• Given the tables of data, you can find the rate of change (the slope of the line) by applying the following formula:
[tex]m=\frac{y_2-y_1}{x_2-x_1_{}_{}}[/tex]Where the following are two points on the line:
[tex]\begin{gathered} (x_1,y_1)_{} \\ (x_2,y_2) \end{gathered}[/tex]- Table A:
Knowing these points:
[tex]\mleft(-2,-3\mright);(0,0)[/tex]You can substitute the corresponding coordinates into the formula and evaluate:
[tex]\begin{gathered} m_A=\frac{-3-0}{-2-0}=\frac{-3}{-2}=\frac{3}{2} \\ _{}_{} \end{gathered}[/tex]- Table B:
Given the points:
[tex]\mleft(0,-1\mright);\mleft(8,7\mright)[/tex]You get:
[tex]m_B=\frac{-1-7}{0-8}=\frac{-8}{-8}=1[/tex]- Table C:
Having:
[tex]\mleft(-1,-2\mright);\mleft(1,2\mright)[/tex]You get that the slope is:
[tex]m_C=\frac{-2-2}{-1-1}=\frac{-4}{-2}=2[/tex]- Table D:
Knowing these two points on the line:
[tex]\mleft(-3,3\mright);\mleft(4,-4\mright)[/tex]You get the following slope:
[tex]m_D=\frac{-4-3}{4-(-3)}=\frac{-7}{4+3}=\frac{-7}{7}=-1[/tex]• By definition, the equation of a Proportional Relationship has the following form:
[tex]y=mx[/tex]Where the slope "m" is the constant.
That indicates that the value of "y" varies proportionally with the value of "x".
As you can notice, its graph is a line that passes through the Origin.
Therefore, knowing the slope of the Proportional Relationships A, C and D, you can set up the following equations for each one of them:
- For Line A:
[tex]y=\frac{3}{2}x[/tex]- For Line C:
[tex]y=2x[/tex]- For Line D:
[tex]\begin{gathered} y=-1x \\ y=-x \end{gathered}[/tex]Hence, the answers are:
• Rates of change:
[tex]\begin{gathered} m_A=\frac{3}{2} \\ \\ m_B=1 \\ \\ m_C=2 \\ \\ m_D=-1 \end{gathered}[/tex]
• Function rules for each Proportional Relationship:
[tex]\begin{gathered} y=\frac{3}{2}x\text{ (Line A)} \\ \\ y=2x\text{ (Line C)} \\ \\ y=-x\text{ (Line D)} \end{gathered}[/tex]
