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 When x = 0, y = 2, so the graph goes through the point (0, 2). When y = 0, x = 2, so the graph goes through the point (2, 0).

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

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

5. Step 3: Examine the behaviour as x tends to infinity For numerically large values of x, y → -2. This means that y = -2 is a horizontal asymptote. Sketching the graph Dividing out gives

6. Step 3: Examine the behaviour as x tends to infinity Sketching the graph For numerically large values of x, y → -2. This means that y = -2 is a horizontal asymptote. Dividing out gives

7. Step 3: Examine the behaviour as x tends to infinity Sketching the graph For large positive values of x, y is slightly greater than -2. So as x → ∞, y → -2 from above. For numerically large values of x, y → -2. This means that y = -2 is a horizontal asymptote. Dividing out gives

8. Step 3: Examine the behaviour as x tends to infinity Sketching the graph For large positive values of x, y is slightly greater than -2. So as x → ∞, y → -2 from above. For numerically large values of x, y → -2. This means that y = -2 is a horizontal asymptote. Dividing out gives

9. Step 3: Examine the behaviour as x tends to infinity Sketching the graph For large negative values of x, y is slightly less than -2. So as x → -∞, y → -2 from below. For numerically large values of x, y → -2. This means that y = -2 is a horizontal asymptote. Dividing out gives

10. Step 3: Examine the behaviour as x tends to infinity Sketching the graph For large negative values of x, y is slightly less than -2. So as x → -∞, y → -2 from below. For numerically large values of x, y → -2. This means that y = -2 is a horizontal asymptote. Dividing out gives

11. Sketching the graph Step 4: Complete the sketch Since the graph only crosses the x axis at (2, 0), we can complete the part of the graph to the left of the asymptote.

12. Step 4: Complete the sketch Since the graph only crosses the x axis at (2, 0), we can complete the part of the graph to the left of the asymptote. Sketching the graph

13. Step 4: Complete the sketch We can also complete the part of the graph to the right of the asymptote, using the points where the graph cuts the axes. Sketching the graph

14. Step 4: Complete the sketch Sketching the graph We can also complete the part of the graph to the right of the asymptote, using the points where the graph cuts the axes. Notice that in fact we did not need to know whether the graph was above or below the horizontal asymptote for numerically large x. The sketch shows the only possibility!