y = ax2 + bx + cQuadratic Equation

QuadraticTerm

ConstantTerm

Objective - To transform quadratic functions.

e ea

+a opens up-a opens down

skinny parabola a >1

wide parabola0< a <1

c

+c shifts up-c shifts down

y-intercept

yGraph f (x) = x2

x y

-32

94

x

-2-10123

410149

Vertex(0,0)Axis of

Symmetryx = 0

y

Graph f (x) = x2 − 5

x y

-3-21

−3( )2 − 5 = 4−2( )2 − 5 = −1

1( )2 5 4x

-10123

Vertex(0,-5)

Axis ofSymmetry

x = 0

−1( ) − 5 = −40( )2 − 5 = −51( )2 − 5 = −42( )2 − 5 = −13( )2 − 5 = 4

y

Graph f (x) = −x2 + 4

x y

-3-21

Vertex(0,4)

Axis ofSymmetry

x = 0− −3( )2 + 4 = −5− −2( )2 + 4 = 0

1( )2 + 4 3x

-10123

− −1( ) + 4 = 3− 0( )2 + 4 = 4− 1( )2 + 4 = 3− 2( )2 + 4 = 0− 3( )2 + 4 = −5

Translations of Quadratic Functions

y

Horizontal Translation

2( ) ( )f xh hx

f (x) = x2

x( ) ( )− = −f xh hx

h < 0 moves left

h > 0 moves right

Translations of Quadratic Functions

y

Vertical Translation

2( ) +kf x x k

f (x) = x2

x

( ) − = +kf x x k

k < 0 moves down

k > 0 moves up

Lesson 5-1

Algebra 2 Slide Show: Teaching Made Easy As Pi, by Mike Mills and James Wenk © 2010

Translations of Quadratic Functions

y

Reflection

f (x) = x2

f (x) = x2

x

− f (x) = −x

Stretches and Compressions

y

Horizontal Stretch

f (x) = x2

2

x2

21 1⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

xb

f xb

b > 1 stretches

0 < b < 1 compresses

Stretches and Compressions

y

Horizontal Compression

f (x) = x2

x ( )2 2• =f xa ax

a > 1 stretches

0 < a < 1 compresses

y = a(x − h)2 + k

Vertex Form of a Quadratic Equation

vertex (h,k)axis of symmetry x = h

Lesson 5-1 (cont.)

Algebra 2 Slide Show: Teaching Made Easy As Pi, by Mike Mills and James Wenk © 2010