New Functions from Old:Scaled and Shifted Functions miscellaneous on-line topics for Calculus Applied to the Real World

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

f(x) = x

f(x) = x2

f(x) = x3

f(x)=
 1x

f(x) = x1/2

f(x) = |x|

(To review these graphs, consult the section on functions and their graphs in Chapter 1 of Applied Calculus)

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

 The graph of f(x) is obtained from the graph of 1x 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."

 The graph of g(x) is obtained from the graph of x1/2 by shifting it up down left right 2 units, and up down left right 1 unit.

Now click on the correct graph of the function f.

Question: Why does the Horizontal Shift Rule work?

Question: Why does the Vertical Shift Rule work?

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) =
 13
x+
 1x
. Select the correct options and press "Check."

The graph of g(x) is obtained from the graph of   x+
1

x
by it by a factor of
 0 1 2 3 4 in the x- y- direction.

Now graph the function

f(x)=
x+ 1x

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.

Example 4 A Scaled and Shifted Function

Let   g(x)   =
 (x - 2)23
+4.
Select the correct options and press "Check."

Starting with f(x) = x2,

Step 1: Replace x by (x2), giving (x2)2.
 This shifts it expands it compresses it 2 units left 2 units right 2 units up 2 units down by a factor of 2

Step 2: Divide the last function by 3, giving (x2)2/3.
 This shifts it expands it compresses it 3 units left 3 units right 3 units up 3 units down by a factor of 3

Step 3: Now add 4 to the last function, to obtain the given function g.
 This shifts it expands it compresses it 4 units left 4 units right 4 units up 4 units down by a factor of 4

Here are the graphs corresponding to these steps.

Original Function
y = x2
Step 1
y = (x2)2
Step 2
 y = (x - 2)23
Step 3
 y = (x - 2)23 + 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.

Last Updated:January, 1998