)= ax 2 + ( 2ah)x + (ah 2 + k) Multiply to expand (x h) 2. Distribute a. Simplify and group terms.

43 5-2 Properties of Quadratic Functions in Standard Form a = a a in standard form is the same as in vertex form. It indicates whether a reflection and/or vertical stretch or compression has been applied.

44 5-2 Properties of Quadratic Functions in Standard Form b = 2ah Solving for h gives. Therefore, the axis of symmetry, x = h, for a quadratic function in standard form is.

45 5-2 Properties of Quadratic Functions in Standard Form c = ah 2 + k Notice that the value of c is the same value given by the vertex form of f when x = 0: f(0) = a(0 h) 2 + k = ah 2 + k. So c is the y-intercept.

46 5-2 Properties of Quadratic Functions in Standard Form These properties can be generalized to help you graph quadratic functions.

47 5-2 Properties of Quadratic Functions in Standard Form Helpful Hint When a is positive, the parabola is happy (U). When the a negative, the parabola is sad ( ). U

48 5-2 Properties of Quadratic Functions in Standard Form Example 2A: Graphing Quadratic Functions in Standard Form Consider the function f(x) = 2x 2 4x + 5. a. Determine whether the graph opens upward or downward. Because a is positive, the parabola opens upward. b. Find the axis of symmetry. The axis of symmetry is given by. Substitute 4 for b and 2 for a. The axis of symmetry is the line x = 1.

49 5-2 Properties of Quadratic Functions in Standard Form Example 2A: Graphing Quadratic Functions in Standard Form Consider the function f(x) = 2x 2 4x + 5. c. Find the vertex. The vertex lies on the axis of symmetry, so the x-coordinate is 1. The y-coordinate is the value of the function at this x-value, or f(1). f(1) = 2(1) 2 4(1) + 5 = 3 The vertex is (1, 3). d. Find the y-intercept. Because c = 5, the intercept is 5.

50 5-2 Properties of Quadratic Functions in Standard Form Example 2A: Graphing Quadratic Functions in Standard Form Consider the function f(x) = 2x 2 4x + 5. e. Graph the function. Graph by sketching the axis of symmetry and then plotting the vertex and the intercept point (0, 5). Use the axis of symmetry to find another point on the parabola. Notice that (0, 5) is 1 unit left of the axis of symmetry. The point on the parabola symmetrical to (0, 5) is 1 unit to the right of the axis at (2, 5).

51 5-2 Properties of Quadratic Functions in Standard Form Example 2B: Graphing Quadratic Functions in Standard Form Consider the function f(x) = x 2 2x + 3. a. Determine whether the graph opens upward or downward. Because a is negative, the parabola opens downward. b. Find the axis of symmetry. The axis of symmetry is given by. Substitute 2 for b and 1 for a. The axis of symmetry is the line x = 1.

52 5-2 Properties of Quadratic Functions in Standard Form Example 2B: Graphing Quadratic Functions in Standard Form Consider the function f(x) = x 2 2x + 3. c. Find the vertex. The vertex lies on the axis of symmetry, so the x- coordinate is 1. The y-coordinate is the value of the function at this x-value, or f( 1). f( 1) = ( 1) 2 2( 1) + 3 = 4 The vertex is ( 1, 4). d. Find the y-intercept. Because c = 3, the y-intercept is 3.

53 5-2 Properties of Quadratic Functions in Standard Form Example 2B: Graphing Quadratic Functions in Standard Form Consider the function f(x) = x 2 2x + 3. e. Graph the function. Graph by sketching the axis of symmetry and then plotting the vertex and the intercept point (0, 3). Use the axis of symmetry to find another point on the parabola. Notice that (0, 3) is 1 unit right of the axis of symmetry. The point on the parabola symmetrical to (0, 3) is 1 unit to the left of the axis at ( 2, 3).

54 5-2 Properties of Quadratic Functions in Standard Form For the function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y- intercept, and (e) graph the function. f(x)= 2x 2 4x Check It Out! Example 2a a. Because a is negative, the parabola opens downward. b. The axis of symmetry is given by. Substitute 4 for b and 2 for a. The axis of symmetry is the line x = 1.

55 5-2 Properties of Quadratic Functions in Standard Form f(x)= 2x 2 4x Check It Out! Example 2a c. The vertex lies on the axis of symmetry, so the x-coordinate is 1. The y-coordinate is the value of the function at this x-value, or f( 1). f( 1) = 2( 1) 2 4( 1) = 2 The vertex is ( 1, 2). d. Because c is 0, the y-intercept is 0.

56 5-2 Properties of Quadratic Functions in Standard Form f(x)= 2x 2 4x Check It Out! Example 2a e. Graph the function. Graph by sketching the axis of symmetry and then plotting the vertex and the intercept point (0, 0). Use the axis of symmetry to find another point on the parabola. Notice that (0, 0) is 1 unit right of the axis of symmetry. The point on the parabola symmetrical to (0,0) is 1 unit to the left of the axis at (0, 2).

57 5-2 Properties of Quadratic Functions in Standard Form g(x)= x 2 + 3x 1. Check It Out! Example 2b For the function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y- intercept, and (e) graph the function. a. Because a is positive, the parabola opens upward. b. The axis of symmetry is given by. Substitute 3 for b and 1 for a. The axis of symmetry is the line.

58 5-2 Properties of Quadratic Functions in Standard Form g(x)= x 2 + 3x 1 Check It Out! Example 2b c. The vertex lies on the axis of symmetry, so the x-coordinate is. The y-coordinate is the value of the function at this x-value, or f( ). f( ) = ( ) 2 + 3( ) 1 = The vertex is (, ). d. Because c = 1, the intercept is 1.

59 5-2 Properties of Quadratic Functions in Standard Form Check It Out! Example2 e. Graph the function. Graph by sketching the axis of symmetry and then plotting the vertex and the intercept point (0, 1). Use the axis of symmetry to find another point on the parabola. Notice that (0, 1) is 1.5 units right of the axis of symmetry. The point on the parabola symmetrical to (0, 1) is 1.5 units to the left of the axis at ( 3, 1).

60 5-2 Properties of Quadratic Functions in Standard Form Substituting any real value of x into a quadratic equation results in a real number. Therefore, the domain of any quadratic function is all real numbers. The range of a quadratic function depends on its vertex and the direction that the parabola opens.

61 5-2 Properties of Quadratic Functions in Standard Form

62 5-2 Properties of Quadratic Functions in Standard Form Caution! The minimum (or maximum) value is the y-value at the vertex. It is not the ordered pair that represents the vertex.

63 5-2 Properties of Quadratic Functions in Standard Form Example 3: Finding Minimum or Maximum Values Find the minimum or maximum value of f(x) = 3x 2 + 2x 4. Then state the domain and range of the function. Step 1 Determine whether the function has minimum or maximum value. Because a is negative, the graph opens downward and has a maximum value. Step 2 Find the x-value of the vertex. Substitute 2 for b and 3 for a.

