### Assignments:

Unfinished Assignment Study Questions for Lesson 25

### Lesson Objectives:

- Graph exponential equations and exponential functions.
- Solve applied problems involving exponential functions.

An exponential function with base a is f(x) = a^x, where x is in the exponent, and it's a real number. And a > 0, to avoid imaginary numbers. And a != 1, to avoid the identity function; since 1^x would equal 1.

Sketch the graph of the function, f(x) = 3^(x+1).

So start by setting up a table with your x and y-values. Let's input -2, -1, 0, 1, and 2. When x is -2, y = 3^(-2+1), which is 3^(-1) or 1/3. When x is -1, y = 3^(-1+1), which is 3^0, or just 1. When x is 0, y = 3^(0+1), or 3^1 which is 3. When x is 1, y = 3^(1+1), which is 3^2, or 9. And when x is 2, y = 3^(2+1), which is equal to 3^3, or 27.

So our graph of f contains the points: (-2, 1/3), (-1, 1), (0, 3), (1,9), and (2, 27), to name a few. And all that's left to do is plot the points and complete the graph. So our graph looks like this.

The graph of f(x) = 3^(x+1) is the graph of f(x) = 3^x shifted to the left 1 unit.

When Charles was born, his grandparents gave him a $5,000 certificate of deposit that earns 3.5% interest, compounded quarterly. If the certificate of deposit matures on his 18th birthday, what amount will be available then? The compound interest formula is A = P(1+r/n)^(nt). A is the final amount, P is the starting amount, r is the interest rate in decimal form, n is the amount of times per year that it's compounded, and t is the number of years. So we're given 5000 as P, and we're given a rate of 3.5%. If we convert 3.5% to decimal form, we get 0.035. And since it's compounded quarterly, we put 4 for n. And since it matures on his 18th birthday, we know that t = 18. And now we solve for A. So A = 9362.36. On his 18th birthday,$9,362.36 will be available.

Sketch the graph of the funciton f(x) = e^(3x).

e was a number which was introduced by a Leonard Euler in 1741. It's equal to 2.71828, etc., and you can find this value on your calculator.

So let's start with a table of our x and our y-values. We'll input -3, -2, -1, 0, and 1. And when we put -3 in, we get 0.00012. And -2 gives us 0.00248. And -1 gives us 0.04979. And 0 gives us 1, and 1 gives us 20.086. So now if we plot these points and complete the graph, we get the following graph.

Notice that the graph of f(x) = e^(3x) is the same as f(x) = e^x, shrunk horizontally by a factor of 3.