Lesson Objective: Draw graphs of exponential functions of the form y = kax  and understand ideas of exponential growth and decay

Starter

• 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

Which Job?

• 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

Year Option A Option B

0 20,000.00 £24,000

1 21,200.00 £25,000

2 22,472.00 £26,000

3 23,820.32 £27,000

4 25,249.54 £28,000

5 26,764.51 £29,000

6 28,370.38 £30,000

7 30,072.61 £31,000

8 31,876.96 £32,000

9 33,789.58 £33,000

10 35,816.95 £34,000

11 37,965.97 £35,000

12 40,243.93 £36,000

13 42,658.57 £37,000

14 45,218.08 £38,000

15 47,931.16 £39,000

16 50,807.03 £40,000

17 53,855.46 £41,000

18 57,086.78 £42,000

19 60,511.99 £43,000

General Formula• 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  

Exponential functions

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 = ax where a > 0 and a ≠1.y = ax where a > 0 and a ≠1.

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.

y = 2x y = 5x y = 0.1x y = 0.5x y = 7x

Let’s examine exponential functions. They are different than any of the other types of functions we’ve studied because the independent variable is in the exponent.

xxf 2

Let’s look at the graph of this function by plotting some points. x 2x

3 8 2 4 1 2 0 1

-1 1/2 -2 1/4 -3 1/8

2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 0 4 6 8

7

123456

8

-2-3-4-5-6-7

2

121 1 f

Recall what a negative exponent means:

BASE

xy 2

Reflected about y-axis This equation could be rewritten in a different form: x

xxy

2

1

2

12

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.

In both cases the graph passes through (0, 1) and (1, a).This is because:

a0 = 1 and a1 = afor all a > 0.

When 0 < a < 1 the graph ofy = ax has the following shape:

y

x

1 1

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

y

x

(1, a)(1, a)

General Formula

• 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

Compound interest

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.

After 8 years the value of the car will be

£24 000 × 0.828 = £4905 (to the nearest pound)

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

There are 8 years between 2010 and 2018.

y = 24000x0.82x  

Exponential Modeling

• Population growth often modeled by exponential function

• Half life of radioactive materials modeled by exponential function

Decreasing Exponentials

• 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 hourAt end of hour Amount remaining

1 100x0.85 = 85

2 100x0.85x0.85 = 72.25

3

4

5

Fill in the rest of the table

Fill in the rest of the table

What is the decay factor?

What is the decay factor?

Decreasing Exponentials

• Completed chart

• Graph

At end of hour Amount Remaining1 85.002 72.253 61.414 52.205 44.376 37.717 32.06

At end of hour Amount Remaining1 85.002 72.253 61.414 52.205 44.376 37.717 32.06

Amount Remaining

0.00

20.00

40.00

60.00

80.00

100.00

0 1 2 3 4 5 6 7 8

At End of Hour

Mg

rem

ain

ing

Amount Remaining

0.00

20.00

40.00

60.00

80.00

100.00

0 1 2 3 4 5 6 7 8

At End of Hour

Mg

rem

ain

ing

Growth Factor = 0.85

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

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 financeCompute 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

General Formula

• 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

xxf 2

xxf 3

xexf