64 5-2 Properties of Quadratic Functions in Standard Form Example 3 Continued Find the minimum or maximum value of f(x) = 3x 2 + 2x 4. Then state the domain and range of the function. Step 3 Then find the y-value of the vertex, The maximum value is. The domain is all real numbers, R. The range is all real numbers less than or equal to

65 5-2 Properties of Quadratic Functions in Standard Form Example 3 Continued Check Graph f(x)= 3x 2 + 2x 4 on a graphing calculator. The graph and table support the answer.

66 5-2 Properties of Quadratic Functions in Standard Form Check It Out! Example 3a Find the minimum or maximum value of f(x) = x 2 6x + 3. Then state the domain and range of the function. Step 1 Determine whether the function has minimum or maximum value. Because a is positive, the graph opens upward and has a minimum value. Step 2 Find the x-value of the vertex.

67 5-2 Properties of Quadratic Functions in Standard Form Check It Out! Example 3a Continued Find the minimum or maximum value of f(x) = x 2 6x + 3. Then state the domain and range of the function. Step 3 Then find the y-value of the vertex, f(3) = (3) 2 6(3) + 3 = 6 The minimum value is 6. The domain is all real numbers, R. The range is all real numbers greater than or equal to 6, or {y y 6}.

68 5-2 Properties of Quadratic Functions in Standard Form Check It Out! Example 3a Continued Check Graph f(x)=x 2 6x + 3 on a graphing calculator. The graph and table support the answer.

69 5-2 Properties of Quadratic Functions in Standard Form Check It Out! Example 3b Find the minimum or maximum value of g(x) = 2x 2 4. Then state the domain and range of the function. Step 1 Determine whether the function has minimum or maximum value. Because a is negative, the graph opens downward and has a maximum value. Step 2 Find the x-value of the vertex.

70 5-2 Properties of Quadratic Functions in Standard Form Check It Out! Example 3b Continued Find the minimum or maximum value of g(x) = 2x 2 4. Then state the domain and range of the function. Step 3 Then find the y-value of the vertex, f(0) = 2(0) 2 4 = 4 The maximum value is 4. The domain is all real numbers, R. The range is all real numbers less than or equal to 4, or {y y 4}.

71 5-2 Properties of Quadratic Functions in Standard Form Check It Out! Example 3b Continued Check Graph f(x)= 2x 2 4 on a graphing calculator. The graph and table support the answer.

72 5-2 Properties of Quadratic Functions in Standard Form Lesson Quiz: Part I Consider the function f(x)= 2x 2 + 6x Determine whether the graph opens upward or downward. upward 2. Find the axis of symmetry. x = Find the vertex. ( 1.5, 11.5) 4. Identify the maximum or minimum value of the function. min.: Find the y-intercept. 7

73 5-2 Properties of Quadratic Functions in Standard Form Lesson Quiz: Part II Consider the function f(x)= 2x 2 + 6x Graph the function. 7. Find the domain and range of the function. D: All real numbers; R {y y 11.5}

74 Solving Quadratic Equations by Graphing and Factoring 5-3

75 5-3 Solving Quadratic Equations by Graphing and Factoring Objectives Solve quadratic equations by graphing or factoring. Determine a quadratic function from its roots.

76 5-3 Solving Quadratic Equations by Graphing and Factoring zero of a function root of an equation binomial trinomial Vocabulary

77 5-3 Solving Quadratic Equations by Graphing and Factoring When a soccer ball is kicked into the air, how long will the ball take to hit the ground? The height h in feet of the ball after t seconds can be modeled by the quadratic function h(t) = 16t t. In this situation, the value of the function represents the height of the soccer ball. When the ball hits the ground, the value of the function is zero.

78 5-3 Solving Quadratic Equations by Graphing and Factoring A zero of a function is a value of the input x that makes the output f(x) equal zero. The zeros of a function are the x-intercepts. Unlike linear functions, which have no more than one zero, quadratic functions can have two zeros, as shown at right. These zeros are always symmetric about the axis of symmetry.

79 5-3 Solving Quadratic Equations by Graphing and Factoring Helpful Hint Recall that for the graph of a quadratic function, any pair of points with the same y-value are symmetric about the axis of symmetry.

80 5-3 Solving Quadratic Equations by Graphing and Factoring Example 1: Finding Zeros by Using a Graph or Table Find the zeros of f(x) = x 2 6x + 8 by using a graph and table. Method 1 Graph the function f(x) = x 2 6x + 8. The graph opens upward because a > 0. The y-intercept is 8 because c = 8. Find the vertex: The x-coordinate of the vertex is.

81 5-3 Solving Quadratic Equations by Graphing and Factoring Example 1 Continued Find the zeros of f(x) = x 2 6x + 8 by using a graph and table. Find f(3): f(x) = x 2 6x + 8 f(3) = (3) 2 6(3) + 8 f(3) = f(3) = 1 Substitute 3 for x. The vertex is (3, 1)

82 5-3 Solving Quadratic Equations by Graphing and Factoring Example 1 Continued Plot the vertex and the y-intercept. Use symmetry and a table of values to find additional points. x f(x) The table and the graph indicate that the zeros are 2 and 4. (2, 0) (4, 0)

83 5-3 Solving Quadratic Equations by Graphing and Factoring Example 1 Continued Find the zeros of f(x) = x 2 6x + 8 by using a graph and table. Method 2 Use a calculator. Enter y = x 2 6x + 8 into a graphing calculator. Both the table and the graph show that y = 0 at x = 2 and x = 4. These are the zeros of the function.

84 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 1 Find the zeros of g(x) = x 2 2x + 3 by using a graph and a table. Method 1 Graph the function g(x) = x 2 2x + 3. The graph opens downward because a < 0. The y-intercept is 3 because c = 3. Find the vertex: The x-coordinate of the vertex is.

85 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 1 Continued Find the zeros of g(x) = x 2 2x + 3 by using a graph and table. Find g(1): g(x) = x 2 2x + 3 g( 1) = ( 1) 2 2( 1) + 3 g( 1) = g( 1) = 4 Substitute 1 for x. The vertex is ( 1, 4)

86 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 1 Continued Plot the vertex and the y-intercept. Use symmetry and a table of values to find additional points. x f(x) ( 3, 0) (1, 0) The table and the graph indicate that the zeros are 3 and 1.

87 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 1 Continued Find the zeros of f(x) = x 2 2x + 3 by using a graph and table. Method 2 Use a calculator. Enter y = x 2 2x + 3 into a graphing calculator. Both the table and the graph show that y = 0 at x = 3 and x = 1. These are the zeros of the function.

88 5-3 Solving Quadratic Equations by Graphing and Factoring You can also find zeros by using algebra. For example, to find the zeros of f(x)= x 2 + 2x 3, you can set the function equal to zero. The solutions to the related equation x 2 + 2x 3 = 0 represent the zeros of the function. The solution to a quadratic equation of the form ax 2 + bx + c = 0 are roots. The roots of an equation are the values of the variable that make the equation true.

