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] .
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the
fraction, the value of the fraction becomes 2/3. Find the original fraction.
2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
Q2
x+(x+2)+(x+4)=60
3x+6=60
3x+6-6=60-6
3x=54
3x/3=54/3
x=18
:. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
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