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**Graphical Transformations**

Vertical and Horizontal Translations Vertical and Horizontal Stretches and Shrinks

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**Take the equation f(x)= x2**

How do you modify the equation to translate the graph of this equation 5 units to the right? units to the left? How do you modify the equation to translate the graph of this equation 3 units down? units up? What if you wanted to translate the graph of this equation 5 units to the left and 3 units down?

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**The parabola has been translated 5 units to the right.**

How is the equation modified to cause this translation?

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**Notice the change in the equation y = x2 to create the horizontal**

shift of 5 units to the right. f(x) = x2 g(x) = (x-5)2

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**The parabola is now translated 5 units to the left.**

How is the equation modified to cause this translation?

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**Notice the change in the graph of the equation y=x2 to**

create a horizontal shift of 5 units to the left. f(x)=x2 h(x)=(x+5)2

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**The parabola has now been translated three units down.**

How is the equation modified to cause this translation?

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**Notice how the equation y = x2 has changed to make the**

Vertical translation of 3 units down. f(x)=x2 q(x)=x2-3

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**The parabola has now been translated 3 units up.**

How is the equation modified to cause this translation?

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**Notice how the equation y = x2 has been changed to make the**

Vertical translation 3 units up. f(x)=x2 r(x)=x2+3

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**Write what you think would be the equation for translating**

the parabola 5 units to the left and 3 units up?

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The equation would be What would the graph would look like?

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**g(x) is the translation of f(x) 5 units to the left and 3 units up.**

f(x)= x2 g(x) = (x+5)2+3

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**Vertical and horizontal stretches and shrinks**

How does the coefficient on the x2 term affect the graph of f(x) = x2? What if we substitute an expression such as 2x into f(x)? How would that affect the graph of f(x) = x2?

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**The parabola has been vertically stretched by a factor of 2.**

Notice how the equation has been modified to cause this stretch.

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**The parabola is vertically shrunk by a factor of ½.**

Notice how the equation has been modified to cause this shrink.

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**By substituting an expression like 2x in for x in f(x) = x2**

gives a different type of shrink. f(2x) = (2x)2. A horizontal shrink by a factor of ½.

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**Suppose we found g(1/2x). The equation would be**

y = (1/2x)2.. How would this affect the graph of the function g(x) = x2? It is a horizontal stretch by a factor of 2.

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**If we were to write some rules for translations of functions**

and stretches/shrinks of functions, what would we write? Horizontal translation: Vertical translation: Vertical stretch: Vertical shrink: Horizontal stretch: Horizontal shrink:

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Parabolas and Modeling

Parabolas and Modeling

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