Functions of the Form <em>y = ax<sup>2</sup> + q</em>
Functions of the form \(y=a{x}^{2}+q\)
Optional Investigation: The effects of \(a\)and \(q\) on a parabola.
Complete the table and plot the following graphs on the same system of axes:
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\({y}_{1}={x}^{2}-2\)
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\({y}_{2}={x}^{2}-1\)
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\({y}_{3}={x}^{2}\)
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\({y}_{4}={x}^{2}+1\)
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\({y}_{5}={x}^{2}+2\)
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\(x\) |
\(-\text{2}\) |
\(-\text{1}\) |
\(\text{0}\) |
\(\text{1}\) |
\(\text{2}\) |
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\({y}_{1}\) |
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\({y}_{2}\) |
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\({y}_{3}\) |
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\({y}_{4}\) |
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\({y}_{5}\) |
Use your results to deduce the effect of \(q\).
Complete the table and plot the following graphs on the same system of axes:
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\({y}_{6}=-2{x}^{2}\)
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\({y}_{7}=-{x}^{2}\)
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\({y}_{8}={x}^{2}\)
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\({y}_{9}=2{x}^{2}\)
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\(x\) |
\(-\text{2}\) |
\(-\text{1}\) |
\(\text{0}\) |
\(\text{1}\) |
\(\text{2}\) |
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\({y}_{6}\) |
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\({y}_{7}\) |
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\({y}_{8}\) |
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\({y}_{9}\) |
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. The turning point of \(f(x)\) is above the \(y\)-axis.
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For \(q<0\), the graph of \(f(x)\) is shifted vertically downwards by \(q\) units. The turning point of \(f(x)\) is below the \(y\)-axis.
The effect of \(a\)
The sign of \(a\) determines the shape of the graph.
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For \(a>0\), the graph of \(f(x)\) is a “smile” and has a minimum turning point at \((0;q)\). The graph of \(f(x)\) is stretched vertically upwards; as \(a\) gets larger, the graph gets narrower.
For \(0
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For \(a<0\), the graph of \(f(x)\) is a “frown” and has a maximum turning point at \((0;q)\). The graph of \(f(x)\) is stretched vertically downwards; as \(a\) gets smaller, the graph gets narrower.
For \(-1
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\(a<0\) |
\(a>0\) |
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\(q>0\) |
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\(q=0\) |
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\(q<0\) |
You can use this Phet simulation to help you see the effects of changing \(a\) and \(q\) for a parabola.
This lesson is part of:
Functions I