Question

if f(x) is a linear function, f(-1)=-2, and f(3)=-3, find an equation for f(x). f(x)= question help: video submit question

Explanation:

Step1: Find the slope

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $x_1=-1,y_1 = - 2,x_2=3,y_2=-3$. So $m=\frac{-3-(-2)}{3-(-1)}=\frac{-3 + 2}{3 + 1}=-\frac{1}{4}$.

Step2: Use the point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$. Using the point $(-1,-2)$ and $m =-\frac{1}{4}$, we have $y-(-2)=-\frac{1}{4}(x - (-1))$.

Step3: Simplify to slope - intercept form

$y + 2=-\frac{1}{4}(x + 1)$. Expand the right - hand side: $y+2=-\frac{1}{4}x-\frac{1}{4}$. Then $y=-\frac{1}{4}x-\frac{1}{4}-2=-\frac{1}{4}x-\frac{1 + 8}{4}=-\frac{1}{4}x-\frac{9}{4}$. Since $y = f(x)$, $f(x)=-\frac{1}{4}x-\frac{9}{4}$.

Answer:

$f(x)=-\frac{1}{4}x-\frac{9}{4}$