Exponential Functions

1. Exponential Functions The equation y = bx is an exponential function provided that b is a positive number other than 1. Exponential functions have variables as exponents. • The domain is the set of real numbers, and the range is the set of positive real numbers • if b > 1, the graph of y = bxrises from left to right and intersects the y-axis at (0, 1). As x decreases, the negative x-axis is a horizontal asymptote of the graph. • If 0 < b < 1, the graph of y = bxfalls from left to right and intersects the y-axis at (0, 1). As x increases, the positive x-axis is a horizontal asymptote of the graph. Generalizations about Exponential Functions

2. Graphs of Exponential Functions Let’s look at the graph of y = 2x That was easy

3. Comparing Graphs of Exponential Functions What happens to the graph of y = bxas the value of b changes? Now, let’s look at the graphs. Let’s look at some tables of values. I think I see a pattern here.

4. Let’s Look at the Other Side What happens to the graph of y = bxwhen b < 1 and the value of b changes? Now, let’s look at the graphs. Let’s look at some tables of values. I knew there was going to be a pattern!

5. Let’s Shift Things Around Let’s take another look at the graph of y = 2x Now, let’s compare this to the graphs of y = (2x)+3 and y = 2(x+3)

6. Translations of Exponential Functions The translation Th, kmaps y = f(x) to y = f(x - h) + k Remember if f(x) = x2, then f(a - 3) = (a - 3)2 Hey, that rings a bell! It looks like that evaluating functions stuff. It’s all starting to come back to me now. Since yandf(x) are the Sam Ting, We can apply this concept to the equation y = bx

7. Let’s take a closer look If y = bx and y = f(x), thenf(x) = bx This is a little confusing, but I’m sure it gets easier. If the translation Th, kmaps y = f(x) to y = f(x - h) + k Then the translation Th, kmaps y = bxto y = b(x - h) + k This will be easier to understand if we put some numbers in here.

8. Now we have a Formula The translation Th, kmaps y = bxto y = b(x - h) + k Let’s try a translation on our basic exponential equation Let’s apply the transformation T3, 1to the equationy = 2x The transformed equation would be y = 2(x - 3) + 1 I’m not ready to push the easy button yet. Let’s look at some other examples first.

9. Let’s look at some graphs Let’s start with the graph of y = 2x Let’s go one step at a time. When the transformation T3, 1is applied to the equationy = 2x we get y = 2(x - 3) + 1 Step 1 y = 2(x - 3) What happened to the graph? Step b y = 2(x - 3) + 1 What happened to the graph now? What conclusions can we make from this example?

10. Let’s look at some other graphs Let’s start with the graph of y = 2x Let’s go one step at a time. When the transformation T-4, -2is applied to the equationy = 2x we get y = 2(x + 4) - 2 Step 1 y = 2(x + 4) What happened to the graph? Step b y = 2(x + 4) - 2 What happened to the graph now? What conclusions can we make from this example?

11. Let’s Summarize Translations The translation Th, kmaps y = bxto y = b(x - h) + k Positive k shifts the graph up k units This translation stuff sounds pretty shifty, but don’t let it scare you. Negative k shifts the graph down k units Positiveh shifts the graph lefth units Negativeh shifts the graph righth units

12. This exponential equation stuff is pretty easy. I feel like jumping for joy! Oh my! I think I’ll just push the easy button. That was easy