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
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.
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Sketching the graph
Step 2 Find the vertical asymptotes
The denominator is zero when x -1
The vertical asymptote is x -1
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Sketching the graph
Step 2 Find the vertical asymptotes
The denominator is zero when x -1
The vertical asymptote is x -1
For now, dont worry about the behaviour of the
graph near the asymptote. You may not need this
information.
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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.
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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.
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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 ? 8, y ? x 2 from above.
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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 ? 8, y ? x 2 from below.
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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 ? 8, y ? x 2 from below.
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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.
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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.
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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.
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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.