Graphing Quadratic Functions

1. Graphing Quadratic Functions Graph quadratic functions Analyze graphs of quadratic functions and write quadratic functions Graph and solve quadratic inequalities Solve systems of Equations

2. Quadratic functions can be used to model real-world phenomena like the motion of a falling object. • Quadratic functions can be used to model the shape of architectural structures such as the supporting cables of a suspension bridge. • Can you think of examples?

4. Compare each graph with y=x2(Think Transformations) • y = 2x2 • y = 2x2 + 3 • y=(x – 2)2 • 8. • y = (x+3)2 • y = 4(x+3)2 • 11.

6. Quadratic Equations/Functions Standard Form: f(x)=ax2 + bx + c Vertex Form: f(x) = a(x – h)2 + k

7. Vertex form y = a(x-h)2 + k • (h,k) vertex • x = h axis of symmetry • Translations to y = x2, vertex(0,0) h units to left if (x+h) h units to right if (x-h) k units up if k is (+) k units down if k is (-)

8. What happens if we change a? • If a > 0, then the graph opens up • If a < 0, then the graph opens down • If |a| > 1, then the graph is narrower than y = x2 • If |a| < 1, then the graph is wider than y = x2 • The ‘a’ value affects the direction and width of the parabola.

9. How to use vertex form to graph • Plot the vertex • Draw the axis of symmetry • Make a table of values showing symmetry around the vertex • Plot points to complete the graph • State the direction and width

10. Graph. 1) f(x) = (x+2)2 – 3

11. Graph: 2) y= 2(x – 1)2 – 6 3)

12. Assignment 2.1 p.64 #’s 5 – 8, 13, 39-41, 46 – 50 Directions for #’s 5 – 8, 13: Describe the transformation, state the vertex, axis of symmetry, direction and width of the parabola, make a table and graph.

13. 2.2 Graphing quadratic functions • Standard Form – f(x)=ax2 + bx + c. • y–intercept is c • Axis of symmetry and x-coordinate of the vertices – 5. The y-coordinate of the vertex is the maximum or minimum value.

14. 2.2 continued • If a > 0 thenthe parabola opens up and has a minimum value. • If a < 0 then it opens down and has a maximum value. • Minimum/Maximum point and min/max value are not the same. Min/Max value is the y-coordinate of the vertex.Min/Max point is the ordered pair (x,y) of the vertex.

15. 1) f(x) = x2 + 8x + 9 • Find the y-intercept • Find the axis of symmetry • Find the vertex • Find the maximum or minimum value • Does the graph open up or down? 9 -8/2 = -4; x=4 (-4,-7) -7, min. Opens up

16. Find the vertex, a.o.s, y-intercept, direction and width of the parabola, table of points and graph. Find the maximum or minimum value. 2) y = 2x2 – 8x + 6

17. 3. f(x) = x2 + 3x – 1 • V (-1.5, -.3.25) • a.o.s x=-1.5 • Y-int = -1 • a=1, Opens up Standard width Minimum = -3.25

18. 2.2 examples 4. f(x) = 2 – 4x + x2 a. Find the y-intercept, the equation of the axis of symmetry, the vertex. b. Make a table of values that includes the vertex. c. Graph the function • 2, x = 2, (2,-2) • b. (0,2),(1,-1),(2,-2), (3,-1),(4,2)

19. 2.2 Classwork 5. f(x) = -x2 + 2x + 3 a. Determine whether the function has a maximum and minimum value. b. State the maximum and minimum value of the function. • Maximum value • 4

20. 2.2 Application 1. A souvenir shop sells about 200 coffee mugs each month for $6 each. The shop owner estimates that for each $.50 increase in the price, he will sell about 10 fewer coffee mugs per month. a. How much should the owner charge for each mug in order to maximize the monthly income from their sales? b. What is the maximum monthly income the owner can expect to make from these items? $8 $1280

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23. 2.3 Solve quadratic equations by graphing • ax2 + bx + c = 0 • The solutions are called the roots or the zeros. The solutions are the x-intercepts. • Solutions of quadratic equations – 1 real solution or 2 real solutions or no real solution.

24. 2.3 examples • Solve x2 -3x – 4 = 0 by graphing. • Solve x2 – 4x = -4 by graphing. • Solve x2 – 6x + 3 = 0 by graphing. {-1,4} {2} {.55, 5.45}

27. Warm-Up Given: 2x2 = x+15 • Determine if the function has a maximum or minimum. Find the maximum or minimum value. 2. Find the roots of the equation. Minimum, -15.125 {-2.5, 3}

28. Two more to look at: 1. Use a quadratic equation to find two real numbers whose sum is 5 and whose product is -14, or show that no such numbers exist. 2. Find two numbers whose sum is 4 and whose product is 5 or show that no such numbers exist.