89 5-3 Solving Quadratic Equations by Graphing and Factoring You can find the roots of some quadratic equations by factoring and applying the Zero Product Property. Reading Math Functions have zeros or x-intercepts. Equations have solutions or roots.

90 5-3 Solving Quadratic Equations by Graphing and Factoring Example 2A: Finding Zeros by Factoring Find the zeros of the function by factoring. f(x) = x 2 4x 12 x 2 4x 12 = 0 (x + 2)(x 6) = 0 x + 2 = 0 or x 6 = 0 x= 2 or x = 6 Set the function equal to 0. Factor: Find factors of 12 that add to 4. Apply the Zero Product Property. Solve each equation.

91 5-3 Solving Quadratic Equations by Graphing and Factoring Example 2A Continued Find the zeros of the function by factoring. Check Substitute each value into original equation. x 2 4x 12 = 0 x 2 4x 12 = 0 ( 2) 2 4( 2) 12 0 (6) 2 4(6)

92 5-3 Solving Quadratic Equations by Graphing and Factoring Example 2B: Finding Zeros by Factoring Find the zeros of the function by factoring. g(x) = 3x x 3x x = 0 3x(x+6) = 0 3x = 0 or x + 6 = 0 Set the function to equal to 0. Factor: The GCF is 3x. Apply the Zero Product Property. x = 0 or x = 6 Solve each equation.

93 5-3 Solving Quadratic Equations by Graphing and Factoring Example 2B Continued Check Check algebraically and by graphing. 3x x = 0 3x x = 0 3(0) (0) ( 6) ( 6)

94 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 2a Find the zeros of the function by factoring. f(x)= x 2 5x 6 x 2 5x 6 = 0 (x + 1)(x 6) = 0 x + 1 = 0 or x 6 = 0 x = 1 or x = 6 Set the function equal to 0. Factor: Find factors of 6 that add to 5. Apply the Zero Product Property. Solve each equation.

95 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 2a Continued Find the zeros of the function by factoring. Check Substitute each value into original equation. x 2 5x 6 = 0 ( 1) 2 5( 1) x 2 5x 6 = 0 (6) 2 5(6)

96 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 2b Find the zeros of the function by factoring. g(x) = x 2 8x x 2 8x = 0 x(x 8) = 0 x = 0 or x 8 = 0 x = 0 or x = 8 Set the function to equal to 0. Factor: The GCF is x. Apply the Zero Product Property. Solve each equation.

97 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 2b Continued Find the zeros of the function by factoring. Check Substitute each value into original equation. x 2 8x = 0 x 2 8x = 0 (0) 2 8(0) 0 (8) 2 8(8)

98 5-3 Solving Quadratic Equations by Graphing and Factoring Any object that is thrown or launched into the air, such as a baseball, basketball, or soccer ball, is a projectile. The general function that approximates the height h in feet of a projectile on Earth after t seconds is given. Note that this model has limitations because it does not account for air resistance, wind, and other realworld factors.

99 5-3 Solving Quadratic Equations by Graphing and Factoring Example 3: Sports Application A golf ball is hit from ground level with an initial vertical velocity of 80 ft/s. After how many seconds will the ball hit the ground? h(t) = 16t 2 + v 0 t + h 0 Write the general projectile function. h(t) = 16t t + 0 Substitute 80 for v 0 and 0 for h 0.

100 5-3 Solving Quadratic Equations by Graphing and Factoring Example 3 Continued The ball will hit the ground when its height is zero. 16t t = 0 16t(t 5) = 0 16t = 0 or (t 5) = 0 t = 0 or t = 5 Set h(t) equal to 0. Factor: The GCF is 16t. Apply the Zero Product Property. Solve each equation. The golf ball will hit the ground after 5 seconds. Notice that the height is also zero when t = 0, the instant that the golf ball is hit.

101 5-3 Solving Quadratic Equations by Graphing and Factoring Example 3 Continued Check The graph of the function h(t) = 16t t shows its zeros at 0 and

102 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 3 A football is kicked from ground level with an initial vertical velocity of 48 ft/s. How long is the ball in the air? h(t) = 16t 2 + v 0 t + h 0 Write the general projectile function. h(t) = 16t t + 0 Substitute 48 for v 0 and 0 for h 0.

103 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 3 Continued The ball will hit the ground when its height is zero. 16t t = 0 16t(t 3) = 0 16t = 0 or (t 3) = 0 t = 0 or t = 3 Set h(t) equal to 0. Factor: The GCF is 16t. Apply the Zero Product Property. Solve each equation. The football will hit the ground after 3 seconds. Notice that the height is also zero when t = 0, the instant that the football is hit.

104 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 3 Continued Check The graph of the function h(t) = 16t t shows its zeros at 0 and

105 5-3 Solving Quadratic Equations by Graphing and Factoring Quadratic expressions can have one, two or three terms, such as 16t 2, 16t t, or 16t t + 2. Quadratic expressions with two terms are binomials. Quadratic expressions with three terms are trinomials. Some quadratic expressions with perfect squares have special factoring rules.

106 5-3 Solving Quadratic Equations by Graphing and Factoring Example 4A: Find Roots by Using Special Factors Find the roots of the equation by factoring. 4x 2 = 25 4x 2 25 = 0 (2x) 2 (5) 2 = 0 (2x + 5)(2x 5) = 0 2x + 5 = 0 or 2x 5 = 0 x = or x = Rewrite in standard form. Write the left side as a 2 b 2. Factor the difference of squares. Apply the Zero Product Property. Solve each equation.

107 5-3 Solving Quadratic Equations by Graphing and Factoring Check Graph the related function f(x) = 4x 2 25 on a graphing calculator. The function appears to have zeros at and. Example 4 Continued

108 5-3 Solving Quadratic Equations by Graphing and Factoring Example 4B: Find Roots by Using Special Factors Find the roots of the equation by factoring. 18x 2 = 48x 32 18x 2 48x + 32 = 0 2(9x 2 24x + 16) = 0 9x 2 24x + 16 = 0 (3x) 2 2(3x)(4) + (4) 2 = 0 (3x 4) 2 = 0 3x 4 = 0 or 3x 4 = 0 x = or x = Rewrite in standard form. Factor. The GCF is 2. Divide both sides by 2. Write the left side as a 2 2ab +b 2. Factor the perfect-square trinomial. Apply the Zero Product Property. Solve each equation.

109 5-3 Solving Quadratic Equations by Graphing and Factoring Example 4B Continued Check Substitute the root into the original equation. 18x 2 = 48x

110 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 4a Find the roots of the equation by factoring. x 2 4x = 4 x 2 4x + 4 = 0 (x 2)(x 2) = 0 x 2 = 0 or x 2 = 0 x = 2 or x = 2 Rewrite in standard form. Factor the perfect-square trinomial. Apply the Zero Product Property. Solve each equation.

