4.9: Putting It Together- Nonlinear Models
Let’s Summarize
This is what we have learned about exponential models:
The general form of an exponential model is y = C · b x .
- Exponential models are nonlinear. More specifically, exponential models predict that y increases or decreases by a constant percentage for each 1-unit increase in x .
- C is the initial value . It is the y -value when x = 0. It is also the y -intercept.
-
b
is the
growth factor
or
decay factor
.
b
is always positive.
- If b is greater than 1, b is a growth factor. In this case, the association is positive, and y is increasing. This makes sense because multiplying by a number greater than 1 increases the initial value. From the growth factor, we can determine the percent increase in y for each additional 1-unit increase in x .
- Similarly, if b is greater than 0 and less than 1, b is a decay factor. In this case, the association is negative, and y is decreasing. From the decay factor, we can determine the percentage decrease in y for each additional 1-unit increase in x .
Let’s compare the general form of an exponential model to the general form for a linear model: y = a + bx .
- In the linear model, a is the initial value . It is the y -value when x = 0. It is also the y -intercept.
- b is the slope . From the slope, we can determine the amount and direction the y -value changes for each additional 1-unit increase in x . When b is positive, there is a positive association, and y increases. When b is negative, there is a negative association, and y decreases.
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