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**Chapter 3: Transformations of Graphs and Data**

Lesson 5: The Graph Scale-Change Theorem Mrs. Parziale

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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 -1 1 2

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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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(c) Replace (x) with 1. Solve the new equation for y and graph it. 2. What happens to the x-coordinates? This is called a horizontal stretch of magnitude 2 . Under what scale change is the new figure a horizontal scale change of the original?

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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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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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**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 __________ size change If a = negative, the graph has been reflected (flipped) over the y-axis If b = negative, the graph has been reflected over the x-axis

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So, What’s the Equation? (d) Find an equation for g(x), the image of f(x) under x y -2 -1 3 1 -3 2 x y

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**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:

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**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):

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Graph It!

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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?

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I. There are 4 basic transformations for a function f(x). y = A f (Bx + C) + D A) f(x) + D (moves the graph + ↑ and – ↓) B) A f(x) 1) If | A | > 1 then.

I. There are 4 basic transformations for a function f(x). y = A f (Bx + C) + D A) f(x) + D (moves the graph + ↑ and – ↓) B) A f(x) 1) If | A | > 1 then.

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