One of the function types that you may see on the New SAT is the exponential function, which students are normally introduced to in Algebra II. These functions, which can be used to model population growth, to compute compound interest, and for other applications in science, have equations in the form:
f(x) = a·b^x
In this equation, a and b are constants and the variable, x, appears as an exponent.
The number a is the initial value (when x = 0). For a population growth problem, it would represent the initial population of whatever group you are looking at. For a compound interest problem, the a represents the amount of the deposit or the loan.
The b number is the growth factor or decay factor. Let’s take a minute to look at how this number b is computed. You should add the percent of growth (or subtract the percent of decay) to 100 percent and then convert that percent to a decimal.
Example: A population increase of 4.5 percent per year.
b = 100% + 4.5% = 104.5% = 1.045
Example: A population decrease of 3 percent per year.
b = 100% – 3% = 97% = 0.97
Here are some practice problems involving exponential growth that you can try on your own. When you’re done, scroll down for the answers.
1) You deposit $500 into a bank account that pays 3% interest compounded annually. Which of the following equations could you use to compute the amount in the account (in dollars), A, at the end of two years?
A) A = 500(.02)³
B) A = 500(.03)²
C) A = 500(1.02)³
D) A = 500(1.03)²
Professor Roberts has been studying ways to encourage the growth of a population of endangered California seals. The current population of the seals is 600, and Professor Roberts believes that the population will grow at 3 percent if a conservation plan is put in place. She models the population, P, of the seals after t years using the function P = a·b^t. What would be the value of b in this function?
Professor Jenkins believes that the population of seals in the problem above will grow at a rate of 3.5 percent rather than at 3 percent. Professor Roberts and Professor Jenkins both use their models to predict the population of seals after ten years. How many more seals will Professor Jenkins’ model predict compared to the amount predicted by Professor Roberts’ model?
Scroll down for answers
b = 100% + 3% = 103% = 1.03
Jenkins: 600(1.035)^10 ≈ 846 seals
Roberts: 600(1.03)^10 ≈ 806 seals
846 – 806 = 40
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