New Functions from Old:
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Let's start by looking at the graphs of some "well-known" functions.
$f(x) = x$ |
$f(x) = x^{2}$ |
$f(x) = x^3$ |
$f(x)=\frac{1}{x}$ |
$f(x) = x^{1/2}$ |
$f(x) = \|x\|$ |
Rule | Example |
Horizontal Shift
Let c be a fixed positive number.
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Here is a picture of the graph of $g(x) = \|x-4\|.$ It is obtained from the graph of $f(x) = \|x\|$ by shifting it to the right $4$ units.
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Vertical Shift
Let c be a fixed positive number.
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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.
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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 $g(x) = (x-2)^{1/2} + 1.$ 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:
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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 $1/c = 2$.
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Vertical Scaling
If $g(x) = cf(x)$ with $c$ positive, then:
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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 $y$-direction by a factor of $c = 3.$
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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 $g(x) = \frac {(x-2) ^2}{3} +4.$ Select the correct options and press "Check."Here are the graphs corresponding to these steps.
Original Function $y = x^2$ |
Step 1 $y = (x-2)^2$ |
Step 2 $y = \frac{(x-2)^2}{3}$ |
Step 3 $y = \frac{(x-2)^2}{3}+4$ |
Finally, we look at reflections.
Rule | Example |
Horizontal Reflection
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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.
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Vertical Reflection
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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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Index of On-Line Topics Exercises for This Topic Everything for Calculus Everything for Finite Math Everything for Finite Math & Calculus Utility: Function Evaluator & Grapher |