(no subject)
JUST in case anyone might need this: (I just felt like talking about asymptotes for a few seconds.)
Vertical asymptotes are defined by any falue of X in any function or relation that will make the denominator zero. You can't divide by zero in Mathematics, right? That's braking the first commandment: Thou shalt not divide by zero. You can have as many of these as the relation darn well pleases.
EXAMPLE:
f(x) = ((x+4)(x+3))/(x(x+4)(x-3))
Sorry... they have to be in calculator mode because I'm too lazy to do HTML.
Alright. The Zero Property of Multiplication says that anything multiplied by zero is zero, right? So if x, x+4, or x-3 is equal to zero, we're screwed. So here's what we do:
Set all of them equal to zero:
x=0 x+4=0 x-3=0
Solve for the variable:
x=0 x=-4 x=3
All of the x values at this step will be your vertical asymptotes... Sorry I don't have a graph to show you, but the lines above are the equations for the vertical asymptotes.
HERE'S WHERE IT GETS TAX-Y:
Horizontal asymptotes are easier and harder. They're easier in the sense that there will ONLY BE ONE, IF ANY. ONE OR ZERO. NEVER TWO OR MORE. ♥. Yes. But it's kind of wierd deciding WHERE they are. First, I need to tell you about degrees:
Degrees
First, if you have a few terms being multiplied (example: x(x+2)^2), multiply them... ... ... Get the thing "simplified". (example: x^3 + 4x^2 + 4x)
Next, take the variable with the highest exponent in that expression. (example: x^3)
And finally, take the exponent of that variable with the highest exponent, and that's the degree of the expression! (example: the degree of x(x+2)^2 is 3)
Now there are three rules from here:
- 1 - If the degree of the numerator is SMALLER than the degree of the denominator, the horizontal asymptote will be zero (the x-axis).
- 2 - If the degree of the numerator is EQUAL to the degree of the denominator, the horizontal asymtote will be the coefficient of the variable with the exponent (normally the leading coefficient) in the numerator over the coefficient of the variable with the largest exponene (again, normally the leading coefficient). Leading coefficient of numerator over leading coefficient of denominator.
- 3 - If the degree of the numerator is EXACTLY ONE DEGREE GREATER than the degree of the denominator, there will be no horizontal asymptote: the asymptote will be oblique. Remember dividing polynomial expressions by other polynomial expressions? This is where that matters. Take the expression of the numerator and use your amazing algebraic skillz to divide it by the expression of the denominator.
EXAMPLE:
((x+4)(x+3)/(x(x+4)(x-3))
"Simplify".
((x^2)+7x+12)/((x^3)+(x^2)-12x)
Divide.
x^2+7x+12 / x^3+x^2-12x
x^2 goes into x^3 x times, so the first number in the answer is x.
Multiply through and subtract.
x^3+x^2-12x - x(x^2+7x+12) = x^3 + x^2 - 12x - x^3 - 7x^2 - 12x = -6x^2-24x
x^2 goes into -6x^2 -6 times, so the second number in the answer is -6.
-6x^2-24x - (-6)(x^2+7x+12) = -6x^2 -24x + 6x^2 + 42x + 72 = 24x+72
x^2 won't go into 24x, so we're done. (We drop the remainder... We don't care about it. It stole our lunch money when we were a kid, and now we just ignore it, even though it wants to apologize and be our friend. We just don't care about it.)
So the answer is x-6.
In any of these problems, your answer should always have a degree of one (since rule 3 [which we're learning about, in case you've forgotten] states that the numerator is EXACTLY ONE DEGREE GREATER than the denominator), thus making it... a line!
So, the resulting line that you get will be your slant asymptote, in this example, the line is y=x-6.
TWO THINGS TO REMEMBER: If the numerator is MORE THAN ONE DEGREE GREATER than the denominator, it's out of our league. If you get one like that, it was a mistake on the part of your math teacher... unless you're in college or graduate school and you're learning about that mess...
If there is no denominator (or the denominator has no variable), then you don't really have to worry about asymptotes.
"Holes" is a term that has been so gracefully assigned by mathematicians to the situation in the event that something in the numerator and the denominator cancel eachother out. When they do, you'll graph the resulting function/relation, but there's a small catch (infinately small, in fact).
The asymptote still exists.
Don't freak out, you don't have to graph around it... Simply note that because the cancelled expression was still in the denominator before you cancelled, it can still make the denominator zero. So you'll graph the resulting function, BUT, there will be a hole at the point where the x value is equal to the value of x that would make the cancelled expression equal zero.
