Section 2.5 Transformations of Functions

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

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.

Example 1

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.

Example 2

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.

Example 3

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).

Example 4

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

Example 5

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).

Example 6

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).

Example 7

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).

Vertical Stretching and Shifting Given the equation y=a*f(x), where 0<a<1, to obtain the graph, take f(x) and shrink the graph vertically by a factor of (1/a). i.e. If the point (x, y) is in the graph of f(x), then the point (x, y/(1/a)) is in the graph of a*f(x).

Example 8

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).

Horizontal Stretching and Shrinking Given the equation y=f(a*x), where 0<a<1, to obtain the graph, take f(x) and stretch the graph horizontally by a factor of (1/a). i.e. If the point (x, y) is in the graph of f(x), then the point ((1/a)*x, y) is in the graph of f(a*x).

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!

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

Example 9

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.

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!