How can you create an equation for a decreasing

geometric sequence?

For example, if your car depreciates in value at an

exponential rate, how do you know what it will be worth in

10 years?

In this lesson you will learn how to create an equation for a

decreasing geometric sequence by making a table

and drawing a graph.

Let’s Review

Exponential functions grow or shrink at a rate proportional to their current

value.

For example, y = (1/3)x-1

x = 1, y = 1x = 2, y = 1/3x = 3, y = 1/9

x = 4, y = 1/27

Let’s Review

Geometric sequence

54

18

6

2

54(1/3)s – 1 x 1/3

x 1/3

x 1/3

Geometric sequences change exponentially. They have a common

ratio between consecutive terms.

Let’s Review

Rates of change can show a decrease.y = -½x + 64

x y

-2 65

-1 64½

0 64

1 63½

2 63

A Common Mistake

Confusing the initial value with the common ratio in the geometric

sequence

2(3)s – 1 initial value

Common ratioForgetting that any number to the zero

power is 1, not 0.

Core Lesson

Step Tears Area of paper

1 0 64

Core Lesson

Step Tears Area of paper

1 0 64

2 1 32

Core Lesson

Step Tears Area of paper

1 0 64

2 1 32

3 2 16

Core Lesson

Step Tears Area of paper

1 0 64

2 1 32

3 2 16

4 3 8

Core Lesson

x½ x ½x ½x ½

Step Tears Area of paper

1 0 64

2 1 32

3 2 16

4 3 8

5 4 4

Core Lesson

Step Tears Area of Paper

Math Work

1 0 64 64

2 1 32 64 x ½

3 2 16 64 x ½ x ½

4 3 8 64 x ½ x ½ x ½

5 4 4 64 x ½ x ½ x ½ x ½

Core LessonStep Tears Area of

PaperMath Work Exponential

Expression

1 0 64 64 64 x ½0

2 1 32 64 x ½ 64 x ½1

3 2 16 64 x ½ x ½ 64 x ½2

4 3 8 64 x ½ x ½ x ½ 64 x ½3

5 4 4 64 x ½ x ½ x ½ x ½

64 x ½4

2(3)s – 1

initial value

commonratio

64(½)s-1

initial value

commonratio

10th tear?

p = 64(½)10 = 1/16

y = abx

Core Lesson

Number of tears

Area of paper

In this lesson have learned how to create an equation for a

decreasing geometric sequence by making a table

and drawing a graph.

Guided Practice

Suppose I buy a car for $1000, and it depreciates by 5% each year. How much will the car be worth in 10 years?

Extension Activities

Place 100 pennies in a cup. Shake the cup and pour out the coins. Take out every coin that lands on “heads”, then record the new population. Do this 15 times. Find an equation to show this exponential decay model.

Extension Activities

Investigate the graphs of y = 64(1/2)x and y = 2-x. Compare and contrast the two graphs. See if you can explain mathematically what you found.

Quick Quiz

Suppose a population of 3,000,000 decreases 1.5% annually. How many people will be left after 10 years?

Quick Quiz

Which of the following situations best matches the equation of the function y = 120(0.9875)x?A population of 120 wolves decreases 98.75% annually.A population of 120 wolves increases 1.25% annually.A population of 120 wolves decreases 1.25% annually.A population of 120 wolves decreases by almost 98 wolves annually.