1. Lesson 9.2: Graph Essential Question: How do you graph general quadratic functions? Common Core CC.9-12.F.BF.3 Graph linear and quadratic functions and show intercepts, maxima, and minima. Warm-up: 3-18-13

2. 9-2 Graph General Quadratic Functions Homework: Part One: 638/3-11; 28-36 Part Two: 638/ 16-26 even; 40-41; 46-58 even

6. a>0a<0 a = 7 b = 2 c = 11 The graph opens upward. a = -3 b = 9 c = 4 The graph opens downward.

7. Does the vertex contain a minimum or a maximum value? a>0 y value in vertex is a MINIMUM a<0 y value in vertex is a MAXIMUM Tell whether the graph has a maximum value or minimum value.

8. Tell whether the graph opens up or down and whether it has a maximum value or minimum value. Tell whether the graph opens up or down and whether it has a maximum value or minimum value. Upward. Minimum. Upward. Minimum. Downward. Maximum. Downward. Maximum. Upward. Minimum. Upward. Minimum.

9. Tell whether the graph opens up or down and whether it has a maximum value or minimum value. Tell whether the graph opens up or down and whether it has a maximum value or minimum value. Upward. Minimum. Upward. Minimum. Downward. Maximum. Downward. Maximum. Upward. Minimum. Upward. Minimum.

10. Axis of symmetry is the line: The x-coordinate of the vertex is on this line. To find the y-coordinate of the vertex, substitute the value of x into the original equation.

11. Vertex: Axis of Symmetry: This is the Minimum.

12. Vertex: Axis of Symmetry: This is the Maximum.

13. downup > < c 0 c

15. Graph y = -2x 2 + 4x + 1 Step 1: Find the axis of symmetry. x = 1 Step 2: Find the y-coordinate of the vertex. Substitute the value of x into the original equation. y = -2(1) 2 + 4(1) + 1 y = 3 Step 3: Find the y-intercept. Substitute 0 for x in the original equation. y = -2(0) 2 + 4(0) + 1 y = 1 Step 4: Choose another value for x on the same side of the vertex. Substitute -1 for x in the original equation. y = -2(-1) 2 + 4(-1) + 1 y = -5 Step 5: Reflect the points across the axis of symmetry and draw the parabola.

16. Graph y = x 2 – 6x + 9 Step One: Find the axis of symmetry. Step Two: Find the y-coordinate of the vertex. Substitute the value of x into the original equation. Step Three: Find the y-intercept. Substitute 0 for x in the original equation. Step Four: Choose another value for x on the same side of the vertex. Substitute for x in the original equation. Step Five: Reflect the points across the axis and draw the parabola. y x 12-10-6-24812 -10 -6 -2 4 8 12 -12-46 -12 -8 2 10 -810 -4 -12 2 6

17. This function has a minimum. The minimum value is y = 3.

18. Upward. Minimum. Upward. Minimum. Downward. Maximum. Downward. Maximum. Upward. Minimum. Upward. Minimum.

19. EXAMPLE 3 Find the minimum or maximum value Tell whether the function f ( x ) = – 3x 2 – 12x + 10 has a minimum value or a maximum value. Then find the minimum or maximum value. SOLUTION Because a = – 3 and – 3 < 0, the parabola opens down and the function has a maximum value. To find the maximum value, find the vertex. x = – = – = – 2 b 2a2a – 12 2(– 3) f(–2) = – 3(–2) 2 – 12(–2) + 10 = 22 Substitute –2 for x. Then simplify. The x- coordinate is – b 2a2a

20. Find the minimum or maximum value EXAMPLE 3 ANSWER The maximum value of the function is f ( – 2 ) = 22.

21. Find the minimum value of a function EXAMPLE 4 The suspension cables between the two towers of the Mackinac Bridge in Michigan form a parabola that can be modeled by the graph of y = 0.000097x 2 – 0.37x + 549 where x and y are measured in feet. What is the height of the cable above the water at its lowest point? SUSPENSION BRIDGES

22. Find the minimum value of a function EXAMPLE 4 SOLUTION The lowest point of the cable is at the vertex of the parabola. Find the x- coordinate of the vertex. Use a = 0.000097 and b = – 0.37. x = – = – ≈ 1910 b 2a2a – 0.37 2(0.000097) Use a calculator. Substitute 1910 for x in the equation to find the y -coordinate of the vertex. y ≈ 0.000097(1910) 2 – 0.37(1910) + 549 ≈ 196

23. EXAMPLE 4 ANSWER The cable is about 196 feet above the water at its lowest point. Find the minimum value of a function

24. GUIDED PRACTICE for Examples 3 and 4 3. Tell whether the function f ( x ) = 6x 2 + 18x + 13 has a minimum value or a maximum value. Then find the minimum or maximum value.  1 2 Minimum value; ANSWER

25. GUIDED PRACTICE for Examples 3 and 4 SUSPENSION BRIDGES 4. The cables between the two towers of the Takoma Narrows Bridge form a parabola that can be modeled by the graph of the equation y = 0.00014x 2 – 0.4x + 507 where x and y are measured in feet. What is the height of the cable above the water at its lowest point ? Round your answer to the nearest foot. ANSWER 221 feet

26. The highest point occurs at the Maximum. The y value of the vertex is 10. The height of the dome is 10 feet. The highest point occurs at the Maximum. The y value of the vertex is 10. The height of the dome is 10 feet.

27. Summary

28. Classwork/Homework 9.2 Practice B worksheet