SECTION 1.5SECTION 1.5

GRAPHING TECHNIQUES;GRAPHING TECHNIQUES;

TRANSFORMATIONSTRANSFORMATIONS

TRANSFORMATIONSTRANSFORMATIONS

Recall our “library” of Recall our “library” of functions.functions.

Here we will learn techniques Here we will learn techniques for graphing a function which is for graphing a function which is “related” to one we already “related” to one we already know how to graph. know how to graph.

HORIZONTAL SHIFTSHORIZONTAL SHIFTS

On the same screen, graph On the same screen, graph each of the following each of the following functions:functions:

YY11 = x = x22

YY22 = (x - 1) = (x - 1)22

YY33 = (x - 3) = (x - 3)22

YY44 = (x + 2) = (x + 2)22

COMPARING y = x 2 and y = (x -

2) 2

COMPARING y = x 2 and y = (x -

2) 2

If we named the first function If we named the first function f(x), we could denote the f(x), we could denote the second one by f(x - 2).second one by f(x - 2).

In general, we can refer to In general, we can refer to any horizontal shift of a any horizontal shift of a function f(x) by using the function f(x) by using the notationnotation

f(x - h)f(x - h)

y = f(x - 2) y = f(x + 3)

y = f(x - 2) y = f(x + 3)

When h is positive (that is, when there When h is positive (that is, when there is a value being subtracted from x) the is a value being subtracted from x) the shift is to the right.shift is to the right.

When h is negative (that is, when there When h is negative (that is, when there is a value being added to x) the shift is is a value being added to x) the shift is to the left.to the left.

VERTICAL SHIFTSVERTICAL SHIFTS

In general, we can refer to any In general, we can refer to any vertical shift of a function f(x) vertical shift of a function f(x) by using the notation:by using the notation:

f(x) + kf(x) + k

y = f(x) + 4 y = f(x) - 1

y = f(x) + 4 y = f(x) - 1

When k is positive, the When k is positive, the shift is upward.shift is upward.

When k is negative, the When k is negative, the shift is downward.shift is downward.

EXAMPLE:EXAMPLE:

The figure shows the graph of The figure shows the graph of f(x). Sketch the graphs of f(x f(x). Sketch the graphs of f(x + 1) and f(x) - 1.+ 1) and f(x) - 1.

- 2- 2 - 1- 1 11 22

y = f(x + 1)y = f(x + 1)

- - 33

- - 22

- - 11

11 22

y = f(x) - 1y = f(x) - 1

VERTICAL STRETCHES VERTICAL STRETCHES

When we compare the graph of y When we compare the graph of y = x= x22 to the graph of y = 2x to the graph of y = 2x22, we , we find the second one is more find the second one is more narrow than the first.narrow than the first.

This is called a vertical stretch. This is called a vertical stretch. All the y-values are being All the y-values are being doubled.doubled.

VERTICAL SHRINKS VERTICAL SHRINKS

When we compare the graph of y When we compare the graph of y = x= x22 to the graph of y = .5x to the graph of y = .5x22, we , we find the second one is wider than find the second one is wider than the first.the first.

This is called a vertical shrink. This is called a vertical shrink. All the y-values are being halved.All the y-values are being halved.

In general, we can denote vertical In general, we can denote vertical stretches and shrinks to a function stretches and shrinks to a function f(x) in the following way:f(x) in the following way:

For For a a > 1, stretch > 1, stretch

y = af(x)y = af(x)

For 0 < For 0 < a a < 1, < 1, shrinkshrink

EXAMPLE:EXAMPLE:

Sketch the graphs of y = 3f(x), Sketch the graphs of y = 3f(x), y = .5f(x), and y = - .5f(x)y = .5f(x), and y = - .5f(x)

y = 3f (x)y = 3f (x)

- - 22

- - 11

11 22

y = .5f (x)

y = - .5f (x)y = - .5f (x)

HORIZONTAL STRETCHES AND

SHRINKS

HORIZONTAL STRETCHES AND

SHRINKSIn general, we can denote In general, we can denote horizontal stretches and shrinks to horizontal stretches and shrinks to a function f(x) in the following way:a function f(x) in the following way:

For For c > 1, shrinkc > 1, shrink

y = f(cx)y = f(cx)

For 0 <For 0 < c < 1, c < 1, stretchstretch

EXAMPLE:EXAMPLE:

Sketch the graphs of y = f(2x) and y = f(.5x)Sketch the graphs of y = f(2x) and y = f(.5x)

y = f (2x)y = f (2x)

y = f (.5x)y = f (.5x)

EXAMPLE:EXAMPLE:

Given f(x) = x Given f(x) = x 33 - 4x, explain - 4x, explain the transformations that will the transformations that will occur to the graph of the occur to the graph of the function for f(2x) + 3function for f(2x) + 3

The graph will be compressed The graph will be compressed horizontally and shifted 3 horizontally and shifted 3 units up.units up.

Graph it!Graph it!

CONCLUSION OF SECTION 1.5CONCLUSION OF SECTION 1.5