The constant ratio of an exponential function is simply the base of the exponent. For an exponential function in the form f(x) = abx, the constant ratio is b.
Here's a more detailed explanation:
Understanding Exponential Functions and the Constant Ratio
An exponential function describes a relationship where a quantity increases or decreases by a constant factor for each unit increase in the input variable. The general form of an exponential function is:
- f(x) = abx
Where:
- f(x) is the output value.
- a is the initial value (the value of the function when x = 0).
- b is the constant ratio or the base.
- x is the input variable.
Identifying the Constant Ratio
The constant ratio b represents the factor by which the output multiplies when the input x increases by 1. This means f(x+1) = b f(x).
Example:
Consider the exponential function f(x) = 3 2x*.
- Here, a = 3 and b = 2.
- The constant ratio is 2. This means that for every increase of 1 in x, the output f(x) doubles.
Let's verify:
- f(1) = 3 21 = 6*
- f(2) = 3 22 = 12*
As you can see, f(2) is twice f(1), confirming that 2 is indeed the constant ratio.
Finding the Constant Ratio from a Table of Values
If you are given a table of values for an exponential function, you can find the constant ratio by dividing any output value by the previous output value (provided the input values are increasing by a constant amount of 1).
Example:
| x | f(x) |
|---|---|
| 0 | 5 |
| 1 | 15 |
| 2 | 45 |
| 3 | 135 |
To find the constant ratio, divide any f(x) by f(x-1):
- 15 / 5 = 3
- 45 / 15 = 3
- 135 / 45 = 3
The constant ratio is 3. The exponential function can be written as f(x) = 5 3x*.
Summary
In an exponential function f(x) = abx, the constant ratio is the base, b. This value indicates the factor by which the output multiplies each time the input increases by 1. You can determine the constant ratio directly from the function's equation or by dividing consecutive output values in a table, assuming the input values increase by 1.
