Explain how to find the average rate of change of a function between two input values.

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The average rate of change between two input values of a function is the slope of the line connecting the coordinate pairs of the inputs and associated outputs.

Let x represent the input so the two inputs are `x_1" and "x_2` . If we let y represent the outputs, then `y_1` is the output associated with `x_1` and `y_2` is associated with `x_2` . Now we can think of the average rate of change as the change in the outputs divided by the change in inputs. Algebraically we can write this as `(Delta Y)/(Delta x)=(y_2-y_1)/(x_2-x_1)` where the Greek letter `Delta` delta represents change.

If we were to plot the input, output coordinate pairs `(x_1,y_1),(x_2,y_2)` in the coordinate plane, then the average rate of change is the slope of the line connecting the points. (Note that this line need not be part of the function except that it contains these two points.)

Suppose we throw a ball such that after one second of flight the ball is 2 meters above the ground, and after 3 seconds of flight the ball is 3 meters above the ground. If we wanted to know the rate at which the ball was rising (in meters per second), then the input is time in seconds, and the output is height in meters, so we have the "points" (1,2) and (3,3). The rate of change will be `m=(3-2)/(3-1)=1/2` or 1 meter every 2 seconds or 1/2 m/s.

Note that this is the average rate of change. As the ball flies, the rate at which it increases in height changes at each moment. But the line between the points is a decent approximation for the function itself. (The line is called a secant for the functions curve.)

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