the slope of the graphed line is $\frac{2}{3}$. which formulas represent the line that is graphed? check all…

Explanation:

Step1: Recall point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. Given $m = \frac{2}{3}$ and points $(1,2)$ and $(4,4)$ on the line. For point $(1,2)$: Substitute into point - slope form, we get $y - 2=\frac{2}{3}(x - 1)$. For point $(4,4)$: Substitute into point - slope form, we get $y - 4=\frac{2}{3}(x - 4)$.

Step2: Recall slope - intercept form

The slope - intercept form is $y=mx + b$. Using the point $(1,2)$ and $m=\frac{2}{3}$, substitute into $y=mx + b$: $2=\frac{2}{3}(1)+b$. Solve for $b$: $2=\frac{2}{3}+b$, then $b=2-\frac{2}{3}=\frac{6 - 2}{3}=\frac{4}{3}$. So the equation is $y=\frac{2}{3}x+\frac{4}{3}$ or $f(x)=\frac{2}{3}x+\frac{4}{3}$.

Answer:

$y - 2=\frac{2}{3}(x - 1)$, $y - 4=\frac{2}{3}(x - 4)$, $f(x)=\frac{2}{3}x+\frac{4}{3}$