Exponential Functions

​Key Features of Exponential Functions

Today, we will learn what an exponential function is and what its key features are.​

If b is...

• Greater than 1 (b>1), the function is exponential GROWTH

• Less than 1 and greater than 0 (0<b<1) the function is exponential DECAY

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​Definition of exponential function

​Growth vs Decay

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Multiple Choice

Which of the following is an exponential function?

• y = x²

• f(x) = 3x³

• y = x³

• f(x) = 3^x

Multiple Choice

Is this exponential growth or decay?

• Growth
• Decay

Multiple Choice

Is the following function an example of decay or growth?

f(x) = 2(1.25)^x

• Exponential Growth

• Exponential Decay

Multiple Choice

Is the pictured graph growth, decay, or linear or none?  

• Exponential Growth
• Exponential Decay

Multiple Choice

Which represents exponential decay?

• A

• B

• C

• D

Domain: all possible x values

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​D= all real numbers

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Range: all possible y values

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R= y>0​

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Asymptote: Imaginary line that the graph approaches, but never crosses​

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y=0​

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​PARENT FUNCTION ONLY

​Key Features

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Y- Intercept: one y-intercept

​The y-intercept is "a"

y-intercept=1​

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X- Intercept​​: NO x-intercepts

​PARENT FUNCTION ONLY

​Key Features

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Multiple Choice

What is the domain of the function shown?

• All real numbers

• $$y>0$$  

• All real numbers greater than 1

• $$y<0$$  

Multiple Choice

What is the range of the exponential function?

• all real numbers

• y > -1

• y < -1

• x > -1

Multiple Choice

What is the y-intercept of the function?

• 2
• 3
• 1
• -2

Multiple Choice

What is the x-intercept of the exponential function? 

• (0,1)
• (0,0)
• There is not one

Multiple Choice

In an exponential function, what does the 'a' represent? 

• SLOPE
• RATE OF CHANGE
• Y-INTERCEPT
• COMMON RATIO

Multiple Choice

What is the equation of the horizontal asymptote for following exponential function?

• y = -4

• y > -4

• y < -4

• (0, -3)

Multiple Choice

The line shown on the graph is an asymptote for the exponential curve shown on the graph.

• TRUE

• FALSE

Multiple Choice

What is the equation of the horizontal asymptote for following exponential function?

• y = 1

• y = 2

• y > 1

• (0, 2)

Multiple Choice

What is the range of the exponential function?

• all real numbers

• y > 3

• y < 3

• x > 3

Multiple Choice

What are the domain and range of the function/graph?

• Domain: All Real Numbers

  Range: All Real Numbers

• Domain: All Real Numbers

  Range: y < 2

• Domain: All Real Numbers

  Range: y > 2

• Domain: x > 2

  Range: All Real Numbers

If the ( ) are bigger than 1 it is growth. If the ( ) are smaller than 1 it is decay.

Multiple Choice

Is the pictured graph growth, decay, or linear or none?  

• Growth
• Decay
• Linear
• None

Multiple Choice

Is the pictured graph growth, decay, or linear or none?  

• Growth
• Decay
• Linear
• None

Multiple Choice

Is the following function and example of decay or growth?

f(x)=2(0.85)^x

• Exponential Decay

• Exponential Growth

Multiple Choice

The expression 12(1.015)^t models the population of elephants in a wildlife refuge after t years since 1975.
Is the population of elephants increasing or decreasing?

• Increasing
• Decreasing

Multiple Choice

Is the following function an example of decay or growth?

f(x) = 2(1.25)^x

• Exponential Growth

• Exponential Decay

Finding the rate of growth or decay from an equation

• Is your 'b' bigger than 1? How much bigger? Times that # by 100 to get your %

• Is your 'b' smaller than 1? How much smaller? (1-b)? Times by 100 to find your %

Multiple Choice

Which of the following functions shows an initial amount of $15 and an increase of 35% each year?

• y = 15(35)^x
• y = 15(1.35)^x
• y = 15(0.35)^x
• y = 35(1.15)^x

Multiple Choice

The amount of ants in a colony, f, that is decaying can be modeled by                      f(x) = 800(.87)^x, where x is the number of days since the decay started. By what percentage is the colony decaying?

• 87%
• 13%
• 800%
• 8.7%

Writing an Exponential Function

• Your initial amount is always the starting #

• ( 1 plus or minus....

• divide the % by 100 to find the rate )

• exponent is always time

Multiple Choice

Write an equation that models the following situation:
Samantha's hair was known to grow very rapidly. It began at a length of 6 in and grew at a rate of 14% a week.

• y=6(0.14)^x
• y=6(1+14)^x
• y=6(1.14)^x
• y=6(0.86)^x

Multiple Choice

A new savings account is opened with $400 and gains 3% every year. What is the exponential function?

• f(x)=400(1+3)^x
• f(x)=400(1-0.03)^x
• f(x)=400(1+0.03)^x
• f(x)=400(0.03)^x