5-Minute Check on Activity 5-45-Minute Check on Activity 5-45-Minute Check on Activity 5-45-Minute Check on Activity 5-4

Click the mouse button or press the Space Bar to display the answers.Click the mouse button or press the Space Bar to display the answers.

Determine the domain, range and horizontal asymptotes for:

1. y = 4x domain: range: HA:

2. y = (1/2)x domain: range: HA:

Compare the transformations between the two functions, f and g:

3. f(x) = 4x and g(x) = 4x + 3

4. f(x) = 4x and g(x) = 4x + 2

5. Determine the growth rate and the growth factor for y = 0.3(1.27)x

6. Determine the decay rate and the decay factor for y = 6(0.87)x

All real > 0 y = 0

All real > 0 y = 0

shifts function up by 3

shifts function left by 2

growth factor = 1.27 growth rate = 0.27 or 27%

decay factor = 0.87 decay rate = -0.13 or -13%

Activity 5 - 5

Cellular Phones

Objectives• Determine the growth and decay factor for an

exponential function represented by a table of values or an equation

• Graph exponential functions defined by y = abx, where a ≠ 0, b > 0 and b ≠ 1

• Identify the meaning of a in y = abx as it relates to a practical situation

• Determine the doubling and halving time

VocabularyDoubling time – the time required for the amount to double

Halving time – also known as half-life, the time required for the amount to decrease by one-half

Activity

During a meeting, you hear the familiar ring of a cell phone. Without hesitation, several of your friends reach into their jacket pockets, brief cases and purses to receive the anticipated call. Although sometimes annoying, cell phones have become part of our way of life. The following table shows the rapid increase in the number of cell phone users in the late 1990s.

Year Cell Phones ( in millions)

1996 44.248

1997 55.312

1998 69.14

1999 86.425

2000 108.031

Is this a linear function?

Why or why not?

No

rate of change is not constant

Activity cont

Calculate the missing pieces in the table below:

YearCell Phones (in millions)

Rate of Change Ratio between Years

1996 44.248

1997 55.312

1998 69.14

1999 86.425

2000 108.031

Is the rate of change (slope) the same?

Is the ratio between consecutive years the same?

0 1.25

11.064 1.25

13.828 1.25

17.285 1.25

21.606 1.25

No

Yes

Activity cont

Does the relationship in the table represent an exponential function?

What is the growth factor?

Set up an equation, N = a∙bt, where N represents the number of cell phones in millions and t represents the number of years since 1996

What is the practical domain of the function N?

Yes

1.25

N = 44.248 (1.25)t

domain, t ≥ 0 (and ≤ 5 from table)

Exponential Functions

Exponential Functions of the form y = a∙bx, where b is > 0 and b ≠ 1

a is called the initial value, y-intercept (0, a) at x = 0

Exponential functions have successive ratios that are constant

The constant ratio is a growth factor, if y-values are increasing (b > 1)

The constant ratio is a decay factor, if y-values are decreasing ( 0 < b < 1)

Doubling time set by growth factor

Half-life is set by the decay factor

Exponential Function Identification

Identify the y-intercept, growth or decay factor, and whether the function is increasing or decreasing

a)f(x) = 5(2)x

b)g(x) = ¾(0.8)x

c)h(x) = ½ (5/6)x

d)f(t) = 3(4/3)x

y-int = 5, gf = 2, increasing

y-int = 3/4 df = 0.8, decreasing

y-int = 1/2 df = 5/6, decreasing

y-int = 3, gf = 4/3, increasing

Doubling and Halving Times

Doubling time – time it takes for y-values to double. It is determined by the growth rate and is the same for all y-values

Half-life – time it takes for y-values to decay by one-half. It is determined by the decay rate and is the same for all y-values

To find using your calculator: Let Y1 = exponential function and Y2 = doubled or halved amount. Graph them and use 2nd TRACE to find Intersection

Exponential Growth Example

An investment account’s balance, B(t), in dollars, is defined by B(t) = 5500(1.12)t, where t is the number of years.

What was the initial investment?

What is the interest rate on the account?

When will the investment double in value?

When will the investment quadruple in value?

initial investment = $5,500

interest rate = 12%

11000 = 5500(1.12)t solve graphically t ≈ 6.12 years

22000 = 5500(1.12)t solve graphically t ≈ 12.23 years

Exponential Decay Example

Chocolate chip cookie freshness decays over time due to exposure to air. If the cookie freshness is defined by f(t) = (0.8)t, find the following information.

What was the initial cookie freshness?

What is the decay rate on the cookies?

When will the cookie’s freshness be halved?

initial cookie freshness was 1

decay rate = 1 – 0.8 = 0.2 or 20%

0.5 = 1(0.8)t solve graphically t ≈ 3.11 days

Summary and Homework

• Summary– Functions defined by y = abx, where a is the initial

value and b is the growth or decay factor are exponential functions

– Y-intercept is (0, a)– Growth factor, b > 1, y-values are increasing– Decay factor, 0 < b < 1, y-values are decreasing– Doubling time is the time for the y-value to double– Half-life is the time for the y-value to be halved

• Homework– pg 576 – 580; problems 2, 3, 6, 7