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?

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)