0. Functions

The accompanying figure shows the graph of a function f(x) with domain [0,2] and range [0,1]. Find the domains and ranges of the following functions, and sketch their graphs.

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c. 2f(x)

Given the equation $y=f(x-3)+2$, what kind of transformation does this represent?

Shifting and Scaling Graphs

Suppose the graph of g is given. Write equations for the graphs that are obtained from the graph of g by shifting, scaling, or reflecting, as indicated.

e. Stretch vertically by a factor of 5

Use the graph of $f$ in the figure to plot the following functions.

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$y=f\(\left\)(x-2\(\right\))$

Shifting Graphs

The accompanying figure shows the graph of y = −x² shifted to two new positions. Write equations for the new graphs.

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Graph the functions in Exercises 37–56.

y = |x − 2|

Graph the functions in Exercises 37–56.

y = (x + 1)²/³

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>

a. $\text{ƒ}(x+1)$

Graph the function y = √|x|.

How do you obtain the graph of $y=f\(\left\)(x+2\(\right\))$ from the graph of $y=f\(\left\)(x\(\right\))$?

Shifting Graphs

Exercises 27–36 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation.

y = x³ Left 1, down 1

Use shifts and scalings to transform the graph of $\text{ƒ}\left(x\right)=\sqrt{x}$  into the graph of g. Use a graphing utility to check your work.

$g\left(x\right)=3\sqrt{x-1}-5$

Shifting Graphs

The accompanying figure shows the graph of y = −x² shifted to four new positions. Write an equation for each new graph.

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The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>

$f\left(2\right(x-1\left)\right)$

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.

$h\left(x\right)=-4x^{{}^2}-4x+12$