Functions of the Form <em>y = ab<sup>x</sup> + q</em>
Functions of the form \(y=a{b}^{x}+q\)
Optional Investigation: The effects of \(a\), \(q\) and \(b\) on an exponential graph.
On the same set of axes, plot the following graphs (\(a=1\), \(q=0\) and \(b\) changes):
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\(y_{1} = 2^{x}\)
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\(y_{2} = \left(\cfrac{1}{2}\right)^{x}\)
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\(y_{3} = 6^{x}\)
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\(y_{4} = \left(\cfrac{1}{6}\right)^{x}\)
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\(-\text{2}\) |
\(-\text{1}\) |
\(\text{0}\) |
\(\text{1}\) |
\(\text{2}\) |
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\(y_{1}=2^{x}\) |
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\(y_{2}=\left(\cfrac{1}{2}\right)^{x}\) |
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\(y_{3}=6^{x}\) |
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\(y_{4}=\left(\cfrac{1}{6}\right)^{x}\) |
Use your results to deduce the effect of \(b\).
On the same set of axes, plot the following graphs (\(b=2\), \(a=1\) and \(q\) changes):
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\({y}_{5}={2}^{x}-2\)
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\({y}_{6}={2}^{x}-1\)
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\({y}_{7}={2}^{x}\)
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\({y}_{8}={2}^{x}+1\)
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\({y}_{9}={2}^{x}+2\)
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\(-\text{2}\) |
\(-\text{1}\) |
\(\text{0}\) |
\(\text{1}\) |
\(\text{2}\) |
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\({y}_{5}={2}^{x}-2\) |
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\({y}_{6}={2}^{x}-1\) |
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\({y}_{7}={2}^{x}\) |
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\({y}_{8}={2}^{x}+1\) |
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\({y}_{9}={2}^{x}+2\) |
Use your results to deduce the effect of \(q\).
On the same set of axes, plot the following graphs (\(b=2\), \(q=0\) and \(a\) changes).
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\({y}_{10}=1 \times {2}^{x}\)
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\({y}_{11}=2\times {2}^{x}\)
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\({y}_{12}=-1 \times {2}^{x}\)
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\({y}_{13}=-2\times {2}^{x}\)
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\(-\text{2}\) |
\(-\text{1}\) |
\(\text{0}\) |
\(\text{1}\) |
\(\text{2}\) |
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\({y}_{10}=1 \times {2}^{x}\) |
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\({y}_{11}=2\times {2}^{x}\) |
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\({y}_{12}=-1 \times {2}^{x}\) |
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\({y}_{13}=-2\times {2}^{x}\) |
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 is shifted vertically upwards by \(q\) units.
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For \(q<0\), the graph is shifted vertically downwards by \(q\) units.
The horizontal asymptote is shifted by \(q\) units and is the line \(y=q\).
The effect of \(a\)
The sign of \(a\) determines whether the graph curves upwards or downwards.
For \(0 < b < 1\):
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For \(a>0\), the graph curves downwards. It reflects the graph about the horizontal asymptote.
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For \(a<0\), the graph curves upwards.
For \(b > 1\):
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For \(a>0\), the graph curves upwards.
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For \(a<0\), the graph curves downwards. It reflects the graph about the horizontal asymptote.
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\(b>1\) |
\(a<0\) |
\(a>0\) |
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\(q>0\) |
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\(q<0\) |
The effect of \(a\) and \(q\) on an exponential graph when \(b > 1\).
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\(0 |
\(a<0\) |
\(a>0\) |
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\(q>0\) |
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\(q<0\) |
The effect of \(a\) and \(q\) on an exponential graph when \(0 < b < 1\).
This lesson is part of:
Functions I