Mathematics » Functions I » Quadratic Functions

Functions of the Form y = ax2 + q

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:

  1. \({y}_{1}={x}^{2}-2\)

  2. \({y}_{2}={x}^{2}-1\)

  3. \({y}_{3}={x}^{2}\)

  4. \({y}_{4}={x}^{2}+1\)

  5. \({y}_{5}={x}^{2}+2\)

\(x\)

\(-\text{2}\)

\(-\text{1}\)

\(\text{0}\)

\(\text{1}\)

\(\text{2}\)

\({y}_{1}\)

     

\({y}_{2}\)

     

\({y}_{3}\)

     

\({y}_{4}\)

     

\({y}_{5}\)

     

Use your results to deduce the effect of \(q\).

Complete the table and plot the following graphs on the same system of axes:

  1. \({y}_{6}=-2{x}^{2}\)

  2. \({y}_{7}=-{x}^{2}\)

  3. \({y}_{8}={x}^{2}\)

  4. \({y}_{9}=2{x}^{2}\)

\(x\)

\(-\text{2}\)

\(-\text{1}\)

\(\text{0}\)

\(\text{1}\)

\(\text{2}\)

\({y}_{6}\)

     

\({y}_{7}\)

     

\({y}_{8}\)

     

\({y}_{9}\)

     

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).

  • 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.

  • 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.

  • 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 \(0<a<1\), as \(a\) gets closer to \(\text{0}\), the graph of \(f(x)\) gets wider.

  • 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 \(-1<a<0\), as \(a\) gets closer to \(\text{0}\), the graph of \(f(x)\) gets wider.

Functions of the Form <em>y = ax<sup>2</sup> + q</em>

 

\(a<0\)

\(a>0\)

\(q>0\)

Functions of the Form <em>y = ax<sup>2</sup> + q</em>Functions of the Form <em>y = ax<sup>2</sup> + q</em>

\(q=0\)

Functions of the Form <em>y = ax<sup>2</sup> + q</em>Functions of the Form <em>y = ax<sup>2</sup> + q</em>

\(q<0\)

Functions of the Form <em>y = ax<sup>2</sup> + q</em>Functions of the Form <em>y = ax<sup>2</sup> + q</em>
Table: The effect of \(a\) and \(q\) on a parabola.

You can use this Phet simulation to help you see the effects of changing \(a\) and \(q\) for a parabola.

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