Question

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

Answer 1

see below

Explanation:

you can first simplify this expression:

`4x^2 - 36x = 4x(x-9)`

`(4x^2-36x)/(x-9) = (4x(x-9))/(x-9) = 4x`

therefore, for all points where `(4x^2-36x)/(x-9)` can be defined, you'll get a graph of `f(x)= 4x`.

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(x) = 4x`.

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).