Chapter 3: Transformations of Graphs and Data Lesson 5: The Graph Scale-Change Theorem Mrs. Parziale.

Chapter 3: Transformations of Graphs and Data

Lesson 5: The Graph Scale-Change Theorem

Mrs. Parziale

Vocabulary:• Vertical stretch: A scale change that makes the original graph taller or shorter

• Horizontal stretch: a scale change that makes the original graph wider or skinnier.

• Scale change: a stretch or shrink applied to the graph vertically or horizontally

• Vertical scale change: The value that changes the vertical values of the graph. • Horizontal scale change: The value that changes the horizontal values of the

graph.

• Size change: When the same vertical and horizontal scale change occurs.

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Example 1:

Consider the graph of (a) Complete the table and graph on the grid:

x y-2-1012

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3 4y x x

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(b) Replace (y) with .

1. Solve the new equation for y and graph it on the same grid at right.

2. What happens to the y-coordinates?

3. This is called a vertical stretch of magnitude 3 .

4. Under what scale change is the new figure a vertical scale change of the original?

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y

(c) Replace (x) with .

1. Solve the new equation for y and graph it.

2. What happens to the x-coordinates?

3. This is called a horizontal stretch of magnitude 2 .

4. Under what scale change is the new figure a horizontal scale change of the original?

2

x

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1 2 3 4 5 6–1–2–3–4–5–6 x

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

What is happening to each part of the graph?

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3( ) 4f x x x

( , ) (2 ,3 )S x y x y

• How is the x changed?• Change: horizontal

stretch two times wider.

• How is the y changed?• Change: vertical stretch

three times the original.

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1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x

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( , ) (2 ,3 )S x y x y

Graph Scale-Change Theorem

In a relation described by a sentence in (x) and (y), the following two processes yield the same graph:

(1) replace (x) by and (y) by in the sentence

(2) apply the scale change __________________ to the

graph of the original relation. Note: If a = b, then you have performed a __________

( , ) ( , )x y ax by

x

a

y

b

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

size change

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So, What’s the Equation?

(d) Find an equation for g(x), the image of f(x) under ( , ) (2 ,3 )S x y x y

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3( ) 4f x y x x x y-2 0-1 30 01 -32 0

x y

Example 2:

• Consider . Find an equation for the function under

• Describe what happens to all of the x values:

• Describe what happens to all of the y values:

y x( , ) , 2

3

xS x y y

• Find the equation for the transformed image by

– Replace (x) with ______________

– Replace (y) with ______________– Now make the new equation (remember to

simplify to y= form):

( , ) , 23

xS x y y

Graph It!

2 3

y x

y x

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Closure - Example 3:

The graph to the right is y = f(x). Draw .

• What should happen to all of the x values?

• What should happen to all of the y values?

3 ( )4

xy f