How do you find the maximum value of a quadratic function?

1. How do you find the maximum value of a quadratic function? For example: y=-3x2+18x+25

2. In this lesson you will learn to rewrite a quadratic function to reveal the maximum value by completing the square.

3. Suppose we have y=-3x2+18x+25 • is negative

4. Add a number inside the bracket and distribute before you subtract to preserve equality! • Preserving the equality of an equation • y=-3x2+18x+25 (-27) + • y=-3(x2-6x )+25 - 9 y=-3 (x-3)2 +52

5. y=-5x2-20x+23 +4 -(-20) y=-5(x2+4x )+23 +43 (x+2)2 y=-5 • A square number is always positive unless it’s zero.

6. y=-5(x+2)2 +43 y=-5( +2)2 +43 -2 =-5(0)2+43 =0+43

7. ADDING a negative number to any number makes that number smaller so the function will have a maximum value when the square term is zero.

8. -5(-3+2)2 Maximum value of • y=-5x2-20x+23 • -5(-2+2)2 43 • is 43 when x=-2 • -5(0+2)2 23 • y=-5(x+2)2 +43 • -5(1+2)2 -2

9. In this lesson you have learned to rewrite a quadratic function to reveal the maximum value by completing the square.

10. Rewrite y=-3x2+24x-36 by completing the square. What is the maximum value of this quadratic function?

11. A pencil is thrown into the air. Its height H in meters after t seconds is H =-2(t - 3)2 +20. a)Sketch a picture of how the graph might look like. b) What was the pencil’s maximum height? At what time did this occur? c) From what height was the pencil thrown?

12. A rancher needs to enclose two adjacent rectangular corrals-one for cattle and one for sheep. If the river forms one side of the corrals, and 390 yd. of fencing is available, Find the largest total area that can be enclosed. a)width = x length=?

13. Recall: Area=l w c)Write the equation for the area of the corral A(x)= d)What is the largest total area that can be enclosed?

14. Rewrite each function to find its maximum value by completing the square. • y=-2x2-8x+9 • y=-4x2+12x+25