111 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 4a Continued Check Substitute the root 2 into the original equation. x 2 4x = 4 (2) 2 4(2)

112 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 4b Find the roots of the equation by factoring. 25x 2 = 9 25x 2 9 = 0 Rewrite in standard form. (5x) 2 (3) 2 = 0 (5x + 3)(5x 3) = 0 5x + 3 = 0 or 5x 3 = 0 x = or x = Write the left side as a 2 b 2. Factor the difference of squares. Apply the Zero Product Property. Solve each equation.

113 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 4b Continued Check Graph the related function f(x) = 25x 2 9 on a graphing calculator. The function appears to have zeros at and

114 5-3 Solving Quadratic Equations by Graphing and Factoring If you know the zeros of a function, you can work backward to write a rule for the function

115 5-3 Solving Quadratic Equations by Graphing and Factoring Example 5: Using Zeros to Write Function Rules Write a quadratic function in standard form with zeros 4 and 7. x = 4 or x = 7 x 4 = 0 or x + 7 = 0 (x 4)(x + 7) = 0 x 2 + 3x 28 = 0 f(x) = x 2 + 3x 28 Write the zeros as solutions for two equations. Rewrite each equation so that it equals 0. Apply the converse of the Zero Product Property to write a product that equals 0. Multiply the binomials. Replace 0 with f(x).

116 5-3 Solving Quadratic Equations by Graphing and Factoring Example 5 Continued Check Graph the function f(x) = x 2 + 3x 28 on a calculator. The graph shows the original zeros of 4 and

117 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 5 Write a quadratic function in standard form with zeros 5 and 5. x = 5 or x = 5 x + 5 = 0 or x 5 = 0 (x + 5)(x 5) = 0 x 2 25 = 0 f(x) = x 2 25 Write the zeros as solutions for two equations. Rewrite each equation so that it equals 0. Apply the converse of the Zero Product Property to write a product that equals 0. Multiply the binomials. Replace 0 with f(x).

118 5-3 Solving Quadratic Equations by Graphing and Factoring Check It Out! Example 5 Continued Check Graph the function f(x) = x 2 25 on a calculator. The graph shows the original zeros of 5 and

119 5-3 Solving Quadratic Equations by Graphing and Factoring Note that there are many quadratic functions with the same zeros. For example, the functions f(x) = x 2 x 2, g(x) = x 2 + x + 2, and h(x) = 2x 2 2x 4 all have zeros at 2 and

120 5-3 Solving Quadratic Equations by Graphing and Factoring Lesson Quiz: Part I Find the zeros of each function. 1. f(x)= x 2 7x 0, 7 2. f(x) = x 2 9x , 5 Find the roots of each equation using factoring. 3. x 2 10x + 25 = x = 15 2x 2 5,

121 Completing the Square 5-4

122 5-4 Completing the Square Objectives Solve quadratic equations by completing the square. Write quadratic equations in vertex form. completing the square Vocabulary

123 5-4 Completing the Square Many quadratic equations contain expressions that cannot be easily factored. For equations containing these types of expressions, you can use square roots to find roots.

124 5-4 Completing the Square Reading Math Read as plus or minus square root of a.

125 5-4 Completing the Square Example 1A: Solving Equations by Using the Square Root Property Solve the equation. 4x = 59 4x 2 = 48 x 2 = 12 Subtract 11 from both sides. Divide both sides by 4 to isolate the square term. Take the square root of both sides. Simplify.

126 5-4 Completing the Square Example 1B: Solving Equations by Using the Square Root Property Solve the equation. x x + 36 = 28 (x + 6) 2 = 28 Factor the perfect square trinomial Take the square root of both sides. Subtract 6 from both sides. Simplify.

127 5-4 Completing the Square Check It Out! Example 1a Solve the equation. 4x 2 20 = 5 4x 2 = x = 4 Add 20 to both sides. Divide both sides by 4 to isolate the square term. Take the square root of both sides. Simplify.

128 5-4 Completing the Square Check It Out! Example 1a Continued Check Use a graphing calculator.

129 5-4 Completing the Square Check It Out! Example 1b Solve the equation. x 2 + 8x + 16 = 49 (x + 4) 2 = 49 Factor the perfect square trinomial. Take the square root of both sides. x = 4 ± 49 Subtract 4 from both sides. x = 11, 3 Simplify.

130 5-4 Completing the Square The methods in the previous examples can be used only for expressions that are perfect squares. However, you can use algebra to rewrite any quadratic expression as a perfect square. You can use algebra tiles to model a perfect square trinomial as a perfect square. The area of the square at right is x 2 + 2x + 1. Because each side of the square measures x + 1 units, the area is also (x + 1)(x + 1), or (x + 1) 2. This shows that (x + 1) 2 = x 2 + 2x + 1.

131 5-4 Completing the Square If a quadratic expression of the form x 2 + bx cannot model a square, you can add a term to form a perfect square trinomial. This is called completing the square.

132 5-4 Completing the Square The model shows completing the square for x 2 + 6x by adding 9 unit tiles. The resulting perfect square trinomial is x 2 + 6x + 9. Note that completing the square does not produce an equivalent expression.

133 5-4 Completing the Square Complete the square for the expression. Write the resulting expression as a binomial squared. x 2 14x + Example 2A: Completing the Square Find. Check x 2 14x + 49 Add. (x 7) 2 Factor. Find the square of the binomial. (x 7) 2 = (x 7)(x 7) = x 2 14x + 49

134 5-4 Completing the Square Example 2B: Completing the Square Complete the square for the expression. Write the resulting expression as a binomial squared. x 2 + 9x + Find. Check Find the square of the binomial. Add. Factor.

135 5-4 Completing the Square Check It Out! Example 2a Complete the square for the expression. Write the resulting expression as a binomial squared. x 2 + 4x + Find. x 2 + 4x + 4 (x + 2) 2 Add. Factor. Check Find the square of the binomial. (x + 2) 2 = (x + 2)(x + 2) = x 2 + 4x + 4

136 5-4 Completing the Square Check It Out! Example 2b Complete the square for the expression. Write the resulting expression as a binomial squared. x 2 4x + Find. x 2 4x + 4 (x 2) 2 Add. Factor. Check Find the square of the binomial. (x 2) 2 = (x 2)(x 2) = x 2 4x + 4

137 5-4 Completing the Square Check It Out! Example 2c Complete the square for the expression. Write the resulting expression as a binomial squared. x 2 + 3x + Find. Check Find the square of the binomial. Add. Factor.

138 5-4 Completing the Square You can complete the square to solve quadratic equations.

139 5-4 Completing the Square Example 3A: Solving a Quadratic Equation by Completing the Square Solve the equation by completing the square. x 2 = 12x 20 x 2 12x = 20 x 2 12x + = 20 + Collect variable terms on one side. Set up to complete the square. Add to both sides. x 2 12x + 36 = Simplify.

