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How Do You Identify Exponential Behavior from a Pattern in the Data?

How do you identify exponential behavior from a pattern in the data?

Summary

  1. As we increase x by 1 in the first table, we multiply f(x) by 3
  2. As we increase x by 1 in the second table, we divide f(x) by 3
  3. Dividing by 3 is the same as multiplying by 1/3
  4. So in the second table, we could also say that we are multiplying f(x) by 1/3 each time
  5. y is the same thing as f(x)
  6. The general form of an exponential function is f(x)=a•bx
  7. When b is greater than 1, you have exponential growth
  8. When 'b' is between 0 and 1, you have exponential decay

Notes

    1. Let's look at the relationship between the 'x' and 'f(x)' values in the table
    2. Maybe we can find a pattern there!
    1. In the first table, the f(x) values increase as the x values increase
    2. x increases by 1 as we go down each line in the table
    3. Each time we add 1 to x, we multiply f(x) by 3
    4. This is a pattern that you see when you have an exponential function
    1. If your x values are increasing at the same rate and you can multiply by the same number to get the next f(x) value each time, you have exponential behavior
    1. In the second table, the f(x) values decrease as the x values increase
    2. Again, x increases by 1 as we go down each line in the table
    3. But this time, our f(x) values get smaller as x gets bigger
    4. Each time we add 1 to x, we divide f(x) by 3
    5. Since dividing by 3 is the same as multiplying by 1/3, we have exponential behavior here as well!
    1. Remember, dividing is the same as multiplying by a reciprocal
    2. To find the reciprocal of a whole number, just take 1 over that number
    3. So the reciprocal of 3 is 1/3
    4. To have exponential behavior, we need to be able to multiply f(x) by a common factor each time x increases by the same amount
    5. Since we're multiplying by 1/3 each time, we have exponential behavior here too!
    1. y is the same thing as f(x)
    2. As x increases at a constant rate, y starts to increase faster and faster
    3. This is a feature of an exponential growth function
    1. y is the same thing as f(x)
    2. As x increases at a constant rate, y decreases quickly at first, and then slows down and starts to level off
    3. This is a feature of an exponential decay function
    1. Let's see how the functions that go with these tables relate to the patterns we found
    1. Remember, the general form of an exponential function is f(x)=a•bx
    2. So in our first function, 'a' is 4 and 'b' is 3
    3. '>' means 'greater than'
    4. When 'b' is greater than 1 in an exponential function, you have exponential growth
    5. When we increase 'x' by 1, we're increasing the exponent by 1 each time
    6. That means we're multiplying the function by 3 once more every time we increase 'x' by 1
    1. Remember, the general form of an exponential function is f(x)=a•bx
    2. So in our second function, 'a' is 4 and 'b' is 1/3
    3. '<' means 'less than'
    4. When 'b' is between 0 and 1 in an exponential function, you have exponential decay