Find and graph the equation for a function,
$\text{\hspace{0.17em}}g(x),$ that reflects
$\text{\hspace{0.17em}}f(x)={1.25}^{x}\text{\hspace{0.17em}}$ about the
y -axis. State its 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
$\text{\hspace{0.17em}}y=0.$
Summarizing translations of the exponential function
Now that we have worked with each type of translation for the exponential function, we can summarize them in
[link] to arrive at the general equation for translating exponential functions.
Translations of the Parent Function
$\text{\hspace{0.17em}}f(x)={b}^{x}$
Translation
Form
Shift
Horizontally
$\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units to the left
Vertically
$\text{\hspace{0.17em}}d\text{\hspace{0.17em}}$ units up
$$f(x)={b}^{x+c}+d$$
Stretch and Compress
Stretch if
$\text{\hspace{0.17em}}\left|a\right|>1$
Compression if
$\text{\hspace{0.17em}}0<\left|a\right|<1$
$$f(x)=a{b}^{x}$$
Reflect about the
x -axis
$$f(x)=-{b}^{x}$$
Reflect about the
y -axis
$$f(x)={b}^{-x}={\left(\frac{1}{b}\right)}^{x}$$
General equation for all translations
$$f(x)=a{b}^{x+c}+d$$
Translations of exponential functions
A translation of an exponential function has the form
$f(x)=a{b}^{x+c}+d$
Where the parent function,
$\text{\hspace{0.17em}}y={b}^{x},$$\text{\hspace{0.17em}}b>1,$ is
shifted horizontally
$\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ units to the left.
stretched vertically by a factor of
$\text{\hspace{0.17em}}\left|a\right|\text{\hspace{0.17em}}$ if
$\text{\hspace{0.17em}}\left|a\right|>0.$
compressed vertically by a factor of
$\text{\hspace{0.17em}}\left|a\right|\text{\hspace{0.17em}}$ if
$\text{\hspace{0.17em}}0<\left|a\right|<1.$
reflected about the
x- axis when
$\text{\hspace{0.17em}}a<0.$
Note the order of the shifts, transformations, and reflections follow the order of operations.
Writing a function from a description
Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.
$f(x)={e}^{x}\text{\hspace{0.17em}}$ is vertically stretched by a factor of
$\text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ , reflected across the
y -axis, and then shifted up
$\text{\hspace{0.17em}}4\text{\hspace{0.17em}}$ units.
We want to find an equation of the general form
$\text{\hspace{0.17em}}f(x)=a{b}^{x+c}+d.\text{\hspace{0.17em}}$ We use the description provided to find
$\text{\hspace{0.17em}}a,$$b,$$c,$ and
$\text{\hspace{0.17em}}d.$
We are given the parent function
$\text{\hspace{0.17em}}f(x)={e}^{x},$ so
$\text{\hspace{0.17em}}b=e.$
The function is stretched by a factor of
$\text{\hspace{0.17em}}2$ , so
$\text{\hspace{0.17em}}a=2.$
The function is reflected about the
y -axis. We replace
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ with
$\text{\hspace{0.17em}}-x\text{\hspace{0.17em}}$ to get:
$\text{\hspace{0.17em}}{e}^{-x}.$
The graph is shifted vertically 4 units, so
$\text{\hspace{0.17em}}d=4.$
The domain is
$\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is
$\text{\hspace{0.17em}}\left(4,\infty \right);\text{\hspace{0.17em}}$ the horizontal asymptote is
$\text{\hspace{0.17em}}y=4.$
Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.
$f(x)={e}^{x}\text{\hspace{0.17em}}$ is compressed vertically by a factor of
$\text{\hspace{0.17em}}\frac{1}{3},$ reflected across the
x -axis and then shifted down
$\text{\hspace{0.17em}}2$ units.
$f(x)=-\frac{1}{3}{e}^{x}-2;\text{\hspace{0.17em}}$ the domain is
$\text{\hspace{0.17em}}\left(-\infty ,\infty \right);\text{\hspace{0.17em}}$ the range is
$\text{\hspace{0.17em}}\left(-\infty ,2\right);\text{\hspace{0.17em}}$ the horizontal asymptote is
$\text{\hspace{0.17em}}y=2.$
General Form for the Translation of the Parent Function
$\text{}f(x)={b}^{x}$
$f(x)=a{b}^{x+c}+d$
Key concepts
The graph of the function
$\text{\hspace{0.17em}}f(x)={b}^{x}\text{\hspace{0.17em}}$ has a
y- intercept at
$\text{\hspace{0.17em}}\left(0,1\right),$ domain
$\text{\hspace{0.17em}}\left(-\infty ,\infty \right),$ range
$\text{\hspace{0.17em}}\left(0,\infty \right),$ and horizontal asymptote
$\text{\hspace{0.17em}}y=0.\text{\hspace{0.17em}}$ See
[link] .
If
$\text{\hspace{0.17em}}b>1,$ the function is increasing. The left tail of the graph will approach the asymptote
$\text{\hspace{0.17em}}y=0,$ and the right tail will increase without bound.
If
$\text{\hspace{0.17em}}0<b<1,$ the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote
$\text{\hspace{0.17em}}y=0.$
The equation
$\text{\hspace{0.17em}}f(x)={b}^{x}+d\text{\hspace{0.17em}}$ represents a vertical shift of the parent function
$\text{\hspace{0.17em}}f(x)={b}^{x}.$
The equation
$\text{\hspace{0.17em}}f(x)={b}^{x+c}\text{\hspace{0.17em}}$ represents a horizontal shift of the parent function
$\text{\hspace{0.17em}}f(x)={b}^{x}.\text{\hspace{0.17em}}$ See
[link] .
Approximate solutions of the equation
$\text{\hspace{0.17em}}f(x)={b}^{x+c}+d\text{\hspace{0.17em}}$ can be found using a graphing calculator. See
[link] .
The equation
$\text{\hspace{0.17em}}f(x)=a{b}^{x},$ where
$\text{\hspace{0.17em}}a>0,$ represents a vertical stretch if
$\text{\hspace{0.17em}}\left|a\right|>1\text{\hspace{0.17em}}$ or compression if
$\text{\hspace{0.17em}}0<\left|a\right|<1\text{\hspace{0.17em}}$ of the parent function
$\text{\hspace{0.17em}}f(x)={b}^{x}.\text{\hspace{0.17em}}$ See
[link] .
When the parent function
$\text{\hspace{0.17em}}f(x)={b}^{x}\text{\hspace{0.17em}}$ is multiplied by
$\text{\hspace{0.17em}}-1,$ the result,
$\text{\hspace{0.17em}}f(x)=-{b}^{x},$ is a reflection about the
x -axis. When the input is multiplied by
$\text{\hspace{0.17em}}-1,$ the result,
$\text{\hspace{0.17em}}f(x)={b}^{-x},$ is a reflection about the
y -axis. See
[link] .
All translations of the exponential function can be summarized by the general equation
$\text{\hspace{0.17em}}f(x)=a{b}^{x+c}+d.\text{\hspace{0.17em}}$ See
[link] .
Using the general equation
$\text{\hspace{0.17em}}f(x)=a{b}^{x+c}+d,$ we can write the equation of a function given its description. See
[link] .
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0
then
4x = 2-3
4x = -1
x = -(1÷4) is the answer.
Jacob
4x-2+3
4x=-3+2
4×=-1
4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3
4x=-3+2
4x=-1
4x÷4=-1÷4
x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?