140 5-4 Completing the Square Example 3A Continued (x 6) 2 = 16 Factor. Take the square root of both sides. x 6 = ±4 x 6 = 4 or x 6 = 4 Simplify. Solve for x. x = 10 or x = 2

141 5-4 Completing the Square Example 3B: Solving a Quadratic Equation by Completing the Square Solve the equation by completing the square. 18x + 3x 2 = 45 x 2 + 6x = 15 Divide both sides by 3. x 2 + 6x + = 15 + Set up to complete the square. Add to both sides. x 2 + 6x + 9 = Simplify.

142 5-4 Completing the Square Example 3B Continued (x + 3) 2 = 24 Factor. Take the square root of both sides. Simplify.

143 5-4 Completing the Square Check It Out! Example 3a Solve the equation by completing the square. x 2 2 = 9x x 2 9x = 2 x 2 9x + = 2 + Collect variable terms on one side. Set up to complete the square. Add to both sides. Simplify.

144 5-4 Completing the Square Check It Out! Example 3a Continued Factor. x 9 2 = ± 89 4 Take the square root of both sides. x = 9 ± 2 89 Simplify.

145 5-4 Completing the Square Check It Out! Example 3b Solve the equation by completing the square. 3x 2 24x = 27 x 2 8x = 9 x 2 8x + = 9 + Divide both sides by 3. Set up to complete the square. Add to both sides. Simplify.

146 5-4 Completing the Square Check It Out! Example 3b Continued Solve the equation by completing the square. Factor. Take the square root of both sides. Simplify. x 4 = 5 or x 4 = 5 Solve for x. x = 1 or x = 9

147 5-4 Completing the Square Recall the vertex form of a quadratic function from lesson 5-1: f(x) = a(x h) 2 + k, where the vertex is (h, k). You can complete the square to rewrite any quadratic function in vertex form. Helpful Hint In Example 3, the equation was balanced by adding to both sides. Here, the equation is balanced by adding and subtracting on one side.

148 5-4 Completing the Square Example 4A: Writing a Quadratic Function in Vertex Form Write the function in vertex form, and identify its vertex. f(x) = x x 12 f(x)=(x x + ) 12 Set up to complete the square. Add and subtract. f(x) = (x + 8) 2 76 Simplify and factor. Because h = 8 and k = 76, the vertex is ( 8, 76).

149 5-4 Completing the Square Example 4A Continued Check Use the axis of symmetry formula to confirm vertex. y = f( 8) = ( 8) ( 8) 12 = 76

150 5-4 Completing the Square Example 4B: Writing a Quadratic Function in Vertex Form Write the function in vertex form, and identify its vertex g(x) = 3x 2 18x + 7 g(x) = 3(x 2 6x) + 7 g(x) = 3(x 2 6x + ) Factor so the coefficient of x 2 is 1. Set up to complete the square. Add. Because is multiplied by 3, you must subtract 3.

151 5-4 Completing the Square Example 4B Continued g(x) = 3(x 3) 2 20 Simplify and factor. Because h = 3 and k = 20, the vertex is (3, 20). Check A graph of the function on a graphing calculator supports your answer.

152 5-4 Completing the Square Write the function in vertex form, and identify its vertex f(x) = x x Check It Out! Example 4a f(x) = (x x + ) Set up to complete the square. Add and subtract. f(x) = (x + 12) Simplify and factor. Because h = 12 and k = 1, the vertex is ( 12, 1).

153 5-4 Completing the Square Check It Out! Example 4a Continued Check Use the axis of symmetry formula to confirm vertex. y = f( 12) = ( 12) ( 12) = 1

154 5-4 Completing the Square Write the function in vertex form, and identify its vertex g(x) = 5x 2 50x g(x) = 5(x 2 10x) Check It Out! Example 4b g(x) = 5(x 2 10x + ) Factor so the coefficient of x 2 is 1. Set up to complete the square. Add. Because is multiplied by 5, you must subtract 5.

155 5-4 Completing the Square Check It Out! Example 4b Continued g(x) = 5(x 5) Simplify and factor. Because h = 5 and k = 3, the vertex is (5, 3). Check A graph of the function on a graphing calculator supports your answer.

156 5-4 Completing the Square Lesson Quiz 1. Complete the square for the expression x 2 15x +. Write the resulting expression as a binomial squared. Solve each equation. 2. x 2 16x + 64 = x 2 27 = 4x Write each function in vertex form and identify its vertex. 4. f(x)= x 2 + 6x 7 5. f(x) = 2x 2 12x 27 f(x) = (x + 3) 2 16; ( 3, 16) f(x) = 2(x 3) 2 45; (3, 45)

157 Complex Numbers and Roots 5-5

158 5-5 Complex Numbers and Roots Warm Up Simplify each expression Find the zeros of each function. 4. f(x) = x 2 18x f(x) = x 2 + 8x 24

159 5-5 Complex Numbers and Roots Objectives Define and use imaginary and complex numbers. Solve quadratic equations with complex roots.

160 5-5 Complex Numbers and Roots imaginary unit imaginary number complex number real part imaginary part complex conjugate Vocabulary

161 5-5 Complex Numbers and Roots You can see in the graph of f(x) = x below that f has no real zeros. If you solve the corresponding equation 0 = x 2 + 1, you find that x =,which has no real solutions. However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as. You can use the imaginary unit to write the square root of any negative number.

162 5-5 Complex Numbers and Roots

163 5-5 Complex Numbers and Roots Example 1A: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Factor out 1. Product Property. Simplify. Multiply. Express in terms of i.

164 5-5 Complex Numbers and Roots Example 1B: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Factor out 1. Product Property. Simplify. 4 6i = 4i 6 Express in terms of i.

165 5-5 Complex Numbers and Roots Check It Out! Example 1a Express the number in terms of i. Factor out 1. Product Property. Product Property. Simplify. Express in terms of i.

166 5-5 Complex Numbers and Roots Check It Out! Example 1b Express the number in terms of i. Factor out 1. Product Property. Simplify. Multiply. Express in terms of i.

167 5-5 Complex Numbers and Roots Check It Out! Example 1c Express the number in terms of i. Factor out 1. Product Property. Simplify. Multiply. Express in terms of i.

168 5-5 Complex Numbers and Roots Example 2A: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. Take square roots. Express in terms of i. Check x 2 = 144 (12i) i ( 1) 144 ( 12i) 2 144i 2 144( 1) x 2 =

169 5-5 Complex Numbers and Roots Example 2B: Solving a Quadratic Equation with Imaginary Solutions Solve the equation. 5x = 0 Check 5x = 0 0 5(18)i ( 1) Add 90 to both sides. Divide both sides by 5. Take square roots. Express in terms of i.

