Chapter 4.2, Problem 19PPE

To determine

The graph that shows the time that the runner has been running as a function of the distance she has travelled

Explanation of Solution

Given information:

The graph that shows the distance a runner has travelled as a function of the amount of time she has been running

Graph:

  

First find the equation of the given graph.

From the graph, use points $\left(6,1\right),\left(12,2\right)$ to find the equation of the given graph.

Equation of a line is given by $y-y_1=\left(\frac{y_2-y_1}{x_2-x_1}\right)(x-x_1)$

Put $\left(x_1,y_1\right)=\left(6,1\right),\left(x_2,y_2\right)=\left(12,2\right)$

  $\begin{array}{l}y-1=\left(\frac{2-1}{12-6}\right)(x-6)\\ \Rightarrow y-1=\frac{1}{6}(x-6)\\ \Rightarrow 6(y-1)=x-6\\ \Rightarrow x=6+6(y-1)\\ \Rightarrow x=6+6y-6\\ \Rightarrow x=6y\end{array}$

To draw the graph that shows the distance a runner has travelled as a function of the amount of time she has been running, exchange $x$ and $y$ in the equation $x=6y$

So, equation becomes $y=6x$

Now draw the graph of equation $y=6x$

First find points that can be used to plot the graph of $y=6x$

At $x=0,y=6(0)=0$

At $x=1,y=6(1)=6$

At $x=-1,y=6(-1)=-6$

At $x=\frac{1}{2},y=6\left(\frac{1}{2}\right)=3$

At $x=\frac{-1}{2},y=6\left(\frac{-1}{2}\right)=-3$

Plot points $(0,0),(1,6),(-1,-6),\left(\frac{1}{2},3\right),\left(\frac{-1}{2},-3\right)$