1. Transformations of Graphs
3.4
Graph functions using vertical and horizontal translations
Graph function using stretching and shrinking
Graph function using reflections
Combine transformations
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2. Vertical Shifts
A graph is shifted up or down. The shape of the graph is not changed—only its position.
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3. Horizontal Shifts A graph is shifted left or right.
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5. Example
Shifts can be combined to translate a graph of y = f(x) both vertically and horizontally.
Shift the graph of y = x2 to the left 3 units and downward 2 units.
y = x y = (x + 3) y = (x + 3)2  2
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6. Example
Find an equation that shifts the graph of f(x) = x2  2x + 3 left 4 units and down 3 units.
Solution
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7. Stretching and Shrinking
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8. Horizontal Stretching and Shrinking
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9. Example
Use the graph of y = f(x) to sketch the graph of each equation.
a) y = 2f(x) b)
Solution
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10. Solution continued b)
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11. Reflections of Graphs
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12. Example
For the function f(x) = x2 + x  2 graph its reflection across the x-axis and across the y-axis.
Solution
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13. Combining Transformations
Transformations of graphs can be combined to create new graphs. For example the graph of y = 3(x + 3)2 + 1 can be obtained by performing four transformations on the graph of y = x2.
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14. Combining Transformations continued
y = 3(x + 3)2 + 1
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15. Example Describe how the graph of the equation
can be obtained by transforming the graph of y = |x|. Then graph the equation.
Solution
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