29. 2.3 solve quadratic equations by factoring • x2 = 6x x2 -6x = 0 x(x-6) = 0 x = 0 x – 6 = 0 x = 6 2. 2x2 + 7x = 15 2x2 + 7x – 15 = 0 (2x – 3)(x + 5) = 0 2x – 3 = 0 x + 5 = 0 x = 3/2 x = -5

30. 2.3 Practice Solve by factoring 1. x2 = -4x 2. 3x2 = 5x + 2 3. x2 – 6x = -9 4. Write a quadratic equation with -2/3 and 6 as its roots. (see example 4 p. 303) {0,-4} {-1/3, 2} {3} 3x2 – 16x – 12 = 0

31. Warm-Up 1. Solve by graphing: 2. Write a quadratic equation and solve by graphing: Find two numbers whose sum is 1 and their product is -6. 3. Graph and find its solutions: 4. Solve by factoring: 5. Write the quadratic equation in standard form with the roots

32. Warm-Up Solve by factoring: 1) 2) 3) 4) Write the quadratic equation in standard form with the roots

33. Warm-Up Given: • Determine the y-intercept, vertex and axis of symmetry. • Does the graph open up or down? Does it have a maximum or minimum value? What is that value? • Solve: • Solve by factoring:

34. Warm-Up Multiply. 3. (x + 3)2 4. (x – 5)2 5. (x + 7)2 6. (3x+2)2 7. (4x – 6)2 Simplify: 1. 2.

35. Warm-Up • Solve by factoring: • Write the quadratic equation in standard form with the roots: Find the value of c that makes each trinomial a perfect square: 3) 4) Write the perfect square for each: 5) 6)

36. 2.4 Completing the square Solve using the square root property 1. x2 + 10x + 25 = 49 (x + 5)2 = 49 x + 5 = + 7 x = -5 + 7 x = -12 x = 2 {-12, 2}

37. continued • x2 – 6x + 9 = 32 (x – 3)2 = 32 x – 3 = + x – 3 = x = 3 + 4 x ≈ 8.7 x ≈ -2.7

38. Complete the square 3. x2 + 8x – 20 = 0 x2 + 8x + ___ = 20 + ___ x2 + 8x + 42 = 20 + 16 (x + 4)2 = 36 x + 4 = +6 x = -4 + 6 x = -10, x = 2 {-10, 2}

39. Your Turn • x2 + 14x – 15 = 0 • x2 – 10x +13 = 0 5. {-15,1}

40. continued 7. 2x2 – 5x + 3 = 0 x2 – (5/2)x + (3/2) = 0 x2 – (5/2)x + ____ = -(3/2) + ___ x2 – (5/2)x + (5/4)2 = –(3/2)+ (25/16) (x – (5/4))2 = (1/16) x – (5/4) = +(1/4) x =+(1/4) + (5/4) x = 3/2 x = 1

41. 2.4 Examples Solve • x2 + 14x – 15 = 0 • x2 – 10x +13 = 0 • x2 + 4x – 12 = 0 {-15,1} {5 + 2 } {-6,2}

42. Warm-Up • Solve by factoring: • x2 +9 = 6x • -2x2 + 12x – 16 = 0 • Solve using the square root property: • x2 + 10x +25 = 49 • Solve by completing the square. • 3x2 – 4x – 2 = 0

43. Warm-Up • Solve by graphing: • Solve by factoring: • Solve by completing the square: {-3, 1} {0,9}

44. Warm-Up • Solve by factoring: 1. 2x2 – 3x = -1 • Solve by completing the square: • x2 – 8x – 65=0 • 2x2 – 18x – 7=0 {1, ½} {-5,13}

45. 2.6 Quadratic Formula and discriminant • Quadratic formula • Discriminant b2 -4ac • b2 – 4ac > 0 2 real roots (rational or irrational?) • b2 -4ac = 0 1 real root • b2 – 4ac < 0 2 imaginary roots

46. 2.6 Examples • Solve x2 – 8x = 33 using the quadratic formula • Solve x2 - 34x + 289 = 0 • Solve x2 – 6x + 2 = 0 • Solve x2 + 13 = 6x

47. 2.6 Examples continued • Find the value of the discriminant and describe the nature and types of roots. • x2 + 6x + 9 = 0 • x2 + 3x + 5 = 0 • x2 + 8x – 4 =0 • x2 – 11x + 10 = 0

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50. Investigation for 6.6 • Graph the following equations and determine the vertex. • Look for a translation of the vertex from the graph of #1.