Exponential and Logarithmic Functions

What do you think of when you hear the word “exponential?”

An exponential function is a function with the general form: y=a(b)x What do a and b mean? Be sure to emphasize that students must write down the general form of an exponential equation.

Graph the following in your calculator and draw a quick sketch of what you see. What’s happening? 1. y = 1(2)x 2. y = 1(5)x 3. y = 1(15)x

Graph the following in your calculator and draw a quick sketch of what you see. What’s happening? 1. y = 1(.8)x 2. y = 1(.5)x 3. y = 1(.1)x

Can you write a sentence about y = a(b)x that explains what you just discovered?

Exponential Growth and Decay y = abx Is b > 1? Then it’s a growth factor! Is 0 < b < 1? Then it’s a decay factor! Make sure to give students ample time to write down the fact that b must be greater than 1 in order for the function to represent exponential growth.

Graph the following in your calculator and draw a quick sketch of what you see. What’s happening? 1. y = 1(3)x 2. y = 5(3)x 3. y = 10(3)x

What did you notice? Write a sentence!

What do you suspect will happen here? 1. y = -1(3)x 2. y = -5(3)x 3. y = -10(3)x

Exponential Growth and Decay y = abx a tells us the y-intercept If it’s negative, the graph becomes negative as well. Make sure to give students ample time to write down the fact that b must be greater than 1 in order for the function to represent exponential growth.

Who wants to act out an exponential function? The distance a frog can jump (y) each year (x) is modeled by the following equation: x y 1 2 3 4 5 6 7 Starting with x=0, jump the distance y in feet. Call a student up to the board and tell them that they will jump the distance y in feet for each year of the growth function. Have a volunteer calculate y for each round starting with x = 0, and record it on the board in a table. After 7 or 8 years (rounds), students cannot jump the distance. Talk about the activity with the students. Talk about how it didn’t take long before students were unable to make the jump. Why do you think it didn’t take long for the student to stop being able to make the jumps? What do you think a graph of these jumps would look like?

Let’s discuss! What are some real world examples of exponential growth and decay?

Exponential Translations y = a∙bx-h+k -h +k h tells us how the function moves left or right (don’t forget that h is being subtracted!) k tells us how the function moves up or down

Example 1 Describe the translation (initial value, growth or decay factor, and vertical and horizontal translations): y = 2∙4x-6-5

Example 2 Describe the translation (initial value, growth or decay factor, and vertical and horizontal translations): y = -3∙2x+2+4

Example 3 How would the function y = 3∙4x be written if it was translated down 2 and right 7?

Example 4 How would the function y = 5∙2x be written if it was translated left 3 and up 9?

Exponential Regression What if we are given a graph or data that looks like it has an exponential trend? How do we find an equation for it?

Exponential Regression Steps Enter the data into L1 and L2. (STAT → Edit…) Use the ExpReg tool. (STAT → CALC) Substitute the value for a and b into the equation y = a(b)x.

Time (mins)    Temp ( ° F)    179.5 5 168.7 8 158.1 11 149.2 15 141.7 18 134.6 22 125.4 25 123.5 30 116.3 34 113.2 38 109.1 42 105.7 45 102.2 50 100.5 The data at the right shows the cooling temperatures of a freshly brewed cup of coffee after it is poured from the pot.  The pot’s temperature is approximately 180° F. Find an exponential regression model for the data.