Section 2.5Section 2.5

Transformations of Transformations of FunctionsFunctions

OverviewOverview

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

ShiftingShifting ReflectingReflecting StretchingStretching

Vertical ShiftingVertical Shifting

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

Example 1Example 1

Example 1Example 1

Example 1Example 1

Vertical Shifts of GraphsVertical Shifts of Graphs

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

Example 2Example 2

Example 2Example 2

Example 2Example 2

Horizontal ShiftingHorizontal Shifting

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

Example 3Example 3

Example 3Example 3

Example 3Example 3

Horizontal Shifts of GraphsHorizontal Shifts of Graphs

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

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

Example 4Example 4

Example 4Example 4

Example 4Example 4

Combining ShiftsCombining Shifts

Suppose I wanted to graph the equation:Suppose I wanted to graph the equation:

How would I do this?How would I do this?

54)( xxf

Work from the variable out!Work from the variable out!

Example 5Example 5

Example 5Example 5

Example 5Example 5

Reflecting GraphsReflecting Graphs

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

Example 6Example 6

Example 6Example 6

Reflecting GraphsReflecting Graphs

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

Example 7Example 7

Example 7Example 7

Vertical Stretching and Vertical Stretching and ShiftingShifting

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

Vertical Stretching and Vertical Stretching and ShiftingShifting

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

Example 8Example 8

Example 8Example 8

Example 8Example 8

Horizontal Stretching and Horizontal Stretching and ShrinkingShrinking

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

Horizontal Stretching and Horizontal Stretching and ShrinkingShrinking

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

Putting It All TogetherPutting It All Together

So how do I graph an equation with So how do I graph an equation with multiple transformations? Does multiple transformations? Does the order in which I do the the order in which I do the transformations matter?transformations matter?

YES!YES!

A more complicated A more complicated exampleexample

Graph the following:Graph the following:

Remember: Stretches First, Remember: Stretches First, Reflections Second, And Shifts Reflections Second, And Shifts Last!Last!

2)1(23)34 xxf

Example 9Example 9

Example 9Example 9

Example 9Example 9

Example 9Example 9

Example 9Example 9

Example 9Example 9

Even and Odd FunctionsEven and Odd Functions

f(x) is EVEN if f(-x) = f(x) for all x in f(x) is EVEN if f(-x) = f(x) for all x in the domain of f. The graph of an the domain of f. The graph of an even function is symmetric with even function is symmetric with respect to the y-axis.respect to the y-axis.

f(x) is ODD if f(-x)=-f(x) for all x in f(x) is ODD if f(-x)=-f(x) for all x in the domain of f. We say odd the domain of f. We say odd function is symmetric with respect function is symmetric with respect to the origin.to the origin.

Even and Odd FunctionsEven and Odd Functions

Big Hint!Big Hint!

If f(x) has all even exponents then If f(x) has all even exponents then f(x) is even!f(x) is even!

If f(x) has all odd exponents then f(x) If f(x) has all odd exponents then f(x) is odd!is odd!