Example:
((x+4)(x-3))/(x(x+4)(x+3))
x+4 appears in both the numerator and the denominator, so they'll cancel out.
(x-3)/(x(x+3)) will therefore be the function to graph, but the asymptote provided by x+4 still exists, but it doesn't deflect the graph, so you set that expression equal to zero and solve for x.
x+4=0 ==> x=-4
So wherever x=-4 on your graph, there will be a hole. It'll be just like the graph (x-3)/(x(x+3)), but the point (-4, -7/4) should be an open-dot thing, signifying that the graph passes right by that point, and there's a hole. A hole. Like a pothole in a street. A hole.
That one might have been a little too hard... Let's give you an easy one:
f(x)=(x(x-2))/(x-2)
The (x-2)s cancel out, making your function f(x)=x with a hole at x=2
This should look like your standard y=x line, but there will be an open-hole dot thing at (2,2), because the x value of 2 would make the denominator equal to zero.
AND THAT'S IT. ALL YOU EVER NEED TO KNOW ABOUT ASYMPTOTES... And I hope I did a good job explaining... Otherwise the certain people that might need to know this, which includes Trig. students (We have an exam on this tomorrow.) and some Algebra 2 students, who are just getting to this mess.
School was much better than expected today... The history test isn't until WEDNESDAY, so that's good... We just goofed around and did practically nothing in Physics... She said we were reviewing, but it was a circus... No one was listening... I just went to sleep. In Trig., we ... did... something, but I forgot exactly what. x.x; Ask Bridgett Nutt or Kathleen if you really care... maybe they remember. Trig. test tomorrow, though. I remember that. In Spanish we read La Catrina and went over the subjunctive tense again. In Anatomy, we dissected a sheep heart (lots of fat). Aaand in Chemistry, we did a lesson on the Kinetic-Energy Theorem and its relation to Solids and Liquids. No homework.
My library books are due... I should get them in sometime...
IN ENGLISH, Mrs. Davis started "EOC Boot Camp", which is a rigorous three-week-long training for the Arkansas End of Course, which before wouldn't matter to her, but due to new legislation, you must be remediated in English if you don't get Proficient or Advanced on this test. I'm not too worried... I know Mrs. Davis wants an advanced, and for her I'll try my hardest, but I know I can get a proficient, and after that, this test doesn't matter at all.
The first writing example she showed our class was a composition made by a sophomore (Her English 10 Honors classes are also training for the EOC.) over a book called The Monkey's Paw. I don't know how well you know this book... but... Oh well, it's irrelevent. She put the paper on the projector, read it, and spent about ten minutes talking about how amazing this writing excerpt is, and how Mrs. Davis keeps telling her she needs to be in AP next year. I want to say "During this long spill about the excellence of this sophomore, Ali looks at me, and I look back at him, and we each give a nod, signifying a mutual understanding.", but it was really after class, he asked me, and I said "I have no doubt in my mind." or something like that. Anyway, after class, we went to Mrs. Davis to find out. Ali asked something like "Who wrote that excerpt that you showed us at the beginning of class?" Mrs. Davis said something like "I'm not at liberty to discuss it." At which point I said something like "Would she happen to be in your seventh period class?" At which Mrs. Davis started laughing a little bit and said "Maybe," signifying an affirmative.
GOOD JOB, CRUST~! ♥ Your English skills are superior. I think that paper might even score a 7 or an 8 if the AP people gave you a prompt like that. And your work was used as an example of an Advanced writing style in front of both AP English classes. Good job! ♥♥♥
MARDI GRAS SALES START TOMORROW. BUY BEADS AND MASKS. I IMPLORE YOU.
Vertical asymptotes are defined by any falue of X in any function or relation that will make the denominator zero. You can't divide by zero in Mathematics, right? That's braking the first commandment: Thou shalt not divide by zero. You can have as many of these as the relation darn well pleases.
EXAMPLE:
f(x) = ((x+4)(x+3))/(x(x+4)(x-3))
Sorry... they have to be in calculator mode because I'm too lazy to do HTML.
Alright. The Zero Property of Multiplication says that anything multiplied by zero is zero, right? So if x, x+4, or x-3 is equal to zero, we're screwed. So here's what we do:
Set all of them equal to zero:
x=0 x+4=0 x-3=0
Solve for the variable:
x=0 x=-4 x=3
All of the x values at this step will be your vertical asymptotes... Sorry I don't have a graph to show you, but the lines above are the equations for the vertical asymptotes.
