Exponential Functions

Exponential Functions

The number of cases of the corona virus represents an exponential function

What is an exponential function?

• A function in which the variable is an exponent

• Non-linear

• Rate of change is not constant over every interval

• Formula is f(x) =  $a\left(b\right)^x$ , where a  $\ne$  0, and b >0, and b  $\ne$   1

How do you make a table of values for an exponential function?

• The x-values in your table will be the numbers that you substitute for the exponent "x"

• The y-values are the values of the base raised to the power of "x" times the coefficient. Sometimes the "a" value, or coefficient is of the base "b" is 1, like in the example

Key Features of Exponential Functions

• x and y intercepts

• increasing or decreasing

• domain and range

• rate of change over an interval

• asymptotes

Key Features of an Exponential Functions

•  the domain is all Real numbers.

•  the range is all positive real numbers (not zero).

• graph has a y-intercept at (0,1). Remember any number to the zero power is 1.

• when b > 1, the graph increases. The greater the base, b, the faster the graph rises from left to right.

• 
  

• when 0 < b < 1, the graph decreases.

• has an asymptote (a line that the graph gets very, very close to, but never crosses or touches). For this graph the asymptote is the x-axis (y = 0).

• graph passes the vertical line test for functions

• graph passes the horizontal line test for functional inverse.

• graph is asymptotic to the x-axis - gets very, very close

• https://mathbitsnotebook.com/Algebra2/Exponential/EXExpFunctions.html

Y Intercepts

• y-intercept is when x = 0

• When the equation is y = b^x, then the y intercept is 1 (when x=0, any number to the 0 power is 1)

• When the equation is y = a(b)^x, then the y-intercept is a(b)⁰ = a(1) or a

• 
  

• 
  

Y-Intercept

• When the equation is y = a(b)^x+h +/- k, then the y-intercept is when x = 0, or "a" (b)^h +/- k

• In the example, f(x) = 2^x+1 - 3. When x = 0, it is 2⁰⁺¹ - 3, or 2-3 = -1.

• The y-intercept is (0, -1)

• 
  

• 
  

X-intercepts/Asymptotes

• The x-intercept occurs when y = 0

• When the equation is f(x) =  $ab^x$  , then there is no x intercept, since a $\ne$   0, and there is no value of x that will make   $b^x$  = 0

• Instead, the graph will approach a horizontal line called an asymptote

X-intercept

• If the graph looks like  $f\left(x\right)\ =\ a\left(b\right)^x$  -k, then the x-intercept can be found by setting f(x) = 0

•  $f\left(x\right)\ =\ 4\left(5\right)^x\ -\ 100$  

• 0 = 4 ( $5^x$ )- 100 

• 100 = 4 ( $5^x$ )    (add 100 to both sides) 

• 25 =  $5^x$  

• 5^2 = 5^x,   2 = x = x-intercept

X-intercept

• If the equation looks like f(x) = a (b)^x + k, we try to solve for the x-intercept by setting y = 0

• f(x) = 4(5)^x + 100

• 0 = 4(5)^x + 100

• -100 = 4 (5)^x (subtract 100)

• -25 = 5^x (divide by 4)

• There is no x-intercept for this function since there is no value of x that will give me a negative number

• This equation has an asymptote at y = 100

Asymptotes

• If the equation looks like  $f\left(x\right)\ =\ ab^x$ + k, where k=0, then the asymptote is y = 0.

• If the equation looks like  $f\left(x\right)\ =\ ab^x+k$ , where k    $\ne$  0, then the asymptote is y = k.

Rate of Change

• Linear functions have a constant rate of change

• No matter which pair of points you choose, the rate of change (slope) is the same

• Exponential functions do not have a constant rate of change, so rate of change is defined for a special interval (values for x)

Rate of Change

• Rate of change would be defined between two x-values.

• To find the rate of change, determine the y-values or output for the given x-values

• Use the slope formula to find the rate of change between those two points

End Value When (b) > 1 (exponential growth )

•  $When\ x\ \rightarrow+\infty,\ y\rightarrow+\infty$  

•  $When\ x\rightarrow-\infty,\$  y--> horizontal asymptote

• As x gets bigger, y gets bigger

• As x gets smaller,

• y --> horizontal asymptote

End Value When 0< b <1

• When x --> +  $+\infty$    , y ---> horizontal asymptote

• When x -->   $-\infty$  , y --> +  $+\infty$  

• As x gets bigger, y approaches some value but never gets there

• As x gets smaller, y approaches + infinity

Multiple Choice

What is the range of this graph?

Hint: The range is the set of y-values.

• y = 6

• (0, -4)

• y > -6

• y < -6

Multiple Choice

Is the table linear, exponential, or neither?

Hint: Do the consecutive y values add the same number, or multiply by the same number?

• Linear

• Exponential

• Nether

Multiple Choice

Is the table linear, exponential, or neither?

Do the consecutive y values add the same number, or multiply by the same number?

• Linear

• Exponential

• Neither

Multiple Choice

What type of pattern do linear functions have?

• Adds by the same number every time

• Multiplies by the same number every time

• Square roots all the inputs

• Squares all the inputs

Multiple Choice

What type of pattern do exponential functions have?

• Adds by the same number every time

• Multiplies by the same number every time

• Square roots all the inputs

• Squares all the inputs

Multiple Choice

What does the y-intercept of the graph represent?

• The amount of grams the material is gaining.
• The amount of grams the material is losing.
• The amount of grams remaining after so many years.
• The initial amount of grams.

Multiple Choice

What does the y-intercept of the graph represent?

• The amount of grams the material is gaining.
• The amount of grams the material is losing.
• The amount of grams remaining after so many years.
• The initial amount of grams.

Multiple Choice

What is the y-intercept of the function f(x)=2(4)^x ?

Hint: Set x = 0. What is y?

• 2

• 4

• (4)^x

• 8

Multiple Choice

What is the y-intercept of the function?

• 1

• 2

• 4

• 3

Multiple Choice

What is a, the starting term, for the function: f(x) = 300(1.16)^x?

Hint: set x = 0. What is y?

• 300

• 1.16

• .16

• x

Multiple Choice

In an exponential function, what does the 'a' represent?

• SLOPE

• RATE OF CHANGE

• Y-INTERCEPT

• COMMON RATIO