Question

How do you sketch the curve of `f(x) = (2x-3) / (2 x -9)^2`?

Answer 1

See explanation section.

Explanation:

`f(x) = (2x-3) / (2 x -9)^2`

Domain: `RR " - " {9/2}`

`x`-intercept: `3/2`
`y` intercept `-1/27`

Symmetry: None

Vertical Asymptote: `x = 9/2`
Horizontal Asymptote: `y = 0` (a.k.a. the `x`-axis)

`f'(x) = (-2(2x+3))/(2x-9)^3`

Sign analysis of `f'` shows that:
`f` is decreasing on `(-oo, -3/2)` and on `(9/2, oo)` and
`f` is increasing on `(-3/2, 9/2)`

Local Minimum `-1/24` (at `x=-3/2`)

`f''(x) = (8(2x+9))/(2x-9)^4`
Sign analysis of `f''` shows that:
`f` is concave down on `(-oo, -9/2)` and
`f` is concave up on `(-9/2, 9/2)` and on `(9/2, oo)`

`(-9/2, -1/27)` is the only inflection point.

The graph looks like this:

graph{y=(2x-3) / (2 x -9)^2 [-6.16, 11.62, -2.91, 5.985]}

(You can scroll in and out using a mouse wheel and also drag the graph around to see features more clearly.)