Exploring Properties of a Family of Curves

The expression "b = 1 to 5.5; stepsize = 0.5" in the Work window represents an iterated constant. This particular expression represents the set of constants {1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5}. When you plot a function expression containing an iterated constant, for example, f(x) = x exp(-b x), a curve will be plotted for each value of the iterated constant.

Plot f(x) = x exp(-b x) on 0x5 and 0y0.4.

Plot of the family f(x) = x exp(-b x)

The data table Maximums[10,2] in the Work window contains the coordinates of the maximum values of the family of curves.

Select the data table Maximums[10,2] in the Work window.
Select Overlay Plot from the Graph menu.

Overlay of Maximums

Does it appear that the points located at the maximum values all lie on a straight line?

Can you find the equation of this straight line?

Select New Function from the Work window and enter the expression for your line into the first cell of the Work window.
Select Overlay Plot from the Graph menu. Does your line pass through the maximums?
The hidden function "MaxLine(x) = Hidden" contains the expression for the line. To see the expression, select the hidden function in the Work window and then select Show Hidden Function from the Options menu.

Overlay of MaxLine(x)

The data table PointsOfInflection[10,2] in the Work window contains the coordinates of the points of inflection of the family of curves.

Select the data table PointsOfInflection[10,2] in the Work window.
Select Overlay Plot from the Graph menu.

Overlay of PointsOfInflection

Does it appear that the points of inflection all lie on a straight line?

Can you find the equation of this straight line?

Select New Function from the Work window and enter the expression for your line into the first cell of the Work window.
Select Overlay Plot from the Graph menu. Does your line pass through the points of inflection?
The hidden function "POILine(x) = Hidden" contains the expression for the line. To see the expression, select the hidden function in the Work window and then select Show Hidden Function from the Options menu.

Overlay of POILine(x)

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