170 5-5 Complex Numbers and Roots Check It Out! Example 2a Solve the equation. x 2 = 36 Take square roots. Express in terms of i. Check x 2 = 36 (6i) i ( 1) 36 x 2 = 36 ( 6i) 2 36i 2 36( 1)

171 5-5 Complex Numbers and Roots Check It Out! Example 2b Solve the equation. x = 0 x 2 = 48 Add 48 to both sides. Take square roots. Express in terms of i. Check x = (48)i ( 1)

172 5-5 Complex Numbers and Roots Check It Out! Example 2c Solve the equation. 9x = 0 9x 2 = 25 Add 25 to both sides. Divide both sides by 9. Take square roots. Express in terms of i.

173 5-5 Complex Numbers and Roots A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i =. The set of real numbers is a subset of the set of complex numbers C. Every complex number has a real part a and an imaginary part b.

174 5-5 Complex Numbers and Roots Real numbers are complex numbers where b = 0. Imaginary numbers are complex numbers where a = 0 and b 0. These are sometimes called pure imaginary numbers. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

175 5-5 Complex Numbers and Roots Example 3: Equating Two Complex Numbers Find the values of x and y that make the equation 4x + 10i = 2 (4y)i true. Real parts 4x + 10i = 2 (4y)i Imaginary parts 4x = 2 Equate the real parts. 10 = 4y Equate the imaginary parts. Solve for x. Solve for y.

176 5-5 Complex Numbers and Roots Check It Out! Example 3a Find the values of x and y that make each equation true. 2x 6i = 8 + (20y)i Real parts 2x 6i = 8 + (20y)i Imaginary parts 2x = 8 Equate the real parts. 6 = 20y Equate the imaginary parts. x = 4 Solve for x. Solve for y.

177 5-5 Complex Numbers and Roots Check It Out! Example 3b Find the values of x and y that make each equation true. 8 + (6y)i = 5x i 6 Real parts 8 + (6y)i = 5x i 6 Imaginary parts 8 = 5x Equate the real parts. Solve for x. Equate the imaginary parts. Solve for y.

178 5-5 Complex Numbers and Roots Example 4A: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. f(x) = x x + 26 x x + 26 = 0 Set equal to 0. x x + = 26 + Rewrite. x x + 25 = Add to both sides. (x + 5) 2 = 1 Factor. Take square roots. Simplify.

179 5-5 Complex Numbers and Roots Example 4B: Finding Complex Zeros of Quadratic Functions Find the zeros of the function. g(x) = x 2 + 4x + 12 x 2 + 4x + 12 = 0 x 2 + 4x + = 12 + x 2 + 4x + 4 = (x + 2) 2 = 8 Set equal to 0. Rewrite. Add to both sides. Factor. Take square roots. Simplify.

180 5-5 Complex Numbers and Roots Check It Out! Example 4a Find the zeros of the function. f(x) = x 2 + 4x + 13 x 2 + 4x + 13 = 0 Set equal to 0. x 2 + 4x + = 13 + x 2 + 4x + 4 = Rewrite. Add to both sides. (x + 2) 2 = 9 Factor. Take square roots. x = 2 ± 3i Simplify.

181 5-5 Complex Numbers and Roots Check It Out! Example 4b Find the zeros of the function. g(x) = x 2 8x + 18 x 2 8x + 18 = 0 Set equal to 0. x 2 8x + = 18 + x 2 8x + 16 = Rewrite. Add to both sides. Factor. Take square roots. Simplify.

182 5-5 Complex Numbers and Roots The solutions and are related. These solutions are a complex conjugate pair. Their real parts are equal and their imaginary parts are opposites. The complex conjugate of any complex number a + bi is the complex number a bi. If a quadratic equation with real coefficients has nonreal roots, those roots are complex conjugates. Helpful Hint When given one complex root, you can always find the other by finding its conjugate.

183 5-5 Complex Numbers and Roots Example 5: Finding Complex Zeros of Quadratic Functions Find each complex conjugate. A i B. 6i 8 + 5i 8 5i Write as a + bi. Find a bi i 0 6i 6i Write as a + bi. Find a bi. Simplify.

184 5-5 Complex Numbers and Roots Check It Out! Example 5 Find each complex conjugate. A. 9 i B. 9 + ( i) Write as a + bi. Write as a + bi. 9 ( i) Find a bi. 9 + i Simplify. Find a bi. C. 8i 0 + ( 8)i Write as a + bi. 0 ( 8)i 8i Find a bi. Simplify.

185 5-5 Complex Numbers and Roots Lesson Quiz 1. Express in terms of i. Solve each equation. 2. 3x = 0 3. x 2 + 8x +20 = 0 4. Find the values of x and y that make the equation 3x +8i = 12 (12y)i true. 5. Find the complex conjugate of

186 QUADRATIC FORMULA 5-6

187 5-6 The Quadratic Formula Objectives Solve quadratic equations using the Quadratic Formula. Classify roots using the discriminant. discriminant Vocabulary

188 5-6 The Quadratic Formula You have learned several methods for solving quadratic equations: graphing, making tables, factoring, using square roots, and completing the square. Another method is to use the Quadratic Formula, which allows you to solve a quadratic equation in standard form. By completing the square on the standard form of a quadratic equation, you can determine the Quadratic Formula.

189 5-6 The Quadratic Formula

190 5-6 The Quadratic Formula Remember! To subtract fractions, you need a common denominator.

191 5-6 The Quadratic Formula The symmetry of a quadratic function is evident in the last step,. These two zeros are the same distance,, away from the axis of symmetry,,with one zero on either side of the vertex.

192 5-6 The Quadratic Formula You can use the Quadratic Formula to solve any quadratic equation that is written in standard form, including equations with real solutions or complex solutions.

193 5-6 The Quadratic Formula Example 1: Quadratic Functions with Real Zeros Find the zeros of f(x)= 2x 2 16x + 27 using the Quadratic Formula. 2x 2 16x + 27 = 0 Set f(x) = 0. Write the Quadratic Formula. Substitute 2 for a, 16 for b, and 27 for c. Simplify. Write in simplest form.

194 5-6 The Quadratic Formula Example 1 Continued Check Solve by completing the square.

195 5-6 The Quadratic Formula Find the zeros of f(x) = x 2 + 3x 7 using the Quadratic Formula. x 2 + 3x 7 = 0 Check It Out! Example 1a Set f(x) = 0. Write the Quadratic Formula. Substitute 1 for a, 3 for b, and 7 for c. Simplify. Write in simplest form.

196 5-6 The Quadratic Formula Check It Out! Example 1a Continued Check Solve by completing the square. x 2 + 3x 7 = 0 x 2 + 3x = 7

197 5-6 The Quadratic Formula Check It Out! Example 1b Find the zeros of f(x)= x 2 8x + 10 using the Quadratic Formula. x 2 8x + 10 = 0 Set f(x) = 0. Write the Quadratic Formula. Substitute 1 for a, 8 for b, and 10 for c. Simplify. Write in simplest form.

