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Section 2.5 Transformations of Functions

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Overview In this section we study how certain transformations of a function affect its graph. We will specifically look at: Shifting Shifting Reflecting Reflecting Stretching Stretching

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Vertical Shifting Adding a constant to a function shifts its graph vertically: upward if the constant is positive and downward if the constant is negative.

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Example 1

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Vertical Shifts of Graphs Given the equation y=f(x)+c, to obtain the graph, take f(x) and shift it c units vertically. i.e. If the point (x, y) is in the graph of f(x), then the point (x, y+c) is in the graph of f(x)+c.

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Example 2

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Horizontal Shifting Adding a constant to the variable shifts its graph horizontally. Adding a positive constant shifts the graph to the left, and adding a negative constant shifts the graph to the right.

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Example 3

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Horizontal Shifts of Graphs Given the equation y=(x-c), to obtain the graph, take f(x) and shift it c units to the right. If the point (x, y) is in the graph of f(x), then the point (x+c, y) is in the graph of f(x-c). Given the equation y=(x+c), to obtain the graph, take f(x) and shift it c units to the left. If the point (x, y) is in the graph of f(x), then the point (x-c, y) is in the graph of f(x+c).

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Example 4

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Combining Shifts Suppose I wanted to graph the equation: How would I do this? Work from the variable out!

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Example 5

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Reflecting Graphs Given the equation y= -f(x), to obtain the graph, take f(x) and reflect it over the x-axis. i.e. if the point (x,y) is in the graph of f(x), then the point (x,-y) is in the graph of –f(x).

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Example 6

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Reflecting Graphs Given the equation y= f(-x), to obtain the graph, take f(x) and reflect it over the y-axis. i.e. if the point (x,y) is in the graph of f(x), then the point (-x,y) is in the graph of f(-x).

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Example 7

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Vertical Stretching and Shifting Given the equation y=a*f(x), where a>1, to obtain the graph, take f(x) and stretch the graph vertically by a factor of a. i.e. If the point (x, y) is in the graph of f(x), then the point (x, a*y) is in the graph of a*f(x).

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Vertical Stretching and Shifting Given the equation y=a*f(x), where 0
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Example 8

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Horizontal Stretching and Shrinking Given the equation y=f(a*x), where a>1, to obtain the graph, take f(x) and shrink the graph horizontally by a factor of a. i.e. If the point (x, y) is in the graph of f(x), then the point (x/a, y) is in the graph of f(a*x).

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Horizontal Stretching and Shrinking Given the equation y=f(a*x), where 0
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Putting It All Together So how do I graph an equation with multiple transformations? Does the order in which I do the transformations matter? YES!

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A more complicated example Graph the following: Remember: Stretches First, Reflections Second, And Shifts Last!

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Example 9

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Even and Odd Functions f(x) is EVEN if f(-x) = f(x) for all x in the domain of f. The graph of an even function is symmetric with respect to the y-axis. f(x) is ODD if f(-x)=-f(x) for all x in the domain of f. We say odd function is symmetric with respect to the origin.

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Even and Odd Functions Big Hint! If f(x) has all even exponents then f(x) is even! If f(x) has all odd exponents then f(x) is odd!

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Vertical and Horizontal Shifts of Graphs. Identify the basic function with a graph as below:

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