Information Section:
This information section objectives are:
1. Ability to find equations for the rotation of images of figures.
2. Ability to state and apply properties of ellipses and hyperbolas that allows for drawing or describing them.
Background...
Another brilliant example of mathematica modleing, which involved ellipses occurred in the late 19th century as part of the work of Francis Galton (1822-1911), an English scientist and statistician.
Galton was attempting to show the influences of heredity on height. He tabulated heights of 928 adult children from 205 families against the mean heights of their parents (he called mid-parent height). He then groupes the data to plot frequencies as shown in the graph in the link below.
He discovered that, in this grouped data, the frequencies with the same values lay roughly on a series of similar ellipses with the same center and whose axes were at the same angle of inclination. Check the points with a frequency 3 that lie just outside of the ellipse drawn on the chart.
Galton showed his findings to a mathematician, J. Hamilton Dickson, who proved Galton's discovery was the result of these heights bein related by a bivariate normal distribution. i.e. a three-dimensional curve with two planes of symmetry in which each cross section parallel to a plane of symmetry is a normal curve.
This work of Galton led to the development of correclation and variance and their connection with regression and the line of best fit. The key item of note from this is that you can determine the equations of ellipses that are rotated so that their axes are no longer horizontal or vertical. The techniques for finding rotation images of graphs are provided in this lesson.
The Graph Rotation Theorem
So,
Now multiply the two matrices on the left side to get the following equation:
Now, if a sentence involving x and y is known, then the above expressions can be substituted for x and y and the result will be a sentence involving x' and y'. The new sentence describes the rotation image. Once the sentence for the rotation image is found, the primes in x' and y' are removed, so that the resulting sentence is in the variables x and y.
Graph-Rotation Theorem:
In a relation described by a sentence in x and y, the following two processes yield the same graph:
2. Applying the rotation of magnitude theta about the origin to the graph of the original equation.
Composites of Rotations and Translations
Review of isometry - a composite of translations, rotations and reflections.
Given this point, the Graph-Translation Theorem and the Graph-Rotation Theorem, together with the reflection over the line x = y, give the means to perform any isometry in the plane on any relation.
Investigation Section
Activity 1 - Using the Graph-Rotation Theorem
Find an equation for the image of the ellipse:
, then
A graph of the two equations is presented here. The blue ellipse shows the original plot. The red ellipse shows the equation rotated by 30 degrees.:
Activity 2 - Using the Graph-Rotation Theorem
Rotate the hyperbola :
In the above graph, the preimage is in blue and the image (rotated) is in red. Notice how the rotated image (hyperbola) is completely within the 2nd and the 4th quadrants. Also notice that its asymptotes are the x-axis and the y-axis.
When the asymptotes of a hyperbola are perpendicular (as they are in this case), the hyperbola is called a rectangular hyperbola.
Each is a rotation image of the other, so they are congruent.
Generally, all rectangular hyperbolas are similar.
Still more generally, two hyperbolas are similar, if and only if the angles between their asymptotes are congruent.
Activity 3 - Composites of Rotation and Translations
Find an equation for the image of the parabola:
This is the graph after the mapping onto the origin. Note that the vertex of the parabola is at (-5, -8).
This next graph shows the result after the mapping and then the rotation by 20 degrees. The vertex is approximately at (-2, -9.23364).
This final graph shows the remapping back to (5, 8). Note that the vertex is approximately (3, -1.25515).
