## Geometer's Sketchpad Investigation of Normal Curves

The curve

is an equation for the normal curve. Use the following steps
to create this curve in GSP and then observe changes to the curve.

#### Step 1

Create Axes in a new sketch. Construct a point *X* on
the x-axis and measure the coordinates of the point. Use the calculate
function to get the x-coordinate value of this point.

#### Step 2

Create two segments, *AB* and *CD*, that are each
smaller than 1 cm in length and measure each segment length.

#### Step 3

Use the calculator to evaluate the equation for the normal
curve (above) with the length of *AB* as the *a*-value
and the length of *CD* as the *b*-value. The x-coordinate
of *X* will be the x-value.

#### Step 4

Select the following in the order listed: the x-coordinate
value for point *X* (the x-value) and the result of the calculation
from Step 3 (the y-value). Then select Plot as (x, y) from the
Graph menu and label the point *N*.

#### Step 5

Select point *N* and point *X*, in that order. Construct
the locus.

If the value for b<1, then this should give a normal curve.
If the value for b>1, play with the scale of the graph until
it is normal.

#### Extensions

Investigate changes in the scale of the graph, the *a*-value,
and the *b*-value. What happens as these values change?

Create a new graph with segments that represent a value for
the standard deviation, s, and the sample size, n. (Create two
segments to represent s and n.) Vary the values of s and n and
see how the shape of the curve varies.

Click here to see a GSP construction
of the curve.

Reference for this investigation:

Kamischke, Ellen, Eric Kamischke, and Jerald Murdock. __Advanced
Algebra Through Data Exploration__. Berkeley, CA: Key Curriculum
Press, 1998.