Question

For each line find the slope between the 2 points given , simply each fraction to prove that the lines have a CONSTANT rate of change :1) Point T : 3) Point R :3) Point S :4) Slope of TR : 5) Slope of RS :6) Slope of TS :7) Describe the SLOPE of the line : 8) Therefore the CONSTANT rate of change is...?

Answer 1

For the information given in the statement and in the graph you have

*Point T has coordinates:

$(-5,0)$

*Point R has coordinates:

$(1,2)$

*Point S has coordinates:

$(7,4)$

Now to find the slopes of the segments you can use the slope formula, that is

$\begin{gathered} m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \text{ Where m is the slope of the line and} \\ (x_1,y_1),(x_2,y_2)\text{ are two points through which the line passes} \end{gathered}$

So, the slope of TR is

$\begin{gathered} (x_1,y_1)=(-5,0) \\ (x_2,y_2)=(1,2) \end{gathered}$$\begin{gathered} m=\frac{2-0}{1-(-5)} \\ m=\frac{2}{1+5} \\ m=\frac{2}{6} \\ \text{ Simplifying} \\ m=\frac{2\cdot1}{2\cdot3} \\ m=\frac{1}{3} \end{gathered}$

Now, the slope of RS is

$\begin{gathered} (x_1,y_1)=(1,2) \\ (x_2,y_2)=(7,4) \end{gathered}$$\begin{gathered} m=\frac{4-2}{7-1} \\ m=\frac{2}{6} \\ m=\frac{1}{3} \end{gathered}$

Now, the slope of TS is

$\begin{gathered} (x_1,y_1)=(-5,0) \\ (x_2,y_2)=(7,4) \end{gathered}$$\begin{gathered} m=\frac{4-0}{7-(-5)} \\ m=\frac{4}{7+5} \\ m=\frac{4}{12} \\ \text{ Simplifying} \\ m=\frac{4\cdot1}{4\cdot3} \\ m=\frac{1}{3} \end{gathered}$

For point 7, you know that a single line passes through two points, and since the segments, TR, RS, and TS have the same slope, that is, 1/3 then the SLOPE of the line is 1/3.

For point 8, you know that the constant rate of change with respect to the variable x of a linear function is the slope of its graph. Therefore, the CONSTANT rate of change is 1/3.