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8.1 Exploring Exponential Models. What is an exponential equation?. An exponential equation has the general form y=ab x. Given the general form y=ab x When b > 1, b is the growth factor. Growth Factor, Decay Factor. When 0 < b < 1, b is the decay factor. Growth or Decay???.

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What is an exponential equation
What is an exponential equation?

An exponential equation has the general form

y=abx


Growth factor decay factor

Given the general form y=abx

When b > 1, b is the growth factor

Growth Factor, Decay Factor

  • When 0 < b < 1, b is the decay factor


Growth or decay
Growth or Decay???

Growth

Decay

Growth

Decay

Growth

Decay



What is an asymptote
What is an asymptote?

“Walking halfway to the wall”

An Asymptote is a line that a graph approaches as x or y increases in absolute value.

In this example, the asymptote is the x axis.


Graphing exponential functions
Graphing Exponential Functions

Complete the table using the integers -3 through 3 for x.



Let s try one
Let’s try one

Complete the table using the integers -3 through 3 for x. Then graph the function.



Let s try one2
Let’s try one

Complete the table using the integers -3 through 3 for x. Then graph the function.



Writing exponential equations
Writing Exponential Equations

  • Find the exponential equation passing through the points (3,20) and (1,5).

Start with the general form.

Choose a point. Substitute for x and y using (3, 20)

Solve for a

Substitute x and y using (1, 5) and a using

Division property of exponents


Writing exponential equations1
Writing Exponential Equations

  • Find the exponential equation passing through the points (3,20) and (1,5).

Simplify

Solve for b

Go back to the equation for a; substitute in b and solve for a


Writing exponential equations2
Writing Exponential Equations

  • Find the exponential equation passing through the points (3,20) and (1,5).

Going back to the general form, substitute in a and b

The exponential equation passing through the points (3,20) and (1,5) is


Let s try one4
Let’s Try One

  • Find the exponential equation passing through the points (2,4) and (3,16).

Start with the general form.

Choose a point. Substitute for x and y using (2, 4)

Solve for a

Substitute x and y using (3, 16) and a using

Division property of exponents


Writing exponential equations3
Writing Exponential Equations

Simplify

Solve for b

Go back to the equation for a; substitute in b and solve for a

Going back to the general form, substitute in a and b

The exponential equation passing through the points (2,4) and (3,16) is


Putting it all together
Putting it all together . . .

  • Find the equation of the exponential function that goes through (1,6) and (0,2). Graph the function.


Modeling growth with an exponential equation
Modeling Growth with an Exponential Equation

  • The growth factor can be found in word problems using b = 1 + r

    where r = rate or amount of increase. You can substitute your new b into your general equation to find the exponential function.


r = rate 3% (write as 0.03)

b = 1 + r = 1.03

x = time

a = amount put into the account ($1,000)


r = 1 (why not 2?)

b = r + 1 = 2

x = time (in hours)

a = the original number in the colony (1,000 bacteria )

b = r + 1, where r is the amount of increase. We are increasing by 100% each time something doubles, so r = 1


r = - 10% (the car is worth 10% less each year)

b = 1 - r = 1 – 0.1 = 0.9

x = time (in years)

a = amount put into the account ($15,000)


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