198 5-6 The Quadratic Formula Check It Out! Example 1b Continued Check Solve by completing the square. x 2 8x + 10 = 0 x 2 8x = 10 x 2 8x + 16 = (x + 4) 2 = 6

199 5-6 The Quadratic Formula Example 2: Quadratic Functions with Complex Zeros Find the zeros of f(x) = 4x 2 + 3x + 2 using the Quadratic Formula. f(x)= 4x 2 + 3x + 2 Set f(x) = 0. Write the Quadratic Formula. Substitute 4 for a, 3 for b, and 2 for c. Simplify. Write in terms of i.

200 5-6 The Quadratic Formula Check It Out! Example 2 Find the zeros of g(x) = 3x 2 x + 8 using the Quadratic Formula. Set f(x) = 0 Write the Quadratic Formula. Substitute 3 for a, 1 for b, and 8 for c. Simplify. Write in terms of i.

201 5-6 The Quadratic Formula The discriminant is part of the Quadratic Formula that you can use to determine the number of real roots of a quadratic equation.

202 5-6 The Quadratic Formula Caution! Make sure the equation is in standard form before you evaluate the discriminant, b 2 4ac.

203 5-6 The Quadratic Formula Example 3A: Analyzing Quadratic Equations by Using the Discriminant Find the type and number of solutions for the equation. x = 12x x 2 12x + 36 = 0 b 2 4ac ( 12) 2 4(1)(36) = 0 b 2 4ac = 0 The equation has one distinct real solution.

204 5-6 The Quadratic Formula Example 3B: Analyzing Quadratic Equations by Using the Discriminant Find the type and number of solutions for the equation. x = 12x x 2 12x + 40 = 0 b 2 4ac ( 12) 2 4(1)(40) = 16 b 2 4ac < 0 The equation has two distinct nonreal complex solutions.

205 5-6 The Quadratic Formula Example 3C: Analyzing Quadratic Equations by Using the Discriminant Find the type and number of solutions for the equation. x = 12x x 2 12x + 30 = 0 b 2 4ac ( 12) 2 4(1)(30) = 24 b 2 4ac > 0 The equation has two distinct real solutions.

206 5-6 The Quadratic Formula Check It Out! Example 3a Find the type and number of solutions for the equation. x 2 4x = 4 x 2 4x + 4 = 0 b 2 4ac ( 4) 2 4(1)(4) = 0 b 2 4ac = 0 The equation has one distinct real solution.

207 5-6 The Quadratic Formula Check It Out! Example 3b Find the type and number of solutions for the equation. x 2 4x = 8 x 2 4x + 8 = 0 b 2 4ac ( 4) 2 4(1)(8) = 16 b 2 4ac < 0 The equation has two distinct nonreal complex solutions.

208 5-6 The Quadratic Formula Check It Out! Example 3c Find the type and number of solutions for each equation. x 2 4x = 2 x 2 4x 2 = 0 b 2 4ac ( 4) 2 4(1)( 2) = 24 b 2 4ac > 0 The equation has two distinct real solutions.

209 5-6 The Quadratic Formula The graph shows related functions. Notice that the number of real solutions for the equation can be changed by changing the value of the constant c.

210 5-6 The Quadratic Formula Properties of Solving Quadratic Equations

211 5-6 The Quadratic Formula Properties of Solving Quadratic Equations

212 5-6 The Quadratic Formula Helpful Hint No matter which method you use to solve a quadratic equation, you should get the same answer.

213 5-6 The Quadratic Formula Lesson Quiz: Part I Find the zeros of each function by using the Quadratic Formula. 1. f(x) = 3x 2 6x 5 2. g(x) = 2x 2 6x + 5 Find the type and member of solutions for each equation. 3. x 2 14x x 2 14x distinct nonreal complex 2 distinct real

214 Operations with Complex Numbers 5-9

215 5-9 Operations with Complex Numbers Warm Up Express each number in terms of i. 1. 9i 2. Find each complex conjugate Find each product

216 5-9 Operations with Complex Numbers Objective Perform operations with complex numbers. Vocabulary complex plane absolute value of a complex number

217 5-9 Operations with Complex Numbers Just as you can represent real numbers graphically as points on a number line, you can represent complex numbers in a special coordinate plane. The complex plane is a set of coordinate axes in which the horizontal axis represents real numbers and the vertical axis represents imaginary numbers.

218 5-9 Operations with Complex Numbers Helpful Hint The real axis corresponds to the x-axis, and the imaginary axis corresponds to the y-axis. Think of a + bi as x + yi.

219 5-9 Operations with Complex Numbers Example 1: Graphing Complex Numbers Graph each complex number. A. 2 3i 1+ 4i B i 4 + i C. 4 + i i D. i 2 3i

220 5-9 Operations with Complex Numbers Graph each complex number. a i Check It Out! Example 1 b. 2i 2i 3 + 2i c. 2 i 2 i 3 + 0i d i

221 5-9 Operations with Complex Numbers Recall that absolute value of a real number is its distance from 0 on the real axis, which is also a number line. Similarly, the absolute value of an imaginary number is its distance from 0 along the imaginary axis.

222 5-9 Operations with Complex Numbers Example 2: Determining the Absolute Value of Complex Numbers Find each absolute value. A i B. 13 C. 7i i 0 +( 7)i 13 7

223 5-9 Operations with Complex Numbers Check It Out! Example 2 Find each absolute value. a. 1 2i b. c. 23i i 23

224 5-9 Operations with Complex Numbers Adding and subtracting complex numbers is similar to adding and subtracting variable expressions with like terms. Simply combine the real parts, and combine the imaginary parts. The set of complex numbers has all the properties of the set of real numbers. So you can use the Commutative, Associative, and Distributive Properties to simplify complex number expressions.

225 5-9 Operations with Complex Numbers Helpful Hint Complex numbers also have additive inverses. The additive inverse of a + bi is (a + bi), or a bi.

226 5-9 Operations with Complex Numbers Example 3A: Adding and Subtracting Complex Numbers Add or subtract. Write the result in the form a + bi. (4 + 2i) + ( 6 7i) (4 6) + (2i 7i) 2 5i Add real parts and imaginary parts.

227 5-9 Operations with Complex Numbers Example 3B: Adding and Subtracting Complex Numbers Add or subtract. Write the result in the form a + bi. (5 2i) ( 2 3i) (5 2i) i (5 + 2) + ( 2i + 3i) 7 + i Distribute. Add real parts and imaginary parts.

228 5-9 Operations with Complex Numbers Example 3C: Adding and Subtracting Complex Numbers Add or subtract. Write the result in the form a + bi. (1 3i) + ( 1 + 3i) (1 1) + ( 3i + 3i) 0 Add real parts and imaginary parts.

229 5-9 Operations with Complex Numbers Check It Out! Example 3a Add or subtract. Write the result in the form a + bi. ( 3 + 5i) + ( 6i) ( 3) + (5i 6i) 3 i Add real parts and imaginary parts.

