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 The only place where this graph cuts either axis is at (0, 0).
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
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
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.
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.
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.
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.
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.
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
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
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
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.
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.
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.
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.
Sketching the graph Step 4: Complete the sketch The sketch can now be completed.
Sketching the graph Step 4: Complete the sketch The sketch can now be completed.