Exponential Functions

In the past few weeks….

The Rule: Exponential Functions The rule for an exponential function looks like f (x) = a (base) x where a ≠ 0 base > 0 but ≠ 1

How to determine the exponential function given the graph of the function There are 3 coordinates that we can read from the graph clearly: (0,3), (1,6) and (2, 12)

How to determine the exponential function given the graph of the function - Con’t Steps: Identify the initial value (y-intercept of the graph) Replace ‘a’ with the initial value in the equation f(x)=a (base) x Solve for the value of the base by using one coordinate from the curve Write out the equation Validate your solution with one of the coordinates from the curve There are 3 coordinates : (0,3), (1,6) and (2, 12) YOUR TURN : Using these steps – determine the equation of this function!

Solution: y-intercept = 3 This is the value of “a” f(x)=3(base)x We’ll use the coordinate (2, 12) to solve for the base: 12 = 3(base)2 base = 2 The equation is: f(x) = 3(2)x Validate using the coordinate (1, 6) 6 = 3(2)1 6 = 6

How to determine the exponential function with a word problem Mary has a secret! Mary tells John her secret and says “don’t tell anyone”…but John tells his friend a minute later. And so….. the total number of people who know the secret doubles every minute. What is the equation of this function? How can we solve this problem?

Let’s make a T.O.V. to help us: Time (min) Calculation # people who know secret 1 2 3 4 5 6 7 8 9

Let’s make a T.O.V. to help us: Time (min) Calculation # people who know secret 1 =1x20 1 1x2 =1x21 2 1x2x2 =1x22 4 3 1x2x2x2 =1x23 8 1x2x2x2x2 =1x24 16 5 1x2x2x2x2x2 =1x25 32 6 1x2x2x2x2x2x2=1x26 64 7 1x2x2x2x2x2x2x2=1x27 128 1x2x2x2x2x2x2x2x2=1x28 256 9 1x2x2x2x2x2x2x2x2x2=1x29 512

So what is the equation of the function? f(x) =1(2)x Since the value of ‘a’ is 1 f(x) = 2x What does the graph of this function look like? Let’s look at the Gizmo: Exponential Functions (Act. A) www.explorelearning.com

Handout: In groups we will try to answer the following questions: 1) How does the value of “a” affect the graph of an exponential function? 2) How does the value of the “base” affect the graph of an exponential function?

How does parameter “a” affect the exponential function? f(x) = 0.5(2) x g(x) = 5(2) x h(x) = 20(2) x i(x) = 1.5(2) x j(x) = -1.5(2) x

How does the “base” value affect the exponential function? k(x) = 2 x m(x) = ( ) x

Example 1- Solution The following T.O.V. are functions: C and E * They are functions because when you look at the pattern of the dependant values (y) they increase by the same product not by a sum or difference.

Example 2 - Solution f(x) = 5400 (1.036) x = 5400 (1.036) 10 = $7691.15