Question

How do you graph `f( x ) = \frac { x + 3} { x ^ { 2} - 2x - 63}`?

Answer 1

asymptotes: `x = -7, x = 9`
`x-`intercept: `(3,0)`
`y-`intercept: `(0, - 1/21)`

Explanation:

firstly, factorise `x^2-2x-63`:

`7+( -9) = -2`

`7 *( -9) = -63`

`x^2-2x-63 = (x+7)(x-9)`

`(x+3)/(x^2-2x-63) = (x-3)/((x+7)(x-9))`

in this form, it is easier to find any asymptotes that there may be.

noting that `n/0=` undefined:

`(x-3)/(x+7)` is undefined
when `x=-7, x+7 = 0`
this means that the graph cannot go through the line `x=-7`.

`(x-3)/(x-9)` is undefined
when `x=9, x-9 = 0`
this means that the graph cannot go through the line `x= 9`.

to plot the `x-`intercept:

use `0/n = 0`

the numerator in the function is `x-3`.

`y = 0` when `x-3=0`, so `y=0` when `x=3`.

this means that the coordinates of the `x-`intercept are `(3,0)`.

to plot the `y-`intercept:

set `x` to `0`.

`(x-3)/((x+7)(x-9)) = (-3)/(7*-9) = -(3)/(63) = -1/21`

when `x = 0`, `y = - 1/21`

this means that the coordinates of the `y-`intercept are `(0, - 1/21)`

graph{(x-3)/((x+7)(x-9)) [-10, 10, -5, 5]}