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 and x = 3 The denominator is zero when x = -2 or x = 3 The vertical asymptotes are x = -2

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

6. Sketching the graph Step 3: Examine the behaviour as x tends to infinity The degree of the denominator is greater than the degree of the numerator, so for numerically large values of x, y → 0. For large positive values of x, x, all three of x, (x (x + 2) and (x (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 The degree of the denominator is greater than the degree of the numerator, so 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 The degree of the denominator is greater than the degree of the numerator, so for numerically large values of x, y → 0. For large negative values of x,x, all three of x, (x (x + 2) and (x (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 The degree of the denominator is greater than the degree of the numerator, so 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 and to the left of x = -2 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

12. Sketching the graph Step 4: Complete the sketch and to the left of x = -2 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

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.