Geometry
Chapter 5
1. Handout 5310
2. Handout 5210
3. Handout 5110
4. Book 5.1: 1-10, 22-25, 27-36, 40-46
5. Handout 5510
6. Book 5.3 page 260: 11-13, 37-39
7. Book 5.4: 1, 3, 4, 16, 18, 42, 43
8. Handout 5520
9. Book 5.5: 1, 4, 7, 10, 13, 16, 19, 21, 22, 24, 25, 34, 35
10. Book 5.R: 23, 24, 26, 27, 29, 30-42 page 284: 5, 6
A rational function, in lowest terms, will have vertical asymptotes at the real zeros of the denominator.
Examples
This above example has the zero 3 in the denominator. The vertical line x = 3 is a boundary for this rational function. Mentally, do you see the line x = 3. It is a boundary, It is an asymptote. This graph also has a horizontal boundary or asymptote. More about that in a few moments.
This above example has two zeros in the denominator: -3 and 4. The vertical lines x = -3 and x = 4 are vertical asymptotes. The curve dares not to touch these vertical asymptotes.
This above example has two zeros in the denominator: 0 and -2. You did see that 0 is a double root, Right? The vertical lines x = 0 and x = -2 are vertical asymptotes. Do you see how the curve rises on both sides of the vertical asymptote x = 0. Because it was a double root, this is a characteristic of that. The curve will either run up or down each side of an asymptote that was a double root or more exactly a root that occurred an even amount of times.
We find the vertical asymptotes by solving a rational function denominator for zero. These also correspond to the domain restrictions for the function.
Vertical asymptotes are sacred ground, horizontal asymptotes are just useful suggestions. You can never touch a vertical asymptote, you can (and often do) touch and even cross horizontal asymptotes.
1. If the degree of the numerator is less than the degree of the denominator, the rational function is a proper rational function and will have the horizontal asymptote y = 0.
Examples
In the above example, the degree of the numerator is less than the degree of the denominator. When this happens, the rational function is proper and the horizontal asymptote is the horizontal line y = 0. The horizontal line y = 0 is a horizontal boundary and in this example it is not very strict — the middle curve crosses the horizontal asymptotes. It will sometimes do this with horizontal asymptotes, never a vertical asymptote. We also have the vertical asymptotes x = 0 and x = -4.
In the above example, the degree of the numerator is also less than the degree of the denominator. Again when this happens, the rational function is proper and the horizontal asymptote is the horizontal line y = 0. The horizontal line y = 0 is a horizontal boundary. We also have the vertical asymptotes x = 1 and x = -2.
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Is this a proper rational function? Why? Yes, the degree of the numerator is less than the degree of the denominator.
The horizontal asymptote is y = 0.
The vertical asymptote is x = -2. And do you see that the curve is rising on both sides of this asymptote. This is indicative of the fact that -2 was a double root. When that happens the curve will either rise or fall on both sides of the vertical asymptote.
The degree of the numerators in these examples is less than the degree of the denominators. When this happens, the rational function is proper and the horizontal asymptote is the horizontal line y = 0.
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2. If the degree of the numerator is greater than or equal to the degree of the denominator, the rational function is improper.
a. If the degree of the numerator is equal to the degree of the denominator, the quotient of the lead coefficients of the numerator and denominator is a horizontal asymptote.
Examples
In the above example, the degree for the numerator and denominator are both first. The horizontal asymptote is equal to the quotient of the lead coefficients. y = 2/1 = 2.
In the above example, the degrees for the numerator and denominator are both second. The horizontal asymptote is equal to the quotient of the lead coefficients. y = 1/1 = 1.
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b. If the degree of the numerator is exactly one more than the degree of the denominator, the rational function has a slant asymptote that you will derive by division techniques.
Examples
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In these two examples, the degree of the numerator is exactly one more than the degree of the denominator. Because of this, use division techniques, either long or synthetic division, to find the slant asymptote. Disregard any remainders.
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c. If the degree of the numerator is greater than one more than the degree of the denominator, the rational function has neither a horizontal nor a slant asymptote.
Since the degree of the numerator is more than one more than the denominator, there is no horizontal asymptote.
Graphing Rational Functions.
Step 1: Graph R using your graphing utility.
Step 2: Find the domain of the rational function.
Step 3: Write R in lowest terms.
Step 4: Locate the intercepts. The y-intercept is R(0). The x-intercepts are the roots of the rational function’s numerator in lowest terms.
Step 5: Identify any symmetry.
Step 6: Locate the vertical asymptotes. The vertical asymptotes are the zeros of the rational function’s denominator in lowest terms. Most of the time the vertical asymptotes will be the vertical lines of the domain concerns. However, if the rational function reduces, you may lose a domain concern.
Step 7: Locate the horizontal or slant asymptotes using previous discussed procedures. Horizontal and slant asymptotes are dependent on the degrees of the numerator and denominator. a) If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. b) If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the line y = the quotient of the lead coefficients. c) If the degree of the numerator is exactly one more than the degree of the denominator, then there is no horizontal asymptotes, there is a slant one instead — use division techniques. and d) If the degree of the numerator is more than one more than the denominator, then there is no horizontal nor slant asymptote.
Step 8: Use these results and your graphing utility’s table feature to graph R by hand.
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Another Example
Triangle Inequality Theorem
The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.
in other words, the Triangle Inequality Theorem, or as I sometimes call it, the Triangle Test, the sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Imagine the bottom angles’ vertices are hinges. And the sides of length 10 and 13 can pivot on those hinges. Because the sum of 10 and 13 is greater than 15, the sides of 10 and 13 will come together at a cute little point above the base and a wonderful triangle is form. Ahhh.
Imagine the bottom angles’ vertices are hinges in the above example. And the sides of length 5 and 7 can pivot on those hinges. Because the sum of 5 and 7 is less than 15, the sides of 5 and 7 will not intersect to form a triangle. Sad.
Imagine the bottom angles’ vertices are hinges in the above example as well. And the sides of length 5 and 10 can pivot on those hinges. Because the sum of 5 and 10 is equal to 15, the sides of 5 and 10 will intersect, but they will intersect only after they are collapse to the bottom segment. This is not a triangle. Sad.
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Try the following examples. If you hold the left click of your mouse down and slide the mouse beside the problems, you will see the answers.
Can the following lengths be those of the three sides of a triangle?
1. 9, 10, 15 Yes, 9 + 10 = 19 which of course is greater than 15.
2. 3, 4, 5 Yes, 3 + 4 > 5.
3. 3, 4, 7 No, 3 + 4 is not greater than 7. Being equal is not good enough. Sorry.
4. 3, 4, 10 No, 3 + 4 is not greater than 10.
5. 7, 7, 10 Here we have an isosceles example. 7 + 7 > 14, yes this is a triangle.
6. 7, 7, 20 7 + 7 is not greater than 20, no triangle.
7. 2, 7, 7 What do we do here? Since the 7s are the larger, this will always be a triangle. The rule is that any two sides’ sum must be greater than the third side. This will always be true.
8. 10, 10, 10 Here we have an equilateral example. These will always be possible. Yes this is a triangle.
Midsegment of a Triangle
A line segment joining the midpoints of two sides of a triangle. A triangle has 3 possible midsegments.
Properties
1. The midsegment is always parallel to the third side of the triangle.
2. The midsegment is always half the length of the third side.
3. A triangle has three possible midsegments.
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Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length.
A shortcut formula for the variance of a probability distribution is
You still have to do an abbreviated table. Plus I think it is easy to cross up the columns you are using. Use it at your own risk. I do think it is quicker though.
