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

Sketching the graph Step 1: Find where the graph cuts the axes When x = 0, y = 4 / 3, so the graph goes through the point (0, 4 / 3 ). y ≠ 0 for any values of x as b 2 –4ac is negative for the top line, so the graph does not cut the x axis.

Step 2: Find the vertical asymptotes The denominator is not zero for any real values of x, so there are no vertical asymptotes. Sketching the graph

Step 3: Examine the behaviour as x tends to infinity Sketching the graph For numerically large values of x as x → , y → 1. Dividing out gives So y = 1 is a horizontal asymptote.

Step 3: Examine the behaviour as x tends to infinity For numerically large values of x, y → 1. For large positive values of x, y is slightly greater than 1. As x → ∞, y → 1 from above. Sketching the graph So y = 1 is a horizontal asymptote. Dividing out gives

Step 3: Examine the behaviour as x tends to infinity Sketching the graph For numerically large values of x, y → 1. For large positive values of x, y is slightly greater than 1. As x → ∞, y → 1 from above. So y = 1 is a horizontal asymptote. Dividing out gives

Step 3: Examine the behaviour as x tends to infinity Sketching the graph For numerically large values of x, y → 1. So y = 1 is a horizontal asymptote. Dividing out gives For large negative values of x, y is slightly less than 1. As x → -∞, y → 1 from below.

Step 3: Examine the behaviour as x tends to infinity Sketching the graph For numerically large values of x, y → 1. So y = 1 is a horizontal asymptote. Dividing out gives For large negative values of x, y is slightly less than 1. As x → ∞, y → 1 from below.

Step 4: Complete the sketch We can now complete the sketch. Sketching the graph

Step 4: Complete the sketch We need to find how the graph acts close to the origin – are there any TP`s? Find y` using the quotient rule Sketching the graph

Solving x = -3 or x = + 1 y = - 5 / 6 or y = + 1.5