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 = 3, so the graph goes through the point (0, 3). When y = 0, there are no real values of x, so the graph does not cut the x-axis.

Sketching the graph Step 2: Find the vertical asymptotes The denominator is zero when x = -1 The vertical asymptote is x = -1

Sketching the graph Step 2: Find the vertical asymptotes The denominator is zero when x = -1 The vertical asymptote is x = -1 For now, don’t worry about the behaviour of the graph near the asymptote. You may not need this information.

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

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

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

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

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

Sketching the graph Step 4: Complete the sketch It is easy to complete the part of the graph to the right of the asymptote, which must pass through the point on the y axis.

Sketching the graph Step 4: Complete the sketch It is easy to complete the part of the graph to the right of the asymptote, which must pass through the point on the y axis.

Sketching the graph Step 4: Complete the sketch We can also complete the part of the graph to the left of the asymptote, remembering that the graph does not cut the x-axis.

Sketching the graph Step 4: Complete the sketch We can also complete the part of the graph to the left of the asymptote, remembering that the graph does not cut the x-axis.