1. Curve sketching
This PowerPoint presentation shows the different stages involved in sketching the graph

2. Sketching the graph Step 1: Find where the graph cuts the axes
The only place where this graph cuts either axis is at (0, 0).

3. Sketching the graph Step 2: Find the vertical asymptotes
The denominator is zero when x = -2 or x = 3
The vertical asymptotes are x = -2

4. Sketching the graph Step 2: Find the vertical asymptotes
The denominator is zero when x = -2 or x = 3
The vertical asymptotes are x = -2
and x = 3

5. Sketching the graph Step 2: Find the vertical asymptotes
The denominator is zero when x = -2 or x = 3
The vertical asymptotes are x = -2
and x = 3
For now, don’t worry about the behaviour of the graph near the asymptotes. You may not need this information.

6. Sketching the graph
Step 3: Examine the behaviour as x tends to infinity
For numerically large values of x, y → 0.
For large positive values of x, all three of x, (x + 2) and (x – 3) are positive, so y is positive.
As x → ∞, y → 0 from above.

7. Sketching the graph
Step 3: Examine the behaviour as x tends to infinity
For numerically large values of x, y → 0.
For large positive values of x, all three of x, (x + 2) and (x – 3) are positive, so y is positive.
As x → ∞, y → 0 from above.

8. Sketching the graph
Step 3: Examine the behaviour as x tends to infinity
For numerically large values of x, y → 0.
For large negative values of x, all three of x, (x + 2) and (x – 3) are negative, so y is negative.
As x → -∞, y → 0 from below.

9. Sketching the graph
Step 3: Examine the behaviour as x tends to infinity
For numerically large values of x, y → 0.
For large negative values of x, all three of x, (x + 2) and (x – 3) are negative, so y is negative.
As x → -∞, y → 0 from below.

10. Sketching the graph Step 4: Complete the sketch
Since the graph only crosses the x axis at the origin, we can complete the part of the graph to the right of x = 3

11. Sketching the graph Step 4: Complete the sketch
Since the graph only crosses the x axis at the origin, we can complete the part of the graph to the right of x = 3
and to the left of x = -2

12. Sketching the graph Step 4: Complete the sketch
Since the graph only crosses the x axis at the origin, we can complete the part of the graph to the right of x = 3
and to the left of x = -2

13. Sketching the graph Step 4: Complete the sketch
Now there is a difficulty. We know that the graph goes through the origin, but we don’t know whether it goes from positive to negative or negative to positive.
Try a value of x between -2 and 0.
You should find that y is positive in this case.

14. Sketching the graph Step 4: Complete the sketch
Now there is a difficulty. We know that the graph goes through the origin, but we don’t know whether it goes from positive to negative or negative to positive.
Try a value of x between -2 and 0.
You should find that y is positive in this case.

15. Sketching the graph Step 4: Complete the sketch
Now there is a difficulty. We know that the graph goes through the origin, but we don’t know whether it goes from positive to negative or negative to positive.
Try a value of x between 0 and 3.
You should find that y is negative in this case.

16. Sketching the graph Step 4: Complete the sketch
Now there is a difficulty. We know that the graph goes through the origin, but we don’t know whether it goes from positive to negative or negative to positive.
Try a value of x between 0 and 3.
You should find that y is negative in this case.

17. Sketching the graph Step 4: Complete the sketch
The sketch can now be completed.

18. Sketching the graph Step 4: Complete the sketch
The sketch can now be completed.