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.