Other Features of Exponential Functions

Other Features of Exponential Functions

Asymptote

In mathematics, an asymptote is a horizontal, vertical, or slanted line that a graph approaches but never touches.

The asymptote of an exponential function is a horizontal line.

The equation of a horizontal asymptote can be written as

y = constant.

Exponential Transformation Equation

• a: vertial stretch or compression

• h: horizontal shift

• k: vertical shift

• The asymptote is determined by the "k." (If there is no k, the asymptote is 0)

Aysmptote of the Exponential parent function

The asymptote of the exponential parent function is the horizontal line y = 0.

Examples of exponential functions and their asymptotes

Multiple Choice

• y = 0

• y = 5

• y = 1

• All real numbers

Multiple Choice

• y = 2

• y = 0

• y = 6

• all real numbers

Fill in the Blanks

Domain and Range of a function

• Domain: the set of all possible inputs (x's)

• The domain of all exponential functions is "All real numbers."

• Range: the set of all possible outputs (y's)

• The range of an exponential function is limited by the function equation.

The asymptote of an exponential function is used to determine the range of that function.

The range of an exponential function can be written as y > asymptote.

Multiple Choice

What is the range of the function?

• All real numbers

• y > 3

• y < 4

• Impossible to determie from the graph.

Y-intercept of an exponential function

The y-intercept of ANY function is the value of that function

when x = 0 or where the graph of that function intersects the y-axis.

Because the value of any base with an exponent of 0 is 1, the y-intercept of an exponential that has not been horizontally shifted is "a+k"

**If there is a horizontal shift, this shortcut cannot be used.**

Multiple Choice

What is the y-intercept?

 $f\left(x\right)\ =\ 6^x$  

• (0,6)

• (0,0)

• (0,1)

Multiple Choice

What is the y-intercept?

 $y\ =\ 2\cdot4^x\ +3$  

• (0,3)

• (0,4)

• (0,5)

• (0,6)

• (0,8)

Multiple Select

Which functions have a y-intercept of 1?

• $y\ =\ \left(\frac{2}{3}\right)^x$

• $y\ =\ \left(\frac{1}{3}\right)^x+1$

• $f\left(x\right)\ =\ 2\cdot3^x\ -1$

• $f\left(x\right)\ =\ 5^x$

• $y\ =\ 4^x\ -3$

Poll

I am confident that I can correctly identify the features of exponential functions.

• strongly agree

• somewhat agree

• disagree

Open Ended

Identify whether the following function is exponential growth or decay.  Then, identify the asymptote, domain, range, y-intercept, and interval of increase/decrease. (Use commas between characteristics.  Using the enter button will submit your response)

 $f\left(x\right)\ =2\left(\frac{1}{5}\right)^x\ +1$