## 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.