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 = 1 x = 2, y = 1/3 x = 3, y = 1/9 x = 4, y = 1/27

Geometric sequence (1/3) s – 1 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 xy ½ ½ 263

A Common Mistake Confusing the initial value with the common ratio in the geometric sequence 2(3) s – 1 initial value Common ratio Forgetting that any number to the zero power is 1, not 0.

Core Lesson StepTearsArea of paper 1064

Core Lesson StepTearsArea of paper

Core Lesson StepTearsArea of paper

Core Lesson StepTearsArea of paper

Core Lesson x½ StepTearsArea of paper

Core Lesson StepTearsArea of Paper Math Work x ½ x ½ x ½ x ½ x ½ x ½ x ½ x ½ x ½ x ½

Core Lesson StepTearsArea of Paper Math WorkExponential Expression x ½ x ½64 x ½ x ½ x ½64 x ½ x ½ x ½ x ½64 x ½ x ½ x ½ x ½ x ½64 x ½ 4 2(3) s – 1 initial value common ratio 64(½) s-1 initial value common ratio 10 th tear? p = 64(½) 10 = 1/16 y = ab x

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.

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?

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