Question

How do I do this question Do the data in the table represent a linear function? If so write a rule for the function

Answer 1

No

Step-by-step explanation

To see if this table represents a linear function, we have to find the slope - also called the average rate of change,

`m=(y_1-y_2)/(x_1-x_2)`

We can find it using the first two points in the table, (-2, -7) and (-1, 1),

`m=(1-(-7))/(-1-(-2))=(1+7)/(-1+2)=(8)/(1)=8`

Let's assume that this is a linear function. Then, the equation would be,

`y=8x+b`

Use the first point in the table to find the y-intercept, b,

`\begin{gathered} -7=8\cdot(-2)+b \\ -7=-16+b\text{ }\Rightarrow\text{ }b=16-7=9 \end{gathered}`

So, we have the equation,

`y=8x+9`

Now, we have to check if all the points in the table satisfy this equation. If they do, then the table represents a linear function and this is the equation,

`\begin{gathered} -7=8(-2)+9 \\ -7=-16+9 \\ -7=-7\text{ }\Rightarrow\text{ }true \end{gathered}`
`\begin{gathered} 1=8(-1)+9 \\ 1=-8+9 \\ 1=1\text{ }\Rightarrow\text{ }true \end{gathered}`
`\begin{gathered} 8=8\cdot0+9 \\ 8=8\text{ }\Rightarrow\text{ }false \end{gathered}`

The third point in the table does not satisfy the linear equation, but all the other points do.

Hence, this table does not represent a linear function.