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Exponential Functions

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Exponential Functions

Lesson Objective:

Draw graphs of exponential functions of the form y = kaxand understand ideas of exponential growth and decay

- Suppose you have a choice of two different jobs at graduation
- Start at 20,000 with a 6% per year increase
- Start at 24,000 with 1000 per year raise

- Which should you choose?
- One is linear growth
- One is exponential growth

- How do we get each nextvalue for Option A?
- When is Option A better?
- When is Option B better?
- Rate of increase a constant 1000
- Rate of increase changing
- Percent of increase is a constant
- Ratio of successive years is 1.06

- All exponential functions have the general format:
y = kax

Where

- k = initial value
- a = growth factor (a>1) or decay factor (0<a<1)
- x = number of time periods

Option A y = 20000x1.06x

Option A y = 24000+1000x

y = 2x

y = 5x

y = 0.1x

y = 0.5x

y = 7x

In an exponential function, the variable is in the index. For example:

The general form of an exponential function to the base a is:

y = axwhere a > 0 and a1.

You have probably heard of exponential increase and decrease or exponential growth and decay.

A quantity that changes exponentially either increases or decreases increasingly rapidly as time goes on.

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Lets examine exponential functions. They are different than any of the other types of functions weve studied because the independent variable is in the exponent.

Lets look at the graph of this function by plotting some points.

x 2x

38

2 4

BASE

1 2

0 1

Recall what a negative exponent means:

-1 1/2

-2 1/4

-3 1/8

Reflected about y-axis

This equation could be rewritten in a different form:

So if the base of our exponential function is between 0 and 1 (which will be a fraction), the graph will be decreasing. It will have the same domain, range, intercepts, and asymptote.

There are many occurrences in nature that can be modeled with an exponential function. To model these we need to learn about a special base.

When 0 < a < 1 the graph of

y = ax has the following shape:

y

(1, a)

(1, a)

x

When b > 1 the graph of y = ax has the following shape:

y

1

1

x

In both cases the graph passes through (0, 1) and (1, a).

This is because:

a0 = 1 and a1 = a

for all a > 0.

- All exponential functions have the general format:
y = kax

Where

- k = initial value
- a = growth factor (a>1) or decay factor (0<a<1)
- x = number of time periods

The value of a new car depreciates at a rate of 15% a year.

The car costs 24 000 in 2010.

How much will it be worth in 2018?

To decrease the value by 18% we multiply it by 0.82.

There are 8 years between 2010 and 2018.

After 8 years the value of the car will be

24 000 0.828 =

4905 (to the nearest pound)

y = 24000x0.82x

- Population growth often modeled by exponential function
- Half life of radioactive materials modeled by exponential function

- Consider a medication
- Patient takes 100 mg
- Once it is taken, body filters medication out over period of time
- Suppose it removes 15% of what is present in the blood stream every hour

Fill in the rest of the table

What is the decay factor?

- Completed chart
- Graph

Growth Factor = 0.85

Note: when growth factor < 1, exponential is a decreasing function

- Each year the local country club sponsors a tennis tournament. Play starts with 128 participants. During each round, half of the players are eliminated.How many players remain after 5 rounds?

Why study exponential functions?

Population growth

Banking and finance

Compute compound interest

Whenever quantities grow or shrink by a constant factor, such as in radioactive decay, Depreciation

Medicine provides another common situation where exponential functions give an appropriate model. If you take some medicine, the amount of the drug in your system generally decreases over time.

An understanding of exponential functions will aid you in analyzing data particularly in growth and decay

- All exponential functions have the general format:
y = kax

Where

- k = initial value
- a = growth factor (a>1) or decay factor (0<a<1)
- x = number of time periods