Calculate and Interpret Slope
Learning Outcomes
- Define slope for a linear function
- Calculate slope given two points
We often need to calculate the slope given input and output values. Given two values for the input, and , and two corresponding values for the output, and —which can be represented by a set of points, and —we can calculate the slope , as follows
where is the vertical displacement and is the horizontal displacement. Note in function notation two corresponding values for the output and for the function are and , so we could equivalently write
The graph below indicates how the slope of the line between the points, and is calculated. Recall that the slope measures steepness. The greater the absolute value of the slope, the steeper the line is.
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Calculating Slope
The slope, or rate of change, of a function can be calculated using the following formula: where and are input values, and are output values.Example
If is a linear function and and are points on the line, find the slope. Is this function increasing or decreasing?Answer: The coordinate pairs are and . To find the rate of change, we divide the change in output by the change in input.
We could also write the slope as . The function is increasing because . As noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or y-coordinate, used corresponds with the first input value, or x-coordinate, used.
Example
The population of a city increased from to between and . Find the change of population per year if we assume the change was constant from to .Answer: The rate of change relates the change in population to the change in time. The population increased by people over the four-year time interval. To find the rate of change, divide the change in the number of people by the number of years.
So the population increased by people per year. Because we are told that the population increased, we would expect the slope to be positive. This positive slope we calculated is therefore reasonable.