# How do you graph f(x) = (4x^2-36x) / (x-9)?

Apr 22, 2018

see below

#### Explanation:

you can first simplify this expression:

$4 {x}^{2} - 36 x = 4 x \left(x - 9\right)$

$\frac{4 {x}^{2} - 36 x}{x - 9} = \frac{4 x \left(x - 9\right)}{x - 9} = 4 x$

therefore, for all points where $\frac{4 {x}^{2} - 36 x}{x - 9}$ can be defined, you'll get a graph of $f \left(x\right) = 4 x$.

however, not all points can be defined.

any number divided by zero is undefined. this means that the $x$-value where the denominator $x - 9$ is $0$ is also undefined.

when $x - 9 = 0$, $x = 9$.

this means that the line cannot touch any point where $x = 9$.

however, in all other ways, it will look like the graph of $f \left(x\right) = 4 x$.

this gives a straight line with gradient $4$, and with a hole where the point on the $x$-axis is $9$:

graph{(4x^2-36x)/(x-9) [2.7, 22.7, 32.56, 42.56]}

if you scroll along the graph, you'll see a directly proportional relationship between $x$ and $y$, where the $y$-coordinate is $4$ times the $x$-coordinate.

if you scroll up to where $x = 9$, you'll see that the coordinates are
($9$, undefined).