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