230 5-9 Operations with Complex Numbers Add or subtract. Write the result in the form a + bi. 2i (3 + 5i) Check It Out! Example 3b (2i) 3 5i ( 3) + (2i 5i) 3 3i Distribute. Add real parts and imaginary parts.

231 5-9 Operations with Complex Numbers Add or subtract. Write the result in the form a + bi. (4 + 3i) + (4 3i) Check It Out! Example 3c (4 + 4) + (3i 3i) 8 Add real parts and imaginary parts.

232 5-9 Operations with Complex Numbers You can also add complex numbers by using coordinate geometry.

233 5-9 Operations with Complex Numbers Example 4: Adding Complex Numbers on the Complex Plane Find (3 i) + (2 + 3i) by graphing. Step 1 Graph 3 i and 2 + 3i on the complex plane. Connect each of these numbers to the origin with a line segment i 3 i

234 5-9 Operations with Complex Numbers Example 4 Continued Find (3 i) + (2 + 3i) by graphing. Step 2 Draw a parallelogram that has these two line 2 + 3i segments as sides. The vertex that is opposite the origin represents the sum of the two complex numbers, 5 + 2i. Therefore, (3 i) + (2 + 3i) = 5 + 2i. 3 i 5 +2i

235 5-9 Operations with Complex Numbers Example 4 Continued Find (3 i) + (2 + 3i) by graphing. Check Add by combining the real parts and combining the imaginary parts. (3 i) + (2 + 3i) = (3 + 2) + ( i + 3i) = 5 + 2i

236 5-9 Operations with Complex Numbers Check It Out! Example 4a Find (3 + 4i) + (1 3i) by graphing. Step 1 Graph 3 + 4i and 1 3i on the complex plane. Connect each of these numbers to the origin with a line segment i 1 3i

237 5-9 Operations with Complex Numbers Check It Out! Example 4a Continued Find (3 + 4i) + (1 3i) by graphing. Step 2 Draw a parallelogram that has these two line segments as sides. The vertex that is opposite the origin represents the sum of the two complex numbers, 4 + i. Therefore,(3 + 4i) + (1 3i) = 4 + i i 1 3i 4 + i

238 5-9 Operations with Complex Numbers Check It Out! Example 4a Continued Find (3 + 4i) + (1 3i) by graphing. Check Add by combining the real parts and combining the imaginary parts. (3 + 4i) + (1 3i) = (3 + 1) + (4i 3i) = 4 + i

239 5-9 Operations with Complex Numbers Check It Out! Example 4b Find ( 4 i) + (2 2i) by graphing. Step 1 Graph 4 i and 2 2i on the complex plane. Connect each of these numbers to the origin with a line segment. 4 i 2 2i 2 2i

240 5-9 Operations with Complex Numbers Check It Out! Example 4b Find ( 4 i) + (2 2i) by graphing. Step 2 Draw a parallelogram that has these two line segments as sides. The vertex that is opposite represents the sum of the two complex numbers, 2 3i. Therefore,( 4 i) + (2 2i) = 2 3i. 4 i 2 3i 2 2i

241 5-9 Operations with Complex Numbers Check It Out! Example 4b Find ( 4 i) + (2 2i) by graphing. Check Add by combining the real parts and combining the imaginary parts. ( 4 i) + (2 2i) = ( 4 + 2) + ( i 2i) = 2 3i

242 5-9 Operations with Complex Numbers You can multiply complex numbers by using the Distributive Property and treating the imaginary parts as like terms. Simplify by using the fact i 2 = 1.

243 5-9 Operations with Complex Numbers Example 5A: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. 2i(2 4i) 4i + 8i 2 4i + 8( 1) 8 4i Distribute. Use i 2 = 1. Write in a + bi form.

244 5-9 Operations with Complex Numbers Example 5B: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. (3 + 6i)(4 i) i 3i 6i i 6( 1) i Multiply. Use i 2 = 1. Write in a + bi form.

245 5-9 Operations with Complex Numbers Example 5C: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. (2 + 9i)(2 9i) 4 18i + 18i 81i ( 1) 85 Multiply. Use i 2 = 1. Write in a + bi form.

246 5-9 Operations with Complex Numbers Example 5D: Multiplying Complex Numbers Multiply. Write the result in the form a + bi. ( 5i)(6i) 30i 2 30( 1) 30 Multiply. Use i 2 = 1 Write in a + bi form.

247 5-9 Operations with Complex Numbers Check It Out! Example 5a Multiply. Write the result in the form a + bi. 2i(3 5i) 6i 10i 2 6i 10( 1) i Distribute. Use i 2 = 1. Write in a + bi form.

248 5-9 Operations with Complex Numbers Check It Out! Example 5b Multiply. Write the result in the form a + bi. (4 4i)(6 i) 24 4i 24i + 4i i + 4( 1) Distribute. Use i 2 = i Write in a + bi form.

249 5-9 Operations with Complex Numbers Check It Out! Example 5c Multiply. Write the result in the form a + bi. (3 + 2i)(3 2i) 9 + 6i 6i 4i 2 9 4( 1) Distribute. Use i 2 = Write in a + bi form.

250 5-9 Operations with Complex Numbers The imaginary unit i can be raised to higher powers as shown below. Helpful Hint Notice the repeating pattern in each row of the table. The pattern allows you to express any power of i as one of four possible values: i, 1, i, or 1.

251 5-9 Operations with Complex Numbers Example 6A: Evaluating Powers of i Simplify 6i 14. 6i 14 = 6(i 2 ) 7 = 6( 1) 7 = 6( 1) = 6 Rewrite i 14 as a power of i 2. Simplify.

252 5-9 Operations with Complex Numbers Example 6B: Evaluating Powers of i Simplify i 63. i 63 = i i 62 = i (i 2 ) 31 Rewrite as a product of i and an even power of i. Rewrite i 62 as a power of i 2. = i ( 1) 31 = i 1 = i Simplify.

253 5-9 Operations with Complex Numbers Check It Out! Example 6a Simplify. Rewrite as a product of i and an even power of i. Rewrite i 6 as a power of i 2. Simplify.

254 5-9 Operations with Complex Numbers Check It Out! Example 6b Simplify i 42. i 42 = ( i 2 ) 21 = ( 1) 21 = 1 Rewrite i 42 as a power of i 2. Simplify.

255 5-9 Operations with Complex Numbers Recall that expressions in simplest form cannot have square roots in the denominator (Lesson 1-3). Because the imaginary unit represents a square root, you must rationalize any denominator that contains an imaginary unit. To do this, multiply the numerator and denominator by the complex conjugate of the denominator. Helpful Hint The complex conjugate of a complex number a + bi is a bi. (Lesson 5-5)

256 5-9 Operations with Complex Numbers Simplify. Example 7A: Dividing Complex Numbers Multiply by the conjugate. Distribute. Use i 2 = 1. Simplify.