Question

Find the linear function f(x) for which both f(−2) = 6 and f(4) = −4. Which one of the following matches the correct function?Group of answer choicesf(x) = -1/2x + 5f(x) = -5/3x + 8/3f(x) = -1/2x - 2f(x) = 2x + 10

Answer 1

A linear function can be written in slope-intercept form, which is

`f(x)=mx+b`

Where m represents the slope and b represents the y-intercept.

We have two points that satisfies this equation. (-2, 6) and (4, -4). If we evaluate both points on the slope-intercept form we're going to create a linear system where the solutions are the values for the slope and y-intercept.

`\begin{gathered} 6=-2m+b \\ -4=4m+b \end{gathered}`

If we subtract the first equation from the second, we're going to have a new equation only for the slope.

`\begin{gathered} (6)-(-4)=(-2m+b)-(4m+b) \\ 6+4=-2m+b-4m-b \\ 10=-6m \\ m=-(10)/(6) \\ m=-(5)/(3) \end{gathered}`

Using this value for the slope on any of the previous equations, we can determinate the other coefficient

`\begin{gathered} 6=-2\cdot(-(5)/(3))+b \\ 6=(10)/(3)+b \\ b=6-(10)/(3) \\ b=(18)/(3)-(10)/(3) \\ b=(8)/(3) \end{gathered}`

Now that we have both coefficients, we can write our line equation.

`y=-(5)/(3)x+(8)/(3)`