Question

Find the function for the line joining the points$\left(-2,1\right)$ and $\left(4,-6\right)$.

Short answer

The function for the line joining the given points is $f\left(x\right)=-\frac{7x+8}{6}$.

Step-by-step solution

Step 1. By two point form of a line.

The equation of a line through the points$\left(x_1,y_1\right),\left(x_2,y_2\right)$ is written as,

$y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Step 2. Find equation of line through the given points.

Using two point form of line for the given points,

$$\begin{gathered} y-1=\frac{-6-1}{4+2}\left(x+2\right)\mathrm{....}\left(\begin{array}{l}\left(x_1,y_1\right)\to \left(-2,1\right)\\ \left(x_2,y_2\right)\to \left(4,-6\right)\end{array}\right) \\ \text{or,}y-1=\frac{-7}{6}\left(x+2\right) \\ 6y-6=-7x-14....\left(\text{Crossmultiply}\right) \\ 6y=-7x-8 \\ y=-\frac{7x+8}{6} \end{gathered}$$

Step 3. DESCRIPTION OF STEPS.

By simplifying the given equation, we obtain$y=-\frac{7x+8}{6}$ from which it can be interpreted that for unique value of$\left(x\right)$ there exists a unique value of $\left(y\right)$. Therefore the given equation can be written in function form as $f\left(x\right)=-\frac{7x+8}{6}$.