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] .
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection