For linear functions, the rate of change is constant, meaning that for each step in the x-direction, the change between y-values is the same. The rate of change for exponential functions, however, is not constant. Consider the following tables with the same x-values.
The difference between the y-values in the left table is constant (+2), but that's not the case for the right table.
The tables represent three different functions. Determine which is exponential.
An exponential function changes by a constant multiplier with each step in the x-direction. Notice that all tables have the same x-values, which increase by 1.
Next, we can analyze the y-values for each table individually. Let's start with the second table. Notice that the y-values increase by 3 each step. This means that the rate of change is constant. Therefore, the function is linear, and thus, not exponential.
Therefore, the function in the first table is the only exponential function.
The graph of a function with a constant rate of change will be a line, since for every step in the x-direction, the change in y is the same. For an exponential function, the change in the y-direction shows a constant multiplier, leading to a curve-shaped graph.
The y-intercept is usually called the initial value, and the factor by which a y-value is multiplied to get to the next is called the constant multiplier.
Use the points on the graph to determine if the function is exponential.
When taking another step to the right, we should land on (3,2) because 2 is half of 4.
For every step in the x-direction, the y-value changes by a multiplier of 0.5. We can assume that this behavior continues for the function that extends infinitely. Thus, the function is exponential.
An exponential function is a nonlinear function that can be written in the following form, where a≠0, b>0, and b≠1. As the independent variable x changes by a constant amount, the dependent variable y multiplied by a constant factor. Therefore, consecutive y-values form a constant ratio.