Question

Write a linear function f with the given values f(-2)=0, f(6)=-4

Answer 1

The Slope-Intercept form of the equation of a line is:

`y=mx+b`

Where "m" is the slope of the line and "b" is the y-intercept.

Knowing that:

`\begin{gathered} f\mleft(-2\mright)=0 \\ f(6)=-4 \end{gathered}`

You can identify that the line passes through these points:

`\begin{gathered} (-2,0) \\ (6,-4) \end{gathered}`

By definition, the value of "x" is zero when the line intersects the y-axis and the value of "y" is zero when the line intersects the x-axis.

To find the slope of a line you can use the following formula:

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

In this case, you can set up that:

`\begin{gathered} y_2=-4 \\ y_1=0 \\ x_2=6 \\ x_1=-2 \end{gathered}`

Then substituting values into the formula and evaluating, you get that the slope of the line is:

`m=(-4-0)/(6-(-2))=(-4)/(6+2)=(-4)/(8)=-(1)/(2)`

You can substitute the slope and the coordinates of the second point into the equation

`y=mx+b`

and then solve for "b":

`\begin{gathered} -4=-(1)/(2)(6)+b \\ \\ -4=-3+b \\ -4+3=b \\ b=-1 \end{gathered}`

Knowing "m" and "b", you can write the following equation of this line in Slope-Intercept form:

`y=-(1)/(2)x-1`

Rewriting it with:

`f(x)=y`

You get that the answer is:

`f(x)=-(1)/(2)x-1`