Graph Transformations

Section 1.1 Graph Transformations

Example 1.1.1.

The table below defines the functions $f\text{,}$ $g\text{,}$ and $h\text{.}$ Express $g(x)$ and $h(x)$ in terms of $f\text{.}$

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$8$	$6$	$4$	$2$	$0$	$-1$	$-2$
$g(x)$	$-8$	$-6$	$-4$	$-2$	$0$	$1$	$2$
$h(x)$	$5$	$3$	$1$	$1$	$-3$	$-4$	$-5$

Answer.

$g(x)=-f(x)$ and $h(x)=f(x)-3\text{.}$

Example 1.1.2.

(a)

If $f(x)=x^2$ and $g(x)=2x^2+5\text{,}$ express $g(x)$ in terms of $f\text{.}$

Answer.

$g(x)=2f(x)+5$

(b)

If $f(x)=x^2$ and $h(x)=(x+5)^2-3\text{,}$ express $h(x)$ in terms of $f\text{.}$

Answer.

$h(x)=f(x+5)-3$

Exercises Exercises

One Function in Terms of Another.

In Exercises 1–4, the table below defines the functions $f\text{,}$ $g\text{,}$ $h\text{,}$ $k\text{,}$ and $l\text{.}$

$x$	$-2$	$-1$	$0$	$1$	$2$
$f(x)$	$0$	$1$	$2$	$3$	$4$
$g(x)$	$4$	$3$	$2$	$1$	$0$
$h(x)$	$0$	$-1$	$-2$	$-3$	$-4$
$k(x)$	$6$	$7$	$8$	$9$	$10$
$l(x)$	$0$	$3$	$6$	$9$	$12$

1.

Express $g(x)$ in terms of $f$ and describe how the graph of $y=f(x)$ can be transformed into the graph of $y=g(x)\text{.}$

2.

Express $h(x)$ in terms of $f$ and describe how the graph of $y=f(x)$ can be transformed into the graph of $y=h(x)\text{.}$

3.

Express $k(x)$ in terms of $f$ and describe how the graph of $y=f(x)$ can be transformed into the graph of $y=k(x)\text{.}$

4.

Express $l(x)$ in terms of $f$ and describe how the graph of $y=f(x)$ can be transformed into the graph of $y=l(x)\text{.}$

5.

The second row in the table below gives values for the function $f\text{.}$ Complete the rest of the table. If you don’t have sufficient information to fill in some of the cells, leave those cells blank.

$x$	$-4$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$
$f(x)$	$-2$	$-1$	$0$	$1$	$2$	$3$	$4$	$5$	$6$
$\frac{1}{2}x$
$-2f(x)$
$f(x)+5$
$f(x+2)$
$f\mathopen{}\left( \frac{1}{2}x \right)\mathclose{}$
$f(2x)$
$f(x-3)$	$\phantom{-0}$	$\phantom{-0}$	$\phantom{-0}$	$\phantom{-0}$	$\phantom{-0}$	$\phantom{-0}$	$\phantom{-0}$	$\phantom{-0}$	$\phantom{-0}$

Find the Transformations.

In Exercises 6–9, first write $g(x)$ in terms of $f\text{.}$ Then compose a sequence of transformations that will transform the graph of $y=f(x)$ into the graph of $y=g(x)\text{.}$

6.

$f(x)=\sqrt{x}\qquad g(x)=\frac{\sqrt{x-7}}{4}$

7.

$f(x)=\frac{1}{x}\qquad g(x)=\frac{2}{x}+3$

8.

$f(x)=x^2\qquad g(x)=-4\mathopen{}\left( \frac{1}{2}x-5 \right)^2\mathclose{}+3$

9.

$f(x)=\sqrt[3]{x}\qquad g(x)=\frac{1}{2}\sqrt[3]{10x+30}-6$

Sketch Transformations.

In Exercises 10–13, use the provided graph of $y=f(x)$ to sketch a graph of each given function.

$The graph of f of x which is shown has the points (-4,4), (-2, 0), (0, 4), (2, 4), (4, -4) all connected.$

10.

$k_1(x)=f(2x)$

$A blank graph.$

11.

$k_2(x)=2f(-2x)-1$

$A blank graph.$

12.

$k_3(x)=-2f(2x+4)$

$A blank graph.$

13.

$k_4(x)=f\mathopen{}\left(\frac{1}{2}x\right)\mathclose{} +2$

$A blank graph.$

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\(x\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)
\(f(x)\)	\(8\)	\(6\)	\(4\)	\(2\)	\(0\)	\(-1\)	\(-2\)
\(g(x)\)	\(-8\)	\(-6\)	\(-4\)	\(-2\)	\(0\)	\(1\)	\(2\)
\(h(x)\)	\(5\)	\(3\)	\(1\)	\(1\)	\(-3\)	\(-4\)	\(-5\)

\(x\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(f(x)\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(g(x)\)	\(4\)	\(3\)	\(2\)	\(1\)	\(0\)
\(h(x)\)	\(0\)	\(-1\)	\(-2\)	\(-3\)	\(-4\)
\(k(x)\)	\(6\)	\(7\)	\(8\)	\(9\)	\(10\)
\(l(x)\)	\(0\)	\(3\)	\(6\)	\(9\)	\(12\)

\(x\)	\(-4\)	\(-3\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)
\(f(x)\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)
\(\frac{1}{2}x\)
\(-2f(x)\)
\(f(x)+5\)
\(f(x+2)\)
\(f\mathopen{}\left( \frac{1}{2}x \right)\mathclose{}\)
\(f(2x)\)
\(f(x-3)\)	\(\phantom{-0}\)	\(\phantom{-0}\)	\(\phantom{-0}\)	\(\phantom{-0}\)	\(\phantom{-0}\)	\(\phantom{-0}\)	\(\phantom{-0}\)	\(\phantom{-0}\)	\(\phantom{-0}\)