1.6 Shifting, Reflecting and Stretching Graphs

How to vertical and horizontal shift To use reflections to graph

Sketch a graph

Shifting (up and down)

How is y = x2 different from y – k = a(x – h)2

y = a(x – h)2 + k

k shifts the graph

up or down.

Shifting (right or left)

How is y = x2 different from y – k = a(x – h)2

y = a(x – h)2 + k

h shifts the graph

Left or Right.

Reflecting

Over the x axis will change y to – y

Over the y axis will change x to - x

How does this effect f(x) = x2 + 2

f(x) = x2 + 2

Over the x axis where f(x) becomes – f(x)

-f(x) = -(x2 + 2) or

–x 2 – 2 (0, 2)

(0, - 2)

f(x) = x2 + 2

Over the y axis where f(x) becomes f(- x)

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

(0, 2)

The graph and the equation are the same. What does this show?

Stretching (Nonrigid transformation)

Shifting and reflection are rigid transformation; they move the graph without changing its shape.

Lets look at the equation f(x) = ax 2

“a” will make the graph fat or skinny

What would happen if a=4 in f(x) = ax2?Would the graph become fat or skinny?

FAT Skinny

Skinny

The values of f(x) become larger as “a” increase.

f(x) = 4x2

(- 3, 12) (3, 12)

(-1 , 4) (1, 4)

(0, 0)

What would make the Graph Fat?

What would make the Graph Fat?

Where “a” is 0 < a < 1.

(-2, 1) ( 2, 1)

(0, 0)

2

4

1)( xxf

So lets look at the equationy = a(x – h)2 + k

h will move the graph Left or Right (must remember (x + 3) means h is – 3)

k will move the graph up or down

a will stretch the graph fat or skinny

Homework

Page 64 – 68

#4, 11, 17, 27, 35, 43, 49, 57, 69, 83

Homework

Page 64 – 68

#9, 13, 21, 31, 39, 45, 53, 65, 71, 81