Functions of the Form <em>y = <sup>a</sup>/<sub>x</sub> + q</em>
Functions of the form \(y=\dfrac{a}{x}+q\)
Optional Investigation: The effects of \(a\) and \(q\) on a hyperbola.
On the same set of axes, plot the following graphs:
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\({y}_{1}=\dfrac{1}{x}-2\)
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\({y}_{2}=\dfrac{1}{x}-1\)
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\({y}_{3}=\dfrac{1}{x}\)
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\({y}_{4}=\dfrac{1}{x}+1\)
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\({y}_{5}=\dfrac{1}{x}+2\)
Use your results to deduce the effect of \(q\).
On the same set of axes, plot the following graphs:
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\({y}_{6}=\dfrac{-2}{x}\)
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\({y}_{7}=\dfrac{-1}{x}\)
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\({y}_{8}=\dfrac{1}{x}\)
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\({y}_{9}=\dfrac{2}{x}\)
Use your results to deduce the effect of \(a\).
The effect of \(q\)
The effect of \(q\) is called a vertical shift because all points are moved the same distance in the same direction (it slides the entire graph up or down).
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For \(q>0\), the graph of \(f(x)\) is shifted vertically upwards by \(q\) units.
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For \(q<0\), the graph of \(f(x)\) is shifted vertically downwards by \(q\) units.
The horizontal asymptote is the line \(y = q\) and the vertical asymptote is always the \(y\)-axis, the line \(x = 0\).
The effect of \(a\)
The sign of \(a\) determines the shape of the graph.
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If \(a > 0\), the graph of \(f(x)\) lies in the first and third quadrants.
For \(a > 1\), the graph of \(f(x)\) will be further away from the axes than \(y = \dfrac{1}{x}\).
For \(0 < a < 1\), as \(a\) tends to \(\text{0}\), the graph moves closer to the axes than \(y = \dfrac{1}{x}\).
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If \(a < 0\), the graph of \(f(x)\) lies in the second and fourth quadrants.
For \(a < -1\), the graph of \(f(x)\) will be further away from the axes than \(y = -\dfrac{1}{x}\).
For \(-1 < a < 0\), as \(a\) tends to \(\text{0}\), the graph moves closer to the axes than \(y=-\dfrac{1}{x}\).
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\(a<0\) |
\(a>0\) |
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\(q>0\) |
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\(q=0\) |
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\(q<0\) |
Table: The effects of \(a\) and \(q\) on a hyperbola.
This lesson is part of:
Functions I