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 = x 2 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 = x 2 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) = x 2 + 2

f(x) = x Over the x axis where f(x) becomes – f(x) -f(x) = -(x 2 + 2) or –x 2 – 2 (0, 2) (0, - 2)

f(x) = x Over the y axis where f(x) becomes f(- x) f(-x) = ( -x) or f(x) = x (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 equationf(x) = ax 2

“a” will make the graph fat or skinny What would happen if a=4 in f(x) = ax 2 ? Would the graph become fat or skinny? FATSkinny

Skinny The values of f(x) become larger as “a” increase. f(x) = 4x 2 (- 3, 12) (3, 12) (-1, 4) (1, 4) (0, 0)

What would make the Graph Fat?

Where “a” is 0 < a < 1. (-2, 1)( 2, 1) (0, 0)

So lets look at the equation y = 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

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