Zooming in

Excerpt: This problem develops the idea that the gradient of a curve can be understood by zooming in to the graph. The collected results at the end can lead to a formula for the gradient function (derivative) of $x^2$ .

Activity

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 $2\,\text{cm}$ (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 $2\,\text{cm}$ (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 $2\,\text{cm}$ (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 $2\,\text{cm}$ (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 $y=x^2$ ?

Things you might have noticed

This interactivity shows what you might have drawn. You can select the value of $b$ and then zoom in to the graph at that point.

[interactive embed (geogebra-iframe): //www.geogebratube.org/material/iframe/id/R89wFrDq/width/429/height/518/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/false/ld/false/sri/true/asb/false/at/html5] — not part of the text import; see the source page.

Download GeoGebra file

You have probably noticed that the graph looks like a straight line when you zoom in far enough.

What is the gradient of the straight line you end up with?
Comparing your results with those of other groups or with the interactivity, how does the gradient depend on the value of $b$ you are using?
Does every graph look like a straight line when you zoom in far enough?

You can explore this more generally with the interactive graph we have provided.

After thinking about this problem, you might like to think about tangents or how you might develop these ideas to work out gradient functions for powers of $x$ .

You might also want to think about how this approach relates to the algebraic point of view explored in Binomials are the answer: the derivative there is found as the limit of the gradients of chords, using the algebraic expression $\dfrac{(x+h)^2-x^2}{(x+h)-x}$ .

Interactive graph

This interactivity offers the ability to explore the effect of zooming in on the graph of a function of your choice. You can also show a second point and the chord between them by clicking “Show triangle”.

[interactive embed (geogebra-iframe): //www.geogebratube.org/material/iframe/id/X8qK6jS8/width/810/height/493/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/false/ld/false/sri/true/asb/false/at/html5] — not part of the text import; see the source page.

Download GeoGebra file

Zooming in

Excerpt: This problem develops the idea that the gradient of a curve can be understood by zooming in to the graph. The collected results at the end can lead to a formula for the gradient function (derivative) of $x^2$ .

Activity

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.

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 $2\,\text{cm}$ (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 $2\,\text{cm}$ (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 $2\,\text{cm}$ (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 $2\,\text{cm}$ (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 $y=x^2$ ?

Things you might have noticed

This interactivity shows what you might have drawn. You can select the value of $b$ and then zoom in to the graph at that point.

[interactive embed (geogebra-iframe): //www.geogebratube.org/material/iframe/id/R89wFrDq/width/429/height/518/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/false/ld/false/sri/true/asb/false/at/html5] — not part of the text import; see the source page.

Download GeoGebra file

You have probably noticed that the graph looks like a straight line when you zoom in far enough.

What is the gradient of the straight line you end up with?
Comparing your results with those of other groups or with the interactivity, how does the gradient depend on the value of $b$ you are using?
Does every graph look like a straight line when you zoom in far enough?

You can explore this more generally with the interactive graph we have provided.

After thinking about this problem, you might like to think about tangents or how you might develop these ideas to work out gradient functions for powers of $x$ .

Interactive graph

[interactive embed (geogebra-iframe): //www.geogebratube.org/material/iframe/id/X8qK6jS8/width/810/height/493/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/false/ld/false/sri/true/asb/false/at/html5] — not part of the text import; see the source page.

Download GeoGebra file