Exponential Equations

Writing Exponential Equations

What is an exponential equation?

a--initial value, y-intercept, never 0

b--growth or decay factor, never 0 or 1 (base)

x--exponent/power

Writing an equation

X-values should be a constant increase. Notice they increase by 1 each time.  Y-values multiply by the same number each time.

To write equation, find the y-intercept (where x=0) and the rate of change for the y-values.
This table is,  $f\left(x\right)=2^x$   ,since the y-int is 1 and it multiplies by 2.

Are these exponential?

• A--Yes, multiplies by 3 each time

• B--No, variable is not exponent

• C--Yes, variable is exponent

• D--No, adds 2, not mulitiply

What is this equation?

Y-values multiply by 3 each time, but there is not y-intercept listed. For x=0, you must divide 15/3 to "work backwards".

x=0 gives y=5

Equation:
 $y=5\left(3\right)^x$  

Multiple Choice

What is the equation for this table?

• $f\left(x\right)=8\left(4\right)^x$

• $f\left(x\right)=4^x$

• $f\left(x\right)=2\left(4\right)^x$

Multiple Choice

What is the equation for this table?

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

• $f\left(x\right)=2^x$

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

Multiple Choice

Write the equation that represents the table below:

• y=½(4)^x
• y=2(4)^x
• y=4(2)^x
• y=4(½)^x

Multiple Choice

What is the pattern you see in this table?

• Multiplying by 1/3

• Subtracting 8

• Multiplying by 3

• Subtracting 3

Multiple Choice

Which of the equations below represent this table?

• y=3⋅(36)^x

• y=36⋅(3)^x

• y=3⋅(⅓)^x

• y=36⋅(⅓)^x

Finding the Equation from a Graph

1) Find the y-intercept----> (0, 1)
a=1

2) Find the next points--->(1, 2) (2, 4) (3,8) and look at how the y-values change. These x2 each time.
b=2

3) Write equation

 $f\left(x\right)=1\left(2\right)^x$  

What about this one?

• Y-intercept is (0,2) ---> a=2

• Look at other points->divided by 3 (same as x $\frac{1}{3}$ )       b=1/3

• Equation is   $f\left(x\right)=2\left(\frac{1}{3}\right)^x$  

Multiple Choice

Which equation matches the graph?

• y=2(2)^x

• y=2(1/2)^x

• y=2^x

• y=1/2(2)^x

Multiple Choice

What is the equation that models this graph?

• y = 2(3)^x
• y = 3(2)^x
• y = 3(½)^x
• y =½(3)^x

Let's Do a Little Review!

Multiple Choice

Suppose a culture of bacteria begins with 5000 cells and dies by 30% each year. Write an equation that represents this situation.

• y=5000(0.7)^x
• y=30(5000)^x
• y=5000(1.3)^x
• y=5000x^x

Multiple Choice

What type of function is f(x)=2(1/7)^x ?

• Exponential Growth
• Linear
• Exponential Decay
• None of the Abovee

Multiple Choice

What is the growth for the function: f(x) = 800(0.85)^x?

• 800

• 0.85

• 0.15

• x

Multiple Choice

Which the following graphs shows an exponential decay equation?

Multiple Choice

Identify the y-intercept (initial value) in the function f(x)=13(.27)^x

• (.27)^x
• .27
• 13
• 3.51

Multiple Choice

The population of Winnemucca, Nevada, can be modeled by P=6191(1.04)^t where t is the number of years since 1990. What percent did the population increase each year?

• 4%
• It doesn't say
• I don't know
• 400%

Multiple Choice

The amount of money in Carmen’s savings account is represented by the equation $A\left(t\right)=3500\left(1.025\right)^t$ where t represents years.  How much will Carmen have in the account after 10 years?

• $4,480.30

• $4,480.29

• $4,592.30

• You can't tell from the equation.

Multiple Choice

The amount of money in Carmen’s savings account is represented by the equation $A\left(t\right)=3500\left(1.025\right)^t$ where t represents years.  How much did Carmen originally deposit?

• $1.025

• $3,500

• 25%

• 1.025%

• You can't tell from the equation.

Multiple Choice

The amount of money in Carmen’s savings account is represented by the equation $A\left(t\right)=3500\left(1.025\right)^t$  , where t represents years. What is the interest rate on the account (as a percent)?

• .025%

• .25%

• 2.5%

• 25%

Multiple Choice

1. The population of a school of fish decreases at a rate of 18% per month. There are currently 500 fish in the school. How many fish will there be in 3 months?

• 276

• 0

• 822

• 476

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

You bought a Boston Whaler in 2004 for $12,500. The boat's value depreciates by 7% a year. How much is the boat worth in 2012?

• $11625
• $6994.77
• $21,477
• $875