Exponential Growth and Decay

​Today is your lucky day!!!

​I have found a bank that promises to double the money you put in every month. You start with $1. Exciting, right?

Fill in the Blanks

Predict how much money you will have at the end of the year. You have 30 seconds.

Draw

Use your trackpad to fill in the table below.

Open Ended

How well did your prediction match the actual amount?

Exponential Functions

An exponential function is a function of the form y = ab^x​,

where a ≠ 0, b > 0, and b ≠ 1. The base is a constant and the exponent is a variable. Exponential functions are non-linear.

Some text here about the topic of discussion

• a is the initial or starting amount​ (y-intercept)

• b is the base and must be larger than 0 to represent growth.

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​

Domain: (-∞, ∞) all real numbers (never changes)

​Range: (0, ∞) or y > 0 (may change with transformations)

​Horizontal asymptote: y = 0 (changes vertical shifts in the graph)

Y-intercept: (0, 1) or (0, a)​

​y = ab^x, b > 1 and a ≠ 0

Exponential Growth

• A is the amount at a specified time.

• P is the investment or starting amount

• r is the interest rate written as a decimal.

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​

• t is the time

• 1 + r is the base and should be greater than 0 to represent growth

​A = P(1 + r)^t

Exponential Growth

Multiple Select

Select all that represent exponential growth.

• $$y\ =\ 8^x$$  

• $$y\ =\ 400\left(.25\right)^x$$  

• $$y\ =\ \left(4\right)^x$$  

• $$y\ =\ 200\left(1+.05\right)^x$$  

Increases from left to right.

Y-intercept: (0, 1)

Horizontal Asymptote: y = 0​

Exponential Growth

• a is the initial or starting amount​ (y-intercept)

• b is the base and must be a decimal between 0 and 1 or a proper fractions.

​

​

Domain: (-∞, ∞) all real numbers (never changes)

​Range: (0, ∞) or y > 0 (may change with transformations)

​Horizontal asymptote: y = 0 (changes vertical shifts in the graph)

Y-intercept: (0, 1) or (0, a)​

​y = ab^x, 0 < b < 1 and a ≠ 0

Exponential Decay

• A is the amount at a specified time.

• P is the investment or starting amount

• r is the interest rate written as a decimal.

​

​

• t is the time

• 1 - r is the base and should be a decimal between 0 and 1 or a proper fraction to represent decay

​A = P(1 - r)^t

Exponential Decay

Decreases from left to right.

Y-intercept: (0, 1)

Horizontal Asymptote: y = 0​

Exponential Decay

Multiple Choice

Does the following equation represent growth or decay?

y = 3(5)^x

• Growth

• Decay

Multiple Choice

Does the following equation represent growth or decay?

$$A\ =\ 400\left(1+.08\right)^5$$  

• Growth

• Decay

Multiple Choice

Does the following equation represent growth or decay?

$$A\ =\ 200\left(1-.25\right)^5$$  

• Growth

• Decay

Multiple Choice

Does the following equation represent growth or decay?

$$A\ =\ 2\left(3\right)^t$$  

• Growth

• Decay

Multiple Choice

Does the following equation represent growth or decay?

$$A\ =\ 2\left(\frac{1}{2}\right)^x$$  

• Growth

• Decay

Multiple Choice

Does the following equation represent growth or decay?

$$A\ =\ 9\left(\frac{5}{2}\right)^x$$  

• Growth

• Decay

Multiple Choice

Does the following equation represent growth or decay?

$$y\ =\ 6\left(0.01\right)^x$$  

• Growth

• Decay

Multiple Choice

Does the following graph represent growth or decay?

 

• Growth

• Decay

Multiple Choice

Does the following graph represent growth or decay?

 

• Growth

• Decay

Multiple Choice

Does the following graph represent growth or decay?

 

• Growth

• Decay

Multiple Choice

What is the domain of all exponential graphs?

• (-∞, ∞)

• (0, ∞)

• 0 < b < 1

• y < 0

Multiple Choice

What is the range of the parent exponential function?

• (-∞, ∞)

• (0, ∞)

• 0 < b < 1

• y < 0

Multiple Choice

What is the y-intercept of the following?

$$y\ =\ 2^x$$  

• (0, 1)

• (1 , 0)

• (0, 2)

• (2, 0)

Multiple Choice

What is the y-intercept of the following?

$$y\ =\ 2\left(2\right)^x$$  

• (0, 1)

• (1 , 0)

• (0, 2)

• (2, 0)