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5.3: Exercises

Last updated

May 2, 2022
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- 5.2: Transformation of Graphs
- 6: Operations on Functions

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Exercise $\PageIndex{1}$

Find a possible formula of the graph displayed below.

$clipboard_e8b4a7bb9bf98f2eca140c515f3c07a73.png$
$clipboard_e47936b8705eab4c58541b4e5a5e698d1.png$
$clipboard_eb276835d303e6b20749fe8b803d853a9.png$
$clipboard_e320eabc5569d2368efc7acc8004070ce.png$
$clipboard_ec5c6c5c5b447df93a0a4ef98c135ebaf.png$
$clipboard_e4d073d30b31290b84b180376b82b6bcf.png$

Answer

$y=|x|+1$
$y=-\sqrt{x}$
$y=(x-2)^{2}+1$
$y=\dfrac{1}{x+2}+2$
$y=-(x+2)^{3}$
$y=-|x-3|+2$

Exercise $\PageIndex{2}$

Sketch the graph of the function, based on the basic graphs in Section 5.1 and the transformations described above. Confirm your answer by graphing the function with the calculator.

$f(x)=|x|-3$
$f(x)=\dfrac 1 {x+2}$
$f(x)=-x^2$
$f(x)=(x-1)^3$
$f(x)=\sqrt{-x}$
$f(x)=4\cdot |x-3|$
$f(x)=-\sqrt{x}+1$
$f(x)=(\dfrac{1}{2}\cdot x)^2+3$

Answer

$clipboard_ee265927e6582495145348c203441d81b.png$
$clipboard_e1abd9d34f7825271d3acd1a9db6d3b98.png$
$clipboard_e9e5d2dac8db02640c11efd79f28edf38.png$
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$clipboard_eedf76527991750ca1c1fd494dcdbaf3d.png$
$clipboard_e1d3e44d4e3219d811af6818a6eb48c49.png$

Exercise $\PageIndex{3}$

Consider the graph of $f(x)=x^2-7x+1$ . Find the formula of the function that is given by performing the following transformations on the graph.

Shift the graph of $f$ down by $4$ .
Shift the graph of $f$ to the left by $3$ units.
Reflect the graph of $f$ about the $x$ -axis.
Reflect the graph of $f$ about the $y$ -axis.
Stretch the graph of $f$ away from the $y$ -axis by a factor $3$ .
Compress the graph of $f$ towards the $y$ -axis by a factor $2$ .

Answer

$y=x^{2}-7 x-3$
$y=(x+3)^{2}-7 \cdot(x+3)+1=x^{2}-x-11$
$y=-x^{2}+7 x-1$
$y=x^{2}+7 x+1$
$y=\dfrac{1}{9} x^{2}-\dfrac{7}{3} x+1$
$y=4 x^{2}-14 x+1$

Exercise $\PageIndex{4}$

How are the graphs of $f$ and $g$ related?

$f(x)=\sqrt{x}, \quad g(x)=\sqrt{x-5}$
$f(x)=|x|, \quad g(x)=2 \cdot|x|$
$f(x)=(x+1)^3 , \quad g(x)=(x-3)^3$
$f(x)=x^2+3x+5, \quad g(x)=(2x)^2+3(2x)^2+5$
$f(x)=\dfrac 1 {x+3}, \quad g(x)=-\dfrac 1 x$
$f(x)= 2 \cdot |x|, \quad g(x)=|x+1|+1$

Answer

shift to the right by $5$
stretched away from the $x$ -axis by a factor $2$
shift to the right by $4$
compressed towards the $y$ -axis by a factor $2$
shifted to the right by $3$ (to get the graph of $y=\dfrac{1}{x}$ ) and then reflected about the $x$ -axis
compressed towards the $x$ -axis by a factor $2$ (you get $y=|x|$ ) then shifted to the left by $1$ and up by $1$