Exponential Functions

1. Exponential Functions Lesson 5.1

2. You have used sequences and recursive rules to model geometric growth or decay of money, populations, and other quantities. • Recursive formulas generate only discrete values. • Growth and decay happen continuously. • We will study explicit formulas for these patterns, which will allow you to model situations involving continuous growth and decay, or to find discrete points without using recursion.

3. Radioactive Decay • This investigation will simulate radioactive decay. • Each person will need a standard six-sided die. • Each standing person represents a radioactive atom in a sample. • The people who sit down at each stage represent the atoms that underwent radioactive decay.

4. Follow these steps • Collect data in the form (stage, number standing). • All members of the class should stand up, except for the recorder. The recorder counts and records the number standing at each stage. • Each standing person rolls a die, and anyone who gets a 1 sits down. • Wait for the recorder to count and record the number of people standing. • Repeat the last two steps until fewer than three students are standing.

6. Graph your data. The graph should remind you of the sequence graphs you studied in Chapter 1. • What type of sequence does this resemble? • Identify uo and the common ratio, r, for your sequence. Complete the table below. • Use the values of uo and r to help you write an explicit formula for your data.

7. 0 u0 u1 U0 (r) U0 (r) 1 2 u2 U0 (r)(r) U0 (r)2 3 U0 (r)(r)(r) U0 (r)3 u3 4 u4 U0 (r)(r)(r)(r) U0 (r)4

8. Graph your explicit formula along with your data. Notice where the value of u0 appears in your equation. Your graph should pass through the original data point (0, u0 ) . Modify your equation so that it passes through (1, u1) , the second data point. (Think about translating the graph horizontally and also changing the starting value.)

10. Experiment with changing your equation to pass through other data points. • Decide on an equation that you think is the best fit for your data. Write a sentence or two explaining why you chose this equation. • What equation with ratio r would you write that contains the point (6, u6) ?

11. Example A • Most automobiles depreciate as they get older. • Suppose an automobile that originally costs $14,000 depreciates by one-fifth of its value every year. • What is the value of this automobile after 2 1/2 years? • When is this automobile worth half of its initial value?

12. Value after 2 ½ years should be between 8960 and 7168,

13. To find when the automobile is worth half of its initial value, or 7,000 replace the y with 7000 and solve for x. 7,000= 14,000(1- 0.2)x 0.5= (1- 0.2)x 0.5 =(0.8)x You don’t yet know how to solve for x when x is an exponent, but you can experiment to find an exponent that produces a value close to 0.5. The value of (0.8)3.106 is very close to 0.5. This means that the value of the car is about $7,000, or half of its original value, after 3.106 years (about 3 years 39 days). This is the half-life of the value of the automobile, or the amount of time needed for the value to decrease to half of the original amount.

14. Example B The functions and are transformations of the parent function . Describe the transformations and sketch the graphs. The graph of g (x) is a vertical dilation of the graph of f (x) by a factor of 1/2. The marked points on the red graph show how the y-values of the corresponding points on the black graph have been multiplied by 1/2 .

15. Example B The functions and are transformations of the parent function . Describe the transformations and sketch the graphs. The graph of h(x) is a horizontal translation of the graph of f (x). The marked points on the red graph show how the corresponding points on the black graph have been moved right 1 unit.