New Functions from Old:

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Let's start by looking at the graphs of some "wellknown" functions.
$f(x) = x$ 
$f(x) = x^{2}$ 
$f(x) = x^3$ 
$f(x)=\frac{1}{x}$ 
$f(x) = \sqrt{x}$ 
$f(x) = \x\$ 
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 $g(x) = x^{21}.$ 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 $f(x) = \frac{1}{x+1}.$ 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 $g(x) = \sqrt{x2} + 1.$ 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 $x$direction 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 $g(x) = 3\sqrt{x}.$ Since $c = 3>1,$ the graph is obtained from that of $f(x) = \sqrt{x}$ by stretching it in the $y$direction by a factor of $c = 3.$

Now one for you.
_{} Example 3 Sketching a Scaled Function
Let $g(x) = \frac{1}{3}\left(x+\frac{1}{x}\right).$ Select the correct options and press "Check."Now graph the function
Here is one that is obtained by several successive operations.
_{} Example 4 A Scaled and Shifted Function Let $g(x) = \frac {(x2) ^2}{3} +4.$ 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 $y = \frac{(x2)^2}{3}$ 
Step 3 $y = \frac{(x2)^2}{3}+4$ 
Finally, we look at reflections.
Rule  Example 
Horizontal Reflection

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 $y$axis.

Vertical Reflection

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 $x$axis.

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