Geometric Sequences, Exponential Equations, Exponential Growth and Decay

 Consider the table below…  What is going on in the table?  Adding 4 each time  This is a linear function!  y = 4x - 1 XY

 Consider the table below…  What is going on in the table?  Multiplying by 3 each time  This is an exponential function!  y = 3 x XY

 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

 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

XY  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.

 The equation to make an exponential function is as follows  y = a * b x  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? XY

 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 * 5 x or y = 5 x XY

 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.

XY -23/4 1 ½  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 * 2 x  Is it exponential growth or decay?  Growth!

XY -21/9 1/  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 * 3 x or y = 3 x  Is it exponential growth or decay?  Growth!

XY  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!

 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 * b x