New Functions from Old:

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Let's start by looking at the graphs of some "wellknown" functions.
f(x) = x 
f(x) = x^{2} 
f(x) = x^{3} 


f(x) = x^{1/2} 
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) = (x3)^{2}? Notice that here we've taken f(x) = x^{2} (one of the functions graphed above) and replaced x by (x3) to get a new function. Well, here are some "shift" rules that tell you the effect of operations like this.
Rule  Example 
Horizontal Shift
Let c be a fixed positive number.

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.

Here is a picture of the graph of g(x) = x^{2} 1. It is obtained from the graph of f(x) = x^{2} by shifting it down 1 unit.

Now here is one for you to do.
_{} Example 1 Sketching a Shifted Function
Let f(x)  =  x + 1  .  Select the correct options and press "Check." 
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
Rule  Example 
Horizontal Scaling
If g(x) = f(cx) with c positive, then:

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) = x^{3} by stretching it in the xdirection by a factor of 1/c = 2.

Vertical Scaling
If g(x) = cf(x)with c positive, then:

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) = x^{1/2} by stretching it in the ydirection by a factor of c = 3.

Now one for you.
_{} Example 3 Sketching a Scaled Function
Let g(x) = 


First graph the function
f(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.
_{} Example 4 A Scaled and Shifted FunctionLet g(x) = 
 Select the correct options and press "Check." 
Here are the graphs corresponding to these steps.
Original Function y = x^{2} 
Step 1 y = (x2)^{2} 
Step 2

Step 3


Finally, we look at reflections.
Rule  Example 
Horizontal Reflection
(In other words, it "flips it about the yaxis.")

Here is a picture of the graph of g(x) =(0.5x)^{3}+1. It is obtained from the graph of f(x) = 0.5x^{3}+1 by reflecting it in the yaxis.

Vertical Reflection
(In other words, it "flips it about the xaxis.") 
Here is a picture of the graph of g(x) = (x^{2} 1). It is obtained from the graph of f(x) = x^{2} 1 by reflecting it in the xaxis.

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