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## Lesson 8-6/8-7/8-8 Exponential Functions

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**Lesson 8-6/8-7/8-8Exponential Functions**Geometric Sequences, Exponential Equations, Exponential Growth and Decay**A look at the past…**• Consider the table below… • What is going on in the table? • Adding 4 each time • This is a linear function! • y = 4x - 1**A look at TODAY…**• Consider the table below… • What is going on in the table? • Multiplying by 3 each time • This is an exponential function! • y = 3x**Today…Exponential Functions**• Exponential functions have a pattern of multiplying... like the table you’ve just seen. • Let’s look at a few more patterns… • 2, 8, 32, 128,…. What is the pattern? • Multiplying by 4 • 0.45, 0.9, 1.8, 3.6,… What is the pattern? • Multiplying by 2 • 8, 20, 50, 125,… What is the pattern? • Multiplying by 2.5 or 2 ½**Look at Patterns**• Linear is adding/subtracting • Adding patterns are called arithmetic • 12, 8, 4, 0….What’s the pattern? • Subtracting 4 • Exponential is multiplying • Multiplying patterns are called geometric • 2, 14, 98, 686…What’s the pattern? • Multiplying by 7**Exponential Functions**• Look at the table. What is the pattern? Is it linear or exponential? • It is exponential. The pattern is multiplying by 5. • How would you write an equation to represent the table? • Let’s look at the equation for exponential functions.**Exponential Function**• The equation to make an exponential function is as follows • y = a * bx • a is where the graph crosses the y axis (the y intercept) • b is by what number you are constantly multiplying • Look at the table again. See if you can find the values for a and b. • What do you think they are?**That’s right!**• a is where it crosses the y-axis so in this case when x = 0, it crosses at 1. • b is what you are constantly multiplying by and that number is 5. • So the equation is y = 1 * 5x or y = 5x**Graphing an Exponential**• Take a minute and type the equation you just formulated into your calculator. • What does the shape of the graph look like? • If I gave you a choice of saying it was either exponential growth or exponential decay, which one would you tell me it was and why? • It is exponential growth because the graph goes up. **Exponential Functions y = a*bx**• Look at the table. What is the pattern? Is it linear or exponential? • It is exponential. The pattern is multiplying by 2. • What is the a? Remember that the a is where it crosses the y-axis at x=0. • What is the b? That is what you are multiplying by each time. • y = 3 * 2x • Is it exponential growth or decay? • Growth! **Exponential Functions y = a*bx**• Look at the table. What is the pattern? Is it linear or exponential? • It is exponential. The pattern is multiplying by 3. • What is the a? Remember that the a is where it crosses the y-axis at x=0. • What is the b? That is what you are multiplying by each time. • y = 1 * 3x or y = 3x • Is it exponential growth or decay? • Growth! **Exponential Functions y = a*bx**• Look at the table. What is the pattern? Is it linear or exponential? • It is exponential. The pattern is multiplying by ½. • What is the a? Remember that the a is where it crosses the y-axis at x=0. • What is the b? That is what you are multiplying by each time. • y = 1 * ½ x or y = ½ x • Is it exponential growth or decay? • Decay! If your b is less that one, it’s decay! The graph goes down!**Okay…let’s summarize!**• Linear Functions • Pattern is add or subtract (arithmetic). • Graph is a line. • Has a slope and a y-intercept. • y = mx + b • Exponential Functions • Pattern is multiplying (geometric). • Graph is a weird curve either going up or down. • Up – Growth • Down – Decay • Has a slope (kind of) and a y – intercept. • y = a * bx