Finding the function of an exponential table involves determining the equation that represents the relationship between the x and y values in the table. Exponential functions have the form y = abx, where b is the constant ratio and a is the initial value. Examining the table, you'll notice that as x increases by a constant value, y increases by a common ratio. Here's how to determine the function:
Steps to Find the Exponential Function
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Identify the Constant Ratio (b):
- Look for a pattern in the y-values. Divide consecutive y-values to see if there's a common ratio.
- If the y-values are not changing by a common ratio, then the table probably doesn't represent an exponential function.
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Find the Initial Value (a):
- The initial value 'a' is the y-value when x = 0. Look for the row in the table where x is 0. The corresponding y-value is your 'a'.
- If x = 0 is not in the table, you can work backwards from an existing point using the constant ratio.
- For example, if you know that when x = 1, y = 6, and b = 2, then when x = 0, y must be 6/2 = 3. So, a = 3.
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Write the Equation:
- Substitute the values you found for 'a' and 'b' into the equation y = abx.
Example
Let's say you have the following table:
| x | y |
|---|---|
| 0 | 5 |
| 1 | 15 |
| 2 | 45 |
| 3 | 135 |
- Constant Ratio (b): 15/5 = 3, 45/15 = 3, 135/45 = 3. So, b = 3.
- Initial Value (a): When x = 0, y = 5. So, a = 5.
- Equation: y = 5 * 3x
Practical Insights
- Non-Integer Values: The constant ratio 'b' doesn't have to be an integer. It can be a fraction or a decimal.
- Decreasing Functions: If 'b' is between 0 and 1 (0 < b < 1), the exponential function will decrease as x increases.
- Real-World Applications: Exponential functions are used to model population growth, compound interest, and radioactive decay.
