3-5: The Graph Scale-Change Theorem Pre Calc I (CP) ___________

SCALE CHANGE = a stretch or shrink applied to the graph

Ex. 1: Think back to translations. How do you think you might write the rule for a scale change that stretches the graph horizontally by a factor of 3 and shrinks the graph to ½ its original height?

(x, y) (3x, )

Ex 2: Consider the graph of

(a) Graph. Label 3 points with coordinates:

X Y

-2 0-1 30 01 -32 0

(b) Replace y with in the equation.

Solve the new equation for y: y = 3(x3 – 4x) . Graph it.

What happens to the y-coordinates? Highs are 3 times higher and the lows are 3 times lower

This is called a vertical stretch of magnitude 3 .

(c) Replace x with .

Solve the new equation for y: y = ( )3 – 4( ) . Graph it.

What happens to the x-coordinates? Horizontal stretch – function is 2 times wider

This is called a horizontal stretch of magnitude 2 .

Ex. 3: Continue looking at .

What would happen if I applied the scale change ?

Graph is twice as wide and 3 times taller______________________

Find an equation for g(x), the image of f(x) under .

To do this, we need to substitute where we see an x in the equation,

and substitute where we see a y in the equation.

= ( )3 – 4( )

Simplify into “y =” form:

y = 3( ( )3 – 4( ) )

Graph this in your graphing calculator to ensure the correct scale change.

Graph Translation Theorem:

The following two processes yield the same graph:

(1) applying the scale change to the graph of the original equation.

i.e. applying the scale change to each individual point applies the scale change to the entire graph

(2) replacing x with and y with in the equation

RECALL: THIS IS WHEN THE TRANSLATION APPEARED TO BE THE OPPOSITE OF WHAT YOU WANTED TO DO!! THE SAME IS TRUE OF SCALE CHANGES!

If a = negative, the graph has been reflected (flipped) over the y-axisIf b = negative, the graph has been reflected over the x-axis

Ex. 4: Consider under the scale change

Describe what happens to all of the x values: graph is 1/3 as wide as it was; horizontal shrink

Describe what happens to all of the y values: graph is twice as tall AND flipped over the x-axis

Find the equation for the transformed image by substituting 3x in for x and y/-2 in for y.

Simplify into “y =” form.

Ex. 5: The graph below is y = f(x). Draw .

What should happen to all of the x values? Horizontal stretch by a factor of 4

What should happen to all of the y values? Graph is 1/3 its original height