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

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