4.1 Linear functions  (Page 3/27)

Interpreting slope as a rate of change

In the examples we have seen so far, the slope was provided to us. However, we often need to calculate the slope given input and output values. Recall that given two values for the input, $x_1$ and $x_2,$ and two corresponding values for the output, $y_1$ and $y_2$ —which can be represented by a set of points, $\left(x_1\text{, }{y}_1\right)$ and $\left(x_2\text{, }{y}_2\right)$ —we can calculate the slope $m.$

Note that in function notation we can obtain two corresponding values for the output $y_1$ and $y_2$ for the function $f,$ $y_1=f\left(x_1\right)$ and $y_2=f\left(x_2\right),$ so we could equivalently write

[link] indicates how the slope of the line between the points, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right),$ is calculated. Recall that the slope measures steepness, or slant. The greater the absolute value of the slope, the steeper the slant is.