Intro to Exponential Functions

Intro to Exponential Functions

Exponential Functions

• The exponent is a variable

• They come in the form f(x) = a(b)^x

• a is the initial value (the y-intercept)

• b is the common multipier (the # being multiplied by)

• The common multiplier has the exponent becuase it's the # being multiplied by itself x number of times (that's why its the multiplier)

• If b is GREATER THAN 1, it will be exponential growth

• If b is LESS THAN 1, it will be exponential decay

Example 1: $f\left(x\right)\ =\ 3\left(\frac{1}{2}\right)^x$  

• The initial value is 3

• The common multipier is 1/2 

• Since 1/2 is LESS than 1, this will be exponential decay

• This pattern would start at 3 and multiply by 1/2 like this:

• 3 , 1.5 , 0.75 , 0.375, . . .

Example 2: $f\left(x\right)\ =\ 5\left(2\right)^x$  

• The initial value is 5

• The common multiplier is 2

• Since 1/2 is GREATER than 1, this will be exponential growth

• This pattern would start at 5 and multiply by 2 like this:

• 5, 10, 20, 40, 80, 160, . . . 

Side Notes:

• The initial will never be negative

• Even if the numbers are getting really small, they will never get all the way to zero, but they'll get close. 

• Dividing by a number is the same as multiplying by the reciprocal:

• Dividing by 5 is the same thing as multiplying by 1/5

Multiple Choice

What is the initial value of this equation?

 $f\left(x\right)=2\left(4\right)^x$  

• 2

• 4

• 1/2

• 1/4

Multiple Choice

What is the common multiplier of this equation?

 $f\left(x\right)=2\left(4\right)^x$  

• 2

• 4

• 1/2

• 1/4

Multiple Choice

Use the initial value and common multiplier to identify the correct pattern for this equation:

 $f\left(x\right)=2\left(4\right)^x$  

• 2, 4, 6, 8, ...

• 2, 8, 32, 128, ...

• 4, 8, 16, 32, ...

• 4, 16, 64, 256, ...

Multiple Choice

What is the common multiplier in the number pattern below?

2, 12, 72, 432, 2592

• 2

• 7

• 6

• 9

Multiple Choice

What is the initial value in the number pattern below?

2, 12, 72, 432, 2592

• 2

• 7

• 6

• 9

Multiple Choice

Use the initial value and the common multiplier to identify the correct equation for this pattern:

2, 12, 72, 432, 2592

• f(x) = 2(1/6)^x

• f(x) = 6(2)^x

• f(x) = 2(6)^x

• f(x) = 6(1/2)^x

Multiple Choice

What is the initial value of this table?

• 4

• 5

• 20

• 3

Multiple Choice

What is the common multiplier of this table?

• 4

• 5

• 1/4

• 3

Multiple Choice

Choose the correct equation for the table using the initial value and common multiplier

• $y=4\left(5\right)^x$

• $y=5\left(20\right)^x$

• $y=5\left(4\right)^x$

• $y=5\left(15\right)^x$

Multiple Choice

Use the initial value and common multiplier to identify the correct equation for this table:

• $g\left(x\right)=\frac{1}{2}\left(16\right)^x$

•  $g\left(x\right)=16\left(2\right)^x$ 

•  $g\left(x\right)=2\left(16\right)^x$ 

• $g\left(x\right)=16\left(\frac{1}{2}\right)^x$

Multiple Choice

Choose the correct equation for an exponential function that starts at 5 and has a common multiplier of 3

•  $f\left(x\right)=\frac{1}{3}\left(5\right)^x$ 

•  $f\left(x\right)=5\left(\frac{1}{3}\right)^x$ 

•  $f\left(x\right)=3\left(5\right)^x$ 

•  $f\left(x\right)=5\left(3\right)^x$ 

Multiple Choice

Use the common multiplier to determine what type of function this equation is:  $f\left(x\right)=2\left(.85\right)^x$ 

• Exponential Decay

• Exponential Growth

Multiple Choice

Use the common multiplier to determine what type of function this equation is:  $f\left(x\right)=7\left(1.54\right)^x$ 

• Exponential Growth

• Exponential Decay

Multiple Choice

Use the common multiplier to determine what type of function this equation is:  $f\left(x\right)=9\left(\frac{2}{3}\right)^x$ 

• Exponential Growth

• Exponential Decay

Multiple Choice

Use the common multiplier to determine what type of function this equation is:  $f\left(x\right)=4\left(\frac{5}{2}\right)^x$ 

• Exponential Growth

• Exponential Decay

Multiple Choice

Which equation is exponential decay?

•  $f\left(x\right)\ =\ \frac{1}{2}\left(4\right)^x$  

•  $f\left(x\right)\ =\ 4\left(\frac{1}{2}\right)^x$  

•  $f\left(x\right)\ =\ 3\left(1.5\right)^x$  

•  $f\left(x\right)\ =\ 1.5\left(3\right)^x$  

Multiple Choice

Which equation is exponential growth?

•  $f\left(x\right)=\ 3\left(0.75\right)^x$  

•  $f\left(x\right)\ =\ 4\left(\frac{1}{2}\right)^x$  

•  $f\left(x\right)\ =\ 3\left(0.5\right)^x$  

•  $f\left(x\right)\ =\ 0.75\left(3\right)^x$  

Multiple Choice

Which pattern does not a common multiplier of 1/4?

• 400, 100, 25, 6.25

• 256, 64, 16, 4, 1

• 80, 20, 5, 1.25

• 80, 40, 20, 10

Multiple Choice

Which function has the biggest common multiplier?

• $y=3\cdot4^x$

• $y=4\cdot3^x$

• $y=1\cdot5^x$

• $y=5\cdot2^x$

Multiple Choice

Which function has the biggest initial value?

• $y=3\cdot4^x$

• $y=4\cdot3^x$

• $y=1\cdot5^x$

• $y=5\cdot2^x$