Let's start by looking at the graphs of some "well-known" functions.

f(x) = x	f(x) = x2	f(x) = x3
f(x) = 1 x	f(x) = x1/2	f(x) = |x|

(To review these graphs, consult the section on Functions and Their Graphs in Chapter 1 of Calculus Applied to the Real World)

But what about more complicated functions? For example, what about f(x) = (x-3)²? Notice that here we've taken f(x) = x² (one of the functions graphed above) and replaced x by (x-3) to get a new function. Well, here are some "shift" rules that tell you the effect of operations like this.

Shift Rules

Rule	Example
Horizontal Shift Let c be a fixed positive number. Replacing x by the quantity (xc) shifts the graph to the right c units. Replacing x by the quantity (x+c) shifts the graph to the left c units.	Here is a picture of the graph of g(x) = |x4|. It is obtained from the graph of f(x) = |x| by shifting it to the right 4 units.
Vertical Shift Let c be a fixed positive number. Replacing f(x) by f(x) + c shifts the graph up c units. Replacing f(x) by f(x) c shifts the graph down c units.	Here is a picture of the graph of g(x) = x2 1. It is obtained from the graph of f(x) = x2 by shifting it down 1 unit.

Now here is one for you to do.

Example 1 Sketching a Shifted Function

Let   f(x)

=

1 x + 1

.

Select the correct options and press "Check."

The graph of f(x) is obtained from the graph of 1 x by shifting it up down left right 0 1 2 3 4 unit(s).

Now click on the correct graph of the function f.

Here is one that is obtained by two successive translations.

Example 2 Multiple Shifts

Let g(x) = (x2)^1/2 + 1. Select the correct options and press "Check."

Now click on the correct graph of the function f.

Question: Why does the Horizontal Shift Rule work?
Answer

Question: Why does the Vertical Shift Rule work?
Answer

In addition to the shift rules, we also have the

Scaling Rules

Rule	Example
Horizontal Scaling If g(x) = f(cx) with c positive, then: If c > 1, the graph of g is the graph of f, compressed in the x-direction by a factor of c. If 0 < c < 1, then the graph is stretched in the x-direction by a factor of 1/c	Here is a picture of the graph of g(x) = (0.5x)3. Since c = 0.5 < 1, the graph is obtained from that of f(x) = x3 by stretching it in the x-direction by a factor of 1/c = 2.
Vertical Scaling If g(x) = cf(x)with c positive, then: If c > 1, the graph of g is the graph of f, stretched in the y-direction by a factor of c. If 0 < c < 1, then the graph is compressed in the y-direction by a factor of 1/c.	Here is a picture of the graph of g(x) = 3(x)1/2. Since c = 3 > 1, the graph is obtained from that of f(x) = x1/2 by stretching it in the y-direction by a factor of c = 3.

Now one for you.

Example 3 Sketching a Scaled Function

Let   g(x) =

1 3

x + 1 x . Select the correct options and press "Check."

The graph of g(x) is obtained from the graph of x + 1 x by compressing stretching it by a factor of 0 1 2 3 4 in the x- y- direction.

First graph the function

f(x)

=

x + 1 x

on your graphing calculator (or here) and then click on the correct graph of the function g.

Here is one that is obtained by several successive operations.

Let   g(x)   =

(x - 2)2 3 + 4.

Select the correct options and press "Check."

Here are the graphs corresponding to these steps.

Original Function y = x2	Step 1 y = (x2)2	Step 2 y = (x - 2)2 3	Step 3 y = (x - 2)2 3 + 4

Finally, we look at reflections.

Reflections

Rule	Example
Horizontal Reflection Replacing x by the quantity (x) reflects the graph in y-axis (In other words, it "flips it about the y-axis.")	Here is a picture of the graph of g(x) =(0.5x)3+1. It is obtained from the graph of f(x) = 0.5x3+1 by reflecting it in the y-axis.
Vertical Reflection Replacing f(x) by f(x) reflects the graph in the x-axis (In other words, it "flips it about the x-axis.")	Here is a picture of the graph of g(x) = (x2 1). It is obtained from the graph of f(x) = x2 1 by reflecting it in the x-axis.