Question

Find the rate of change for each table of data write a function rule for each proportional relationship

Answer 1

• Given the tables of data, you can find the rate of change (the slope of the line) by applying the following formula:

`m=\frac{y_2-y_1}{x_2-x_1_{}_{}}`

Where the following are two points on the line:

`\begin{gathered} (x_1,y_1)_{} \\ (x_2,y_2) \end{gathered}`

- Table A:

Knowing these points:

`\mleft(-2,-3\mright);(0,0)`

You can substitute the corresponding coordinates into the formula and evaluate:

`\begin{gathered} m_A=(-3-0)/(-2-0)=(-3)/(-2)=(3)/(2) \\ _{}_{} \end{gathered}`

- Table B:

Given the points:

`\mleft(0,-1\mright);\mleft(8,7\mright)`

You get:

`m_B=(-1-7)/(0-8)=(-8)/(-8)=1`

- Table C:

Having:

`\mleft(-1,-2\mright);\mleft(1,2\mright)`

You get that the slope is:

`m_C=(-2-2)/(-1-1)=(-4)/(-2)=2`

- Table D:

Knowing these two points on the line:

`\mleft(-3,3\mright);\mleft(4,-4\mright)`

You get the following slope:

`m_D=(-4-3)/(4-(-3))=(-7)/(4+3)=(-7)/(7)=-1`

• By definition, the equation of a Proportional Relationship has the following form:

`y=mx`

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:

`y=(3)/(2)x`

- For Line C:

`y=2x`

- For Line D:

`\begin{gathered} y=-1x \\ y=-x \end{gathered}`

Hence, the answers are:

• Rates of change:

`\begin{gathered} m_A=(3)/(2) \\ \\ m_B=1 \\ \\ m_C=2 \\ \\ m_D=-1 \end{gathered}`

• Function rules for each Proportional Relationship:

`\begin{gathered} y=(3)/(2)x\text{ (Line A)} \\ \\ y=2x\text{ (Line C)} \\ \\ y=-x\text{ (Line D)} \end{gathered}`