Exponential Functions In the past few weeks Functions

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Exponential Functions

Exponential Functions

In the past few weeks…. Functions Zero-degree polynomial Function First-degree polynomial Function y =constant

In the past few weeks…. Functions Zero-degree polynomial Function First-degree polynomial Function y =constant y=ax + b Second -degree Polynomial Function y=ax 2 Inverse Function y=k x

Properties of Functions Domain & Range Variation: Increasing Decreasing Constant Extrema: Minimum & Maximum

Properties of Functions Domain & Range Variation: Increasing Decreasing Constant Extrema: Minimum & Maximum Sign: Positive & Negative Intercepts: Zeroes & Initial Value

The Rule: Exponential Functions • The rule for an exponential function looks like f

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

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

How to determine the exponential function given the graph of the function - Con’t Steps: 1) Identify the initial value -intercept of the graph) (y 2) Replace ‘a’ with the initial value in the equation f(x)=a (base) x 3) Solve for the value of the base by using one coordinate from the curve There are 3 coordinates : (0, 3), (1, 6) and (2, 12) 4) Write out the equation 5) Validate your solution with one of the coordinates from the curve YOUR TURN : Using these steps – determine the equation of this function!

Solution: 1. y-intercept = 3 2. f(x)=3(base)x 3. We’ll use the coordinate (2, 12)

Solution: 1. y-intercept = 3 2. f(x)=3(base)x 3. We’ll use the coordinate (2, 12) to solve for the base: 12 = 3(base)2 base = 2 3. The equation is: f(x) = 3(2)x 4. Validate using the coordinate (1, 6) 6 = 3(2)1 6=6 This is the value of “a”

How to determine the exponential function with a word problem 1. Mary has a

How to determine the exponential function with a word problem 1. 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 0 1

Let’s make a T. O. V. to help us: Time (min) Calculation 0 1 2 3 4 5 6 7 8 9 # people who know secret

Let’s make a T. O. V. to help us: Time (min) 0 1 2

Let’s make a T. O. V. to help us: Time (min) 0 1 2 3 4 5 6 7 8 9 Calculation 1 =1 x 20 1 x 2 =1 x 21 1 x 2 x 2 =1 x 22 1 x 2 x 2 x 2 =1 x 23 1 x 2 x 2 =1 x 24 1 x 2 x 2 x 2 =1 x 25 1 x 2 x 2 x 2=1 x 26 1 x 2 x 2 x 2 x 2=1 x 27 1 x 2 x 2 x 2 x 2=1 x 28 1 x 2 x 2 x 2=1 x 29 # people who know secret 1 2 4 8 16 32 64 128 256 512

So what is the equation of the function? f(x) =1(2)x Since the value of

So what is the equation of the function? f(x) =1(2)x Since the value of ‘a’ is 1 f(x) = 2 x 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

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)

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)

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

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)

Example 2 - Solution f(x) = 5400 (1. 036) x = 5400 (1. 036) 10 = $7691. 15