Compare consecutive $x$-values and their corresponding $f(x)$-values. How much does $x$ increase each time? How much does $f(x)$ increase each time?

The slope is the change in $f(x)$ divided by the change in $x$. Use any pair of points from the table to compute this ratio, and then relate that slope to $m$ in $f(x)=mx-28$.

Pick one point from the table, such as $(10,82)$, and substitute $x$ and $f(x)$ into $f(x)=mx-28$. Solve the resulting equation for $m$.

In Desmos, type the expression (82+28)/10 (which comes from rearranging $82=10m-28$ to $m=(82+28)/10$). The value Desmos outputs is the value of $m$ for this linear function.

The function is given by $f(x)=mx-28$, which is the equation of a line. In a linear equation of the form $y=mx+b$, the number $m$ is the slope, or the rate at which $f(x)$ changes when $x$ increases.

Look at how $x$ and $f(x)$ change between any two consecutive rows.

• From $x=10$ to $x=15$, $x$ increases by $5$.
• From $f(10)=82$ to $f(15)=137$, $f(x)$ increases by $137-82=55$.

So, over that interval, the slope (rate of change) is

You can quickly check another interval (for example, from $x=15$ to $x=20$) and you will see the change is still $55$ in $f(x)$ when $x$ changes by $5$, confirming the slope is constant.

Now simplify the fraction for the slope:

This slope is the same as the coefficient $m$ in $f(x)=mx-28$. Therefore, $m=11$.