6.2 Graphs of exponential functions  (Page 5/6)

Try It

Find and graph the equation for a function, $g(x),$ that reflects $f(x)={1.25}^x$ about the y -axis. State its domain, range, and asymptote.

The domain is $\left(-\infty ,\infty \right);$ the range is $\left(0,\infty \right);$ the horizontal asymptote is $y=0.$

Got questions? Get instant answers now!

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.

Key equations

Key concepts

• The graph of the function $f(x)=b^x$ has a y- intercept at $\left(0, 1\right),$ domain $\left(-\infty , \infty \right),$ range $\left(0, \infty \right),$ and horizontal asymptote $y=0.$ See [link] .
• If $b>1,$ the function is increasing. The left tail of the graph will approach the asymptote $y=0,$ and the right tail will increase without bound.
• If $0<b<1,$ the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote $y=0.$
• The equation $f(x)=b^x+d$ represents a vertical shift of the parent function $f(x)=b^x.$
• The equation $f(x)=b^{x+c}$ represents a horizontal shift of the parent function $f(x)=b^x.$ See [link] .
• Approximate solutions of the equation $f(x)=b^{x+c}+d$ can be found using a graphing calculator. See [link] .
• The equation $f(x)=a{b}^x,$ where $a>0,$ represents a vertical stretch if $\left|a\right|>1$ or compression if $0<\left|a\right|<1$ of the parent function $f(x)=b^x.$ See [link] .
• When the parent function $f(x)=b^x$ is multiplied by $-1,$ the result, $f(x)=-b^x,$ is a reflection about the x -axis. When the input is multiplied by $-1,$ the result, $f(x)=b^{-x},$ is a reflection about the y -axis. See [link] .
• All translations of the exponential function can be summarized by the general equation $f(x)=a{b}^{x+c}+d.$ See [link] .
• Using the general equation $f(x)=a{b}^{x+c}+d,$ we can write the equation of a function given its description. See [link] .