I wanted to check the work I've done in these problems. This is a 7 step process to draw the graph of a function.

The steps to follow are:

- Determine domain
- Determine the x and y intercepts
- Find the vertical and horizontal asymptotes
- Determine the interval of decreaing and increasing.
- Determine local maximum and minimum values
- Determine the interval of concave up and concave down. Also find the inflection points.
- Sketch the graph.

So here are the problems that I've done so far, I haven't finished them all the way but I just want to make sure I'm doing them right so far:

1. y = x/x^2-9

- Domain is all real numbers except 3 and -3.
- y intercept: x = 3, x = -3 Mr F says:
**No.** The y-intercept is got by substituting x = 0. You get y = 0. So y-intercept at (0, 0). - x intercept: y = 0 Mr F says:
**No.** This is the equation of a line. It doesn't represent a point. **Points have coordinates**, not equations. The x-intercept is (0, 0), NOT y = 0. - Vertical Aysmptote: x = -3 , x = 3 Horizontal Asymptote: y = 0
- Increasing on: (-(infinity), 3) Decreasing: (3, (infinity)) Mr F says:
**No.** There are three intervals you need to consider: (-oo, -3), (-3 3) and (3, +oo). - No local max or min
- concave up: (3, (infinity)) concave down: (-(infinity), -3) Mr F says: There are two other intervals you need to consider ..... Note that there's a stationary point of inflection at (0, 0).

2. y = x^2/x^2+9

- Domain is all real numbers
- y intercept: x=0 x intercept: y=0 Mr F says: See above.
- No V.A. H.A.: y=1
- Increasing on: (0, (infinity)) Decreasing on: (-(infinity),0)
- No local max, local min at x=0 Mr F says: Coordinates are required! (0, 0).
- concave up: (-1/4, 1/4) concave down: (-(infinity),-1/4)U(1/4, (infinity)) Mr F says:
**No.** The x-coordinates of the points of inflection are $\displaystyle {\color{red}x = \pm \sqrt{3}}$. You're using the **y-coordinates** of the points of inflection.

3. y= x/x^3-1

- Domain, all real numbers except 1.
- V.A. at x=1 H.A. at y=0
- Mr F says: What about x- and y-intercepts?
- increasing: (-(infinity), 1) Mr F says: How can this be correct when there's a local maximum with x-coordinate $\displaystyle {\color{red}x = -\sqrt[3]{\frac{1}{2}}}$?? It is also logically inconsistent if there was a maximum at x = -1.
- decreasing; (1,(infinity)
- local max at x=-1 Mr F says: No. See above.

4. y=x/(x^2-1)^1/2

- Domain: (-(infinity),-1)U(1, (infinity))
- V.A.: x=-1 and x=1 H.A.: y=0
- _
- Couldn't find any intervals of increase or decrease Mr F says: Look harder. There is. Decreasing for x < -1. Increasing for x > 1.
- No local max or min