## 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:

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

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

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

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

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

\(x\) | \(-\text{2}\) | \(-\text{1}\) | \(\text{0}\) | \(\text{1}\) | \(\text{2}\) |

\({y}_{1}\) | |||||

\({y}_{2}\) | |||||

\({y}_{3}\) | |||||

\({y}_{4}\) | |||||

\({y}_{5}\) |

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

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

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

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

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

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

\(x\) | \(-\text{2}\) | \(-\text{1}\) | \(\text{0}\) | \(\text{1}\) | \(\text{2}\) |

\({y}_{6}\) | |||||

\({y}_{7}\) | |||||

\({y}_{8}\) | |||||

\({y}_{9}\) |

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.

\(a<0\) | \(a>0\) | |

\(q>0\) | ||

\(q=0\) | ||

\(q<0\) |

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