## Final Examination

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Due to space considerations, this write up contains three different web pages.

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### Ia. Hyperbola

The graph of xy = c is a hyperbola, with the cordinate axes as the asymptotes. The graph below is the graph of xy = 9. How can the graph be modified so that the center of the hyperbola is not at the origin? If the equations for a hyperbola centered at the point (h,k), that opens either vertically or horizontally, are examined and expanded, they contain quadratic terms, linear terms and a constant term.

In all the equations, h and k are constants that represent the coordinates of the center of the hyperbola, a and b are also constants. 2a defines the difference of the focal radii and b is defined in the equation where c is the distance from the center to the foci. The equation xy = c, has a term of degree two, and a constant term, what it is missing is linear terms. Equations in the form xy = ax + by + c will be examined to determine the effect of adding linear terms. The first equation that we will examine is xy = x + y + 1.

This graph is the graph of a hyperbola whose center has been shifted from the origin. Let us briefly examine the math behin this shift. The equation starts as xy = c, and the center is shifted by substituting (x-h) and (y-k) for x and y.

From the last equation, the graphed hyperbola should be centered at (1,1). In the terminology being used, k=a, h=b and the new constant ic c- hk. There should be one vertical asymptote and one horizontal asymptote, with equations y=1 and x=1. If the lines y=1 and x=1 are graphed, they appear to be the asymptotes of the original graph.

The effects of changing a, b and c should now be examined. When the center (h,k) is changed, the a and b change in the resulting equation. If a*b = - c, the resulting equation can be factored and solved for the variables x and y. The example below shows the factoring xy = -3x + 2y + 6.

Whenthe equation xy = -3x+ 2y+6 is graphed, there should be a vertical line, x = 2, and a vertical line, y =-3. As can be seen in the graph below, this does occur. The preceding example is graphed in the color green. Several other equations have been graphed demonstrating this behavior.

If a*b = - c, the result is two perpendicular lines intersecting at (b,a). When a*b < - c and a*b > - c will also be examined. If a*b > - c the graph will open towards the first and third quadrants, while if a*b < - c, the graph graph will open towards the second and fourth quadrants. The closer the product ab is to -c, the closer the foci of the hyperbolas are to each other. As ab and c grow further apart, the hyperbolas' foci spread out. This can be seen in the red, green and blue graphs, as well as in the brown and purple graphs.

The majority of the equations graphed so far in this investigation have had a=b. The relationship between a and b and the shape of the graph will be examined. The coefficients a and b are give the coordinates of the center of the hyperbola, (b,a). If either a or b is changed, the two transformations of the graph occur. The first transformation that occurs is that either the x or y coordinate of the center has changed. The other result is that the product a*b has changed, so that the distance between the foci has changed as well. This effect can be seen in the red and green hyperbolas, as well as in the brown and blue hyperbols graphed below.

If the foci and difference between the foci are set, the hyperbola can be graphed. An example is foci (-5,-5) and (3,3) and difference of focal radii=8. The center should be at the point (-1,-1), the midpoint along the segment between the foci. Using the definition of a hyperbola, the center can be found.

After completing the math above, the center is at (-1,-1). The graph of the hyperbola is red, and the foci of the hyperbola are the blue points.

The previous mathematical derivation could be applied with variables for the coordinates of the foci and difference of focal radii to determine a general formula for a hyperbola whose foci are not on a horizontal or vertical line.The same mathematical method could be applied to ellipses to determine the general equation of an ellipse whose foci are not on a horizontal or vertical line. The same method could also be used to find the general formula for a parabola whose vertex, and focus are not on a vertical or horizontal line.

### Ib. Box Maximizations

Most of the population would like to get the maximum return possible, with as little work or materials as necessary. An example problem that can be considered is the maximization of the volume of a box, given a flat sheet of cardboard to begin with. The box is created by removing the squares from each of the corners and folding up the sides. In the diagram below, the entire rectangle is the original sheet of material, the red squares at the corners are the material that would be removed and the white lines are where the folds would occur to create the sides.

Geometer SketchPad was used to create the image above. GSP, if the construction is done properly, will be able to animate the diagram to show the changing of the size of the square. The diagram below shows the relative sizes of the areas for the maximum volume of the box.

Another useful feature of GSP is that it will make tables of measured values and calculations. One draw back is that it is extremely difficult to get exact values this way. Unless the problem happens to have nice neat solutions, the best GSP can do is approximate the answers. The table below shows the values that I was able to measure using GSP, notice the inaccuracy of the answers when a volume of 400 cubic inches was desired. 513.04 was the largest volume that I was able to measure.

We need to look to methods that will overcome the weaknesses of GSP. If an algebraic graphing program is used, by entering the correct equation, we can get a graph that plots volume as a function of the size of a cut. One nice feature of a graphical output is that it shows the pattern of the change in the volume all at once. The graph can be interpreted using a straigth edge to approximate the values of the volume and cut size.

One nice feature of a graphing program is that the range of the axis can be manipulated allowing the region of interest to be focused upon. Another feature is that lines can be graphed at exact values helping us with the interpretation of the graph.

In the graph below, the green line would help us interpret when the volume of the box is 400 cubic inches and the blue line would help us interpret the maximum volume.

A third way that this problem can be examined is by using a spread sheet. The spread sheet can be set up so that it will calculate the volume of a box given original dimensions and the size of a cut. A nice feature of the spread sheets is that the results can also be graphed to be interpreted graphically.

The spread sheet does not have to be graphed for the results to be identified. In the cells that are used to calculate the volume, the highest value can be identified. The size of the cut needed for any given volume will be displayed numerically.

A nice feature of the spread sheet is that by varying the first cut and by changing the size of the cuts, greater accuracy can be employed. In the cells above, notice that the increment in the size of the cut is only increased by 0.0000001 inches, and that the original cut size is fairly close to desired value. This close examination can be done for any value of cut size. The following graph shows the Volume of the Box as a function of the Cut Size.

Care needs to be exercised when the results are interpreted in any of the three methods. By calculating the size of the cut to 7 decimal places and the volume of the box to 12 decimal places in the spreadsheet, we may be ignoring significant digits of the original dimensions. The uses of thw information being calculated also needs to be considered. If an item is being constructed using a yard stick and scissors, the accuracy is limited to one decimal place, maybe two if we are lucky.

Final Examination

Click on one of the four choices to jump to the discussion of that question.

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