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

For example:

y=-3x2+18x+25

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

completing the square.

Let’s ReviewLet’s Review

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

Let’s ReviewA Common Mistake

Preserving the equality of an equation

y=-3x2+18x+25

y=-3(x2-6x )+25+ -9y=-3 (x-3)2

(-27)+52

Add a number inside the bracket and distribute before you subtract to preserve equality!

Let’s ReviewCore Lesson

y=-5x2-20x+23

y=-5(x2+4x )+23+4 -(-20)y=-5 (x+2)2 +43

A square number is always positive unless it’s zero.

Let’s ReviewCore Lesson

y=-5(x+2)2 +43y=-5( +2)2 +43-2

=-5(0)2+43=0+43

Let’s ReviewCore Lesson

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.

Let’s ReviewCore Lesson x -5(x+2)2

+43y

-5(-2+2)2

-5(1+2)2

-5(-3+2)2

-5(0+2)2 23

-2

43

Maximum value ofy=-5x2-20x+23is 43 when x=-2

y=-5(x+2)2 +43

In this lesson you have learned to rewrite a quadratic function to

reveal the maximum value by completing the square.

Let’s ReviewGuided Practice

Rewrite y=-3x2+24x-36 by completing the square.

What is the maximum value of this quadratic function?

Let’s ReviewExtension ActivitiesA 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?  

Let’s ReviewExtension ActivitiesA 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=?

Let’s ReviewExtension ActivitiesRecall:Area=l w

c)Write the equation for the area of the corralA(x)=

d)What is the largest total area that can be enclosed?

Let’s ReviewQuick Quiz

Rewrite each function to find its maximum value by completing the square. 1. y=-2x2-8x+9

2. y=-4x2+12x+25