Curve sketching

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Curve sketching
This PowerPoint presentation shows the different
stages involved in sketching the graph
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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).
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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
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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
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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, dont worry about the behaviour of the
graph near the asymptotes. You may not need this
information.
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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 ? 8, y ? 0 from above.
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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 ? 8, y ? 0 from above.
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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 ? -8, y ? 0 from below.
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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 ? -8, y ? 0 from below.
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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
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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
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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
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Sketching the graph
Step 4 Complete the sketch
Now there is a difficulty. We know that the graph
goes through the origin, but we dont 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.
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Sketching the graph
Step 4 Complete the sketch
Step 4 Complete the sketch
Now there is a difficulty. We know that the graph
goes through the origin, but we dont 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.
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Sketching the graph
Step 4 Complete the sketch
Step 4 Complete the sketch
Now there is a difficulty. We know that the graph
goes through the origin, but we dont 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.
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Sketching the graph
Step 4 Complete the sketch
Now there is a difficulty. We know that the graph
goes through the origin, but we dont 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.
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Sketching the graph
Step 4 Complete the sketch
The sketch can now be completed.
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Sketching the graph
Step 4 Complete the sketch
The sketch can now be completed.