Page 766 - Algebra 1

9. Generalize How does changing the value of k affect the graph of the function?
10. Formulate As the number of folds increase, what happens to the number of regions on the folded paper? What is the b-value for each equation? Write an equation in the form y = k(b)x to model situations in which y doubles as x increases.
11. For any function y = k(b)x, what does k represent in any situation when x = 0?
The period of time required for a quantity to double in size or value is called doubling time. The equation will be of the form y = k(2)x.
Just as data can grow exponentially, some data can model exponential decay. Exponential decay is a situation where a quantity always decreases by the same percent in a given time period.
Carbon-14 dating is used to find the approximate age of animal and plant material after it has decomposed. The half-life of carbon-14 is 5730 years. So, every 5730 years half of the carbon-14 in a substance decomposes. Find the amount remaining from a sample containing 100 milligrams of carbon-14 after four half lives.
12. How many years are there in four half-lives? 13. Create and complete a table like the one below.
Caution
Do not divide the original amount of a substance by 3 to calculate the amount of a substance left after three half-lives.
Number of Half-Lives
Number of Years
Amount of Carbon-14 Remaining (mg)
0
0
100
1
5730
2
11,460
3
17,190
4
22,920
Hint
Since x usually represents time in decay equations, x > 0.
14. How much of the sample remains after 22,920 years?
Exponential decay is modeled by the function f(x) = kb x, where k > 0 and 0 < b < 1. Since the value of b is a positive number less than 1, as x increases, the value of f(x) decreases by b.
An exponential decay function can model the amount of a substance in the body over time. Many diabetes patients take insulin. The exponential
_1 x
function f(x) = 100 (2 ) describes the percent of insulin in the body after
x half-lives. The half-life of a substance is the time it takes for one-half of the substance to decay into another substance.
15. About what percent of insulin would be left in the body after 8 half-lives?
16. Write Describe the effect that the b-value has on the amount of substance remaining as the number of half-lives x increases.
Math Reasoning
Analyze Why does f(x) decrease as x increases?
Investigation 11 751