Question

if $f(x)$ is a linear function, $f(-5)=3$, and $f(2)=5$, find an equation for $f(x)$. $f(x)=$

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, $x_1=-5,y_1 = 3,x_2=2,y_2 = 5$. So $m=\frac{5 - 3}{2-(-5)}=\frac{2}{7}$.

Step2: Use the point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$. We can use the point $(x_1,y_1)=(-5,3)$ and $m = \frac{2}{7}$. Then $y-3=\frac{2}{7}(x + 5)$.

Step3: Convert to slope - intercept form

Expand the right - hand side: $y-3=\frac{2}{7}x+\frac{10}{7}$. Add 3 to both sides: $y=\frac{2}{7}x+\frac{10}{7}+3=\frac{2}{7}x+\frac{10 + 21}{7}=\frac{2}{7}x+\frac{31}{7}$. Since $y = f(x)$, we have $f(x)=\frac{2}{7}x+\frac{31}{7}$.

Answer:

$f(x)=\frac{2}{7}x+\frac{31}{7}$