Stretches and compressions of the parent function
f (
x ) =
b^{
x }
For any factor
$\text{\hspace{0.17em}}a>0,$ the function
$\text{\hspace{0.17em}}f(x)=a{\left(b\right)}^{x}$
is stretched vertically by a factor of
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ if
$\text{\hspace{0.17em}}\left|a\right|>1.$
is compressed vertically by a factor of
$\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ if
$\text{\hspace{0.17em}}\left|a\right|<1.$
has a
y -intercept of
$\text{\hspace{0.17em}}\left(0,a\right).$
has a horizontal asymptote at
$\text{\hspace{0.17em}}y=0,$ a range of
$\text{\hspace{0.17em}}\left(0,\infty \right),$ and a domain of
$\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ which are unchanged from the parent function.
Graphing the stretch of an exponential function
Sketch a graph of
$\text{\hspace{0.17em}}f(x)=4{\left(\frac{1}{2}\right)}^{x}.\text{\hspace{0.17em}}$ State the domain, range, and asymptote.
Before graphing, identify the behavior and key points on the graph.
Since
$\text{\hspace{0.17em}}b=\frac{1}{2}\text{\hspace{0.17em}}$ is between zero and one, the left tail of the graph will increase without bound as
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ decreases, and the right tail will approach the
x -axis as
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases.
Since
$\text{\hspace{0.17em}}a=4,$ the graph of
$\text{\hspace{0.17em}}f(x)={\left(\frac{1}{2}\right)}^{x}\text{\hspace{0.17em}}$ will be stretched by a factor of
$\text{\hspace{0.17em}}4.$
Plot the
y- intercept,
$\text{\hspace{0.17em}}\left(0,4\right),$ along with two other points. We can use
$\text{\hspace{0.17em}}\left(-1,8\right)\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\left(1,2\right).$
Draw a smooth curve connecting the points, as shown in
[link] .
The domain is
$\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is
$\text{\hspace{0.17em}}\left(0,\infty \right);\text{\hspace{0.17em}}$ the horizontal asymptote is
$\text{\hspace{0.17em}}y=0.$
Sketch the graph of
$\text{\hspace{0.17em}}f(x)=\frac{1}{2}{\left(4\right)}^{x}.\text{\hspace{0.17em}}$ State the domain, range, and asymptote.
The domain is
$\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is
$\text{\hspace{0.17em}}\left(0,\infty \right);\text{\hspace{0.17em}}$ the horizontal asymptote is
$\text{\hspace{0.17em}}y=0.\text{\hspace{0.17em}}$
In addition to shifting, compressing, and stretching a graph, we can also reflect it about the
x -axis or the
y -axis. When we multiply the parent function
$\text{\hspace{0.17em}}f(x)={b}^{x}\text{\hspace{0.17em}}$ by
$\text{\hspace{0.17em}}\mathrm{-1},$ we get a reflection about the
x -axis. When we multiply the input by
$\text{\hspace{0.17em}}\mathrm{-1},$ we get a
reflection about the
y -axis. For example, if we begin by graphing the parent function
$\text{\hspace{0.17em}}f(x)={2}^{x},$ we can then graph the two reflections alongside it. The reflection about the
x -axis,
$\text{\hspace{0.17em}}g(x)={\mathrm{-2}}^{x},$ is shown on the left side of
[link] , and the reflection about the
y -axis
$\text{\hspace{0.17em}}h(x)={2}^{-x},$ is shown on the right side of
[link] .
Reflections of the parent function
f (
x ) =
b^{
x }
The function
$\text{\hspace{0.17em}}f(x)=-{b}^{x}$
reflects the parent function
$\text{\hspace{0.17em}}f(x)={b}^{x}\text{\hspace{0.17em}}$ about the
x -axis.
has a
y -intercept of
$\text{\hspace{0.17em}}\left(0,-1\right).$
has a range of
$\text{\hspace{0.17em}}\left(-\infty ,0\right)$
has a horizontal asymptote at
$\text{\hspace{0.17em}}y=0\text{\hspace{0.17em}}$ and domain of
$\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ which are unchanged from the parent function.
The function
$\text{\hspace{0.17em}}f(x)={b}^{-x}$
reflects the parent function
$\text{\hspace{0.17em}}f(x)={b}^{x}\text{\hspace{0.17em}}$ about the
y -axis.
has a
y -intercept of
$\text{\hspace{0.17em}}\left(0,1\right),$ a horizontal asymptote at
$\text{\hspace{0.17em}}y=0,$ a range of
$\text{\hspace{0.17em}}\left(0,\infty \right),$ and a domain of
$\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ which are unchanged from the parent function.
Writing and graphing the reflection of an exponential function
Find and graph the equation for a function,
$\text{\hspace{0.17em}}g(x),$ that reflects
$\text{\hspace{0.17em}}f(x)={\left(\frac{1}{4}\right)}^{x}\text{\hspace{0.17em}}$ about the
x -axis. State its domain, range, and asymptote.
Since we want to reflect the parent function
$\text{\hspace{0.17em}}f(x)={\left(\frac{1}{4}\right)}^{x}\text{\hspace{0.17em}}$ about the
x- axis, we multiply
$\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ by
$\text{\hspace{0.17em}}-1\text{\hspace{0.17em}}$ to get,
$\text{\hspace{0.17em}}g(x)=-{\left(\frac{1}{4}\right)}^{x}.\text{\hspace{0.17em}}$ Next we create a table of points as in
[link] .
$x$
$-3$
$-2$
$-1$
$0$
$1$
$2$
$3$
$$g(x)=-(\frac{1}{4}{)}^{x}$$
$-64$
$-16$
$-4$
$-1$
$-0.25$
$-0.0625$
$-0.0156$
Plot the
y- intercept,
$\text{\hspace{0.17em}}\left(0,\mathrm{-1}\right),$ along with two other points. We can use
$\text{\hspace{0.17em}}\left(\mathrm{-1},\mathrm{-4}\right)\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\left(1,\mathrm{-0.25}\right).$
Draw a smooth curve connecting the points:
The domain is
$\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is
$\text{\hspace{0.17em}}\left(-\infty ,0\right);\text{\hspace{0.17em}}$ the horizontal asymptote is
$\text{\hspace{0.17em}}y=0.$
Solve for the first variable in one of the equations, then substitute the result into the other equation.
Point For:
(6111,4111,−411)(6111,4111,-411)
Equation Form:
x=6111,y=4111,z=−411x=6111,y=4111,z=-411
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?