What happens if we “zoom in” to a graph?

What happens if we “zoom in” to a point on the graph of \(y=x^2\)?

In this activity, we are going to zoom in to different points on the graph of \(y=x^2\) to explore what happens when we do so.

This activity is best done in groups of three or four to share the workload. You will need graph paper.

There are four graph-drawing tasks, A, B, C and D. If you have four people in your group, take one task each; if you have only three, then share tasks A, B and C between you, and only do task D if you have time.

Your teacher may give you a value for \(b\), otherwise you can choose your own value for \(b\) from the choices \(-2\), \(-1.5\), \(-1\), \(-0.5\), \(0\), \(0.5\), \(1\), \(1.5\) and \(2\). **Everyone in the group should use the same value of \(b\).** You can work on the tasks simultaneously, and then compare your results at the end.

#### Task A

Complete the table of values of the function \(y=x^2\) and use this to draw a graph of \(y=x^2\) on graph paper for values of \(x\) between \(-3\) and \(3\).

Use a scale of \(\quantity{2}{cm}\) (one big square) per unit **on each axis**.

\(x\) | \(y=x^2\) |
---|---|

\(-3\) | |

\(-2.5\) | |

\(-2\) | |

\(-1.5\) | |

\(-1\) | |

\(-0.5\) | |

\(0\) | |

\(0.5\) | |

\(1\) | |

\(1.5\) | |

\(2\) | |

\(2.5\) | |

\(3\) |

When you have drawn the graph, draw a box on it between \(x=b–0.1\) and \(x=b+0.1\) to show where the graph for task B is.

#### Task B

Complete the table of values of the function \(y=x^2\) and use this to draw a graph of \(y=x^2\) on graph paper for values of \(x\) between \(x=b-0.1\) and \(x=b+0.1\).

Use a scale of \(\quantity{2}{cm}\) (one big square) per \(0.1\) unit **on each axis**.

\(x\) | \(x\) | \(y=x^2\) |
---|---|---|

\(b-0.1\phantom{00}\) | ||

\(b-0.075\) | ||

\(b-0.05\phantom{0}\) | ||

\(b-0.025\) | ||

\(b\phantom{{}+0.000}\) | ||

\(b+0.025\) | ||

\(b+0.05\phantom{0}\) | ||

\(b+0.075\) | ||

\(b+0.1\phantom{00}\) |

When you have drawn the graph, draw a box on it between \(x=b–0.01\) and \(x=b+0.01\) to show where the graph for task C is.

#### Task C

Complete the table of values of the function \(y=x^2\) and use this to draw a graph of \(y=x^2\) on graph paper for values of \(x\) between \(x=b-0.01\) and \(x=b+0.01\).

Use a scale of \(\quantity{2}{cm}\) (one big square) per \(0.01\) unit **on each axis**.

\(x\) | \(x\) | \(y=x^2\) |
---|---|---|

\(b-0.01\phantom{00}\) | ||

\(b-0.0075\) | ||

\(b-0.005\phantom{0}\) | ||

\(b-0.0025\) | ||

\(b\phantom{{}+0.0000}\) | ||

\(b+0.0025\) | ||

\(b+0.005\phantom{0}\) | ||

\(b+0.0075\) | ||

\(b+0.01\phantom{00}\) |

If there are four people in your group, then when you have drawn the graph, draw a box on it between \(x=b–0.001\) and \(x=b+0.001\) to show where the graph for task D is.

#### Task D

Complete the table of values of the function \(y=x^2\) and use this to draw a graph of \(y=x^2\) on graph paper for values of \(x\) between \(x=b-0.001\) and \(x=b+0.001\).

Use a scale of \(\quantity{2}{cm}\) (one big square) per \(0.001\) unit **on each axis**.

\(x\) | \(x\) | \(y=x^2\) |
---|---|---|

\(b-0.001\phantom{00}\) | ||

\(b-0.00075\) | ||

\(b-0.0005\phantom{0}\) | ||

\(b-0.00025\) | ||

\(b\phantom{{}+0.00000}\) | ||

\(b+0.00025\) | ||

\(b+0.0005\phantom{0}\) | ||

\(b+0.00075\) | ||

\(b+0.001\phantom{00}\) |

### Questions

Line up the graphs, so that you can see the progression as you zoom in on the point where \(x=b\). What do you notice?

How do your graphs compare with those of other groups (who are using different values of \(b\))?

What do you think might happen if we had started with a different curve instead of \(y=x^2\)?