how can you create an equation for a decreasing geometric sequence? for example, if your car...
TRANSCRIPT
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.
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
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
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.