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

For example:

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

completing the square.

Let’s ReviewLet’s Review

Suppose we have

is positive

Let’s ReviewLet’s Review

Complete or exact square

(x+a)2

(x+a)(x+a)

X2+2ax+a2

Let’s ReviewA Common Mistake

Forgetting to preserve equality

=(x2-10x ) +32

=(x-5)2+32

y=(x-5)2 + 7

Add and subtract the same number to keep the equality

Let’s ReviewCore Lesson

y=x2-12x+49

=(x2-12x )+49

=(x-6)2

square - always positive unless it’s zero

Let’s ReviewCore Lesson

ADDING a positive number to any

number makes that number bigger so the function will have a minimum value when the square term is zero.

Let’s ReviewCore Lesson

y

=(6-6)2+13

smallest number

Let’s ReviewCore Lesson x (x-6)2 +13 y

(1-6)2+13

(6-6)2+13

(7-6)2+13

(0-6)2+13The minimum value

is 13 when x=6 y=(x-6)2 + 13

7 14

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

reveal the minimum value by completing the square

Let’s ReviewGuided Practice

Rewrite y=x2+8x+3 by completing the square.

What is the minimum value of this quadratic function?

Let’s ReviewExtension Activities

Partner matching cards. Write each polynomial and ordered pair on index cards. Shuffle all 12 cards then match each function with its equivalent form and corresponding minimum value.

y=x2+16x+64 y=(x+8)2 (-8,0)y=x2-2x-9 y=(x-1)2-10 (1,-10)y=x2+26x+68 y=(x+13)2-101 (-13,-101y=x2+18x-4 y=(x+9)2-85 (-9/-85)

Let’s ReviewExtension Activities

Write a quadratic function whose graph has the given characteristics

1. Minimum: (6,1) point on the graph: (4,5)

2. Points on the graph: (1,7), (4,-2), (5,-1)

3. X-intercepts (5,0) (-4,0)

Let’s ReviewQuick Quiz

Rewrite each function to find its minimum value by completing the square. 1. y=x2-3x+9

2. y=x2+12x+24