# How do I find an equation of the line using function notation that goes through (5,8) parallel to f(x)= 3x- 8?

Jun 23, 2015

$y - 8 = 3 \left(x - 5\right)$ or, in standard form, $3 x - y = 7$
or in functional notation $f \left(x\right) = 3 x - 7$

#### Explanation:

$f \left(x\right) = 3 x - 8$ is the equation of a line in slope-intercept form with a slope of $3$.

All lines parallel to $f \left(x\right) = 3 x - 8$ have the same slope.

Temporarily, writing $y$ in place of $f \left(x\right)$ :
the equation of a line through $\left(\hat{x} , \hat{y}\right) = \left(5 , 8\right)$ with a slope of $m = 3$ can be written in point slope form as:
$\textcolor{w h i t e}{\text{XXXX}}$$\left(y - \hat{y}\right) = m \left(x - \hat{x}\right)$
or
$\textcolor{w h i t e}{\text{XXXX}}$$\left(y - 8\right) = 3 \left(x - 5\right)$

We can simplify this:
multiplying through the right side:
$\textcolor{w h i t e}{\text{XXXX}}$$y - 8 = 3 x - 15$
subtracting $y$ from both sides:
$\textcolor{w h i t e}{\text{X8XXX}}$$- 8 = 3 x - 15 - y$
adding $15$ to both sides
$\textcolor{w h i t e}{\text{XXXX}}$$7 = 3 x - y$

Or, going back to $y - 8 = 3 x - 15$ and restoring $f \left(x\right)$ for $y$:
$\textcolor{w h i t e}{\text{XXXX}}$$f \left(x\right) - 8 = 3 x - 15$
$\textcolor{w h i t e}{\text{XXXX}}$$f \left(x\right) = 3 x - 7$