October 3rd copyright2009merrydavidson

Happy Summer Birthday to:

Libby Harper

Quadratic Functions

GENERAL form:

If a>0 it opens UPIf a<0 it opens DOWN

STANDARD or VERTEX form:

where is the vertex.

2( )f x ax bx c

2( ) ( )f x a x h k

0a , ,a b c

( , )h k

2x2 + 3x - 5

2( ) 2( 3) 1f x x

Is an

element of

All quadratic functions are

symmetric over a line called the axis of symmetry.

x = 3

f(x) = a(x – h)2 + k

Practice finding the vertex and axis of symmetry…..

y = 2(x – 3)2 + 5

y = -3(x + 4)2 - 1

(3, 5)

x = 3(-4, -1)

x = -4

standard/vertex

form

f(x) = ax2 + bx + c

Practice finding the vertex and axis of symmetry….. Vertex: (-b/2a,f(-b/2a))

y = 2x2 + 8x - 3

y = -3x2 – 12x + 4

(-2, -11)

x = -2

(- 2, 16)

x = - 2

general

form

Graph: Plot the vertex.

Draw in the axis of symmetry.

y = (x – 3)2 + 5

y = -(x + 4)2 - 1

(3, 5) x = 3

(-4, -1) x = -4

Graph: Plot the vertex.

Draw in the axis of symmetry.

y = 2(x – 3)2 + 5

(3, 5) x = 3

Stretch of 2

Changing to Standard Form by “Completing the Square”

f(x) = x2 + 8x + 11Step 1: Group the 1st 2 terms

Step 2: Add & subtract blanks

Step 3: ½ the middle term squared

Step 4: factor

Step 5: simplify

f(x) = (x2 + 8x) + 11

f(x) = (x2 + 8x + _) + 11 - _42 42

f(x) = (x + 4)2 + 11 - 16

f(x) = (x + 4)2 - 5

f(x) = 2x2 + 8x + 7Step 1: Group the 1st 2 terms

Step 2: Factor out the 2

f(x) = (2x2 + 8x) + 7

f(x) = 2(x2 + 4x) + 7

f(x) = 2(x2 + 4x + _) + 7 - _(2)22 22

f(x) = 2(x + 2)2 + 7 - 8

f(x) = 2(x + 2)2 - 1

f(x) = -x2 - 4x + 21

f(x) = -(x2 + 4x + _) + 21 - _(-1)22 22

f(x) = -(x + 2)2 + 21 + 4

f(x) = -(x + 2)2 + 25

Find the x-intercepts of a quadratic function

f(x) = x2 - 6x + 8

Step 1: Set = 0Step 2: Factor

Step 3: Set each

factor = 0

Step 4: solve each partStep 5: intercepts are points

0 = x2 - 6x + 80 = (x – 4)(x – 2)

x– 4=0 x – 2=0

x = 4 x = 2(4,0)(2,0)

General Form

f(x) = ax2 + bx + c

Standard/vertex form

f(x) = a(x-h)2 +k

Vertex: (-b/2a,f(-b/2a)) (h, k)

AOS: x = -b/2a x = h

Easier to find Easier to graph &

x-intercepts find vertex

Find the equation of the quadratic function that goes through the point (2,3) with a vertex at (4,-5).

y = a(x - h)2 +k3 = a(2 - h)2 +k3 = a(2 - 4)2 + -5 solve for “a”

a = 2

f(x) = 2(x – 4)2 - 5

3 = 4a - 5 8 = 4a

Find the equation of the quadratic function that goes through the point (-2,-2) with a vertex at (-1,0).

f(x) = -2(x + 1)2

The height y (in feet) of a ball thrown by a child is given by

where x is the horizontal

distance (in feet) from where the ball is thrown. How high is the ball when

it is at its maximum height?

214

8y x x

Sketch a graph.

What are you

trying to find?

The y value of the vertex

128

x

2

bx

a

11

2( )8

x

214

8y x x

1( 4) 4x 114

x

(4, y) find y

21(4) 4 4 68

y

6 ft high

Minimizing Cost

A local newspaper has daily production costs of

C = 55,000 – 108x + 0.06x2 where C = total cost in $ and x is the number of newspapers printed.

How many newspapers should be printed each day to yield a minimum cost?

Graph in calculator/play with window

What are you looking for? X value of the vertex