HERE'S WHERE IT GETS TAX-Y:
Horizontal asymptotes are easier and harder. They're easier in the sense that there will ONLY BE ONE, IF ANY. ONE OR ZERO. NEVER TWO OR MORE. ♥. Yes. But it's kind of wierd deciding WHERE they are. First, I need to tell you about degrees:
Degrees
First, if you have a few terms being multiplied (example: x(x+2)^2), multiply them... ... ... Get the thing "simplified". (example: x^3 + 4x^2 + 4x)
Next, take the variable with the highest exponent in that expression. (example: x^3)
And finally, take the exponent of that variable with the highest exponent, and that's the degree of the expression! (example: the degree of x(x+2)^2 is 3)
Now there are three rules from here:
- 1 - If the degree of the numerator is SMALLER than the degree of the denominator, the horizontal asymptote will be zero (the x-axis).
- 2 - If the degree of the numerator is EQUAL to the degree of the denominator, the horizontal asymtote will be the coefficient of the variable with the exponent (normally the leading coefficient) in the numerator over the coefficient of the variable with the largest exponene (again, normally the leading coefficient). Leading coefficient of numerator over leading coefficient of denominator.
- 3 - If the degree of the numerator is EXACTLY ONE DEGREE GREATER than the degree of the denominator, there will be no horizontal asymptote: the asymptote will be oblique. Remember dividing polynomial expressions by other polynomial expressions? This is where that matters. Take the expression of the numerator and use your amazing algebraic skillz to divide it by the expression of the denominator.
EXAMPLE:
((x+4)(x+3)/(x(x+4)(x-3))
"Simplify".
((x^2)+7x+12)/((x^3)+(x^2)-12x)
Divide.
x^2+7x+12 / x^3+x^2-12x
x^2 goes into x^3 x times, so the first number in the answer is x.
Multiply through and subtract.
x^3+x^2-12x - x(x^2+7x+12) = x^3 + x^2 - 12x - x^3 - 7x^2 - 12x = -6x^2-24x
x^2 goes into -6x^2 -6 times, so the second number in the answer is -6.
-6x^2-24x - (-6)(x^2+7x+12) = -6x^2 -24x + 6x^2 + 42x + 72 = 24x+72
x^2 won't go into 24x, so we're done. (We drop the remainder... We don't care about it. It stole our lunch money when we were a kid, and now we just ignore it, even though it wants to apologize and be our friend. We just don't care about it.)
So the answer is x-6.
In any of these problems, your answer should always have a degree of one (since rule 3 [which we're learning about, in case you've forgotten] states that the numerator is EXACTLY ONE DEGREE GREATER than the denominator), thus making it... a line!
So, the resulting line that you get will be your slant asymptote, in this example, the line is y=x-6.
TWO THINGS TO REMEMBER: If the numerator is MORE THAN ONE DEGREE GREATER than the denominator, it's out of our league. If you get one like that, it was a mistake on the part of your math teacher... unless you're in college or graduate school and you're learning about that mess...
If there is no denominator (or the denominator has no variable), then you don't really have to worry about asymptotes.
"Holes" is a term that has been so gracefully assigned by mathematicians to the situation in the event that something in the numerator and the denominator cancel eachother out. When they do, you'll graph the resulting function/relation, but there's a small catch (infinately small, in fact).
The asymptote still exists.
Don't freak out, you don't have to graph around it... Simply note that because the cancelled expression was still in the denominator before you cancelled, it can still make the denominator zero. So you'll graph the resulting function, BUT, there will be a hole at the point where the x value is equal to the value of x that would make the cancelled expression equal zero.
Example:
((x+4)(x-3))/(x(x+4)(x+3))
x+4 appears in both the numerator and the denominator, so they'll cancel out.
(x-3)/(x(x+3)) will therefore be the function to graph, but the asymptote provided by x+4 still exists, but it doesn't deflect the graph, so you set that expression equal to zero and solve for x.
x+4=0 ==> x=-4
So wherever x=-4 on your graph, there will be a hole. It'll be just like the graph (x-3)/(x(x+3)), but the point (-4, -7/4) should be an open-dot thing, signifying that the graph passes right by that point, and there's a hole. A hole. Like a pothole in a street. A hole.
That one might have been a little too hard... Let's give you an easy one:
f(x)=(x(x-2))/(x-2)
The (x-2)s cancel out, making your function f(x)=x with a hole at x=2
This should look like your standard y=x line, but there will be an open-hole dot thing at (2,2), because the x value of 2 would make the denominator equal to zero.
AND THAT'S IT. ALL YOU EVER NEED TO KNOW ABOUT ASYMPTOTES... And I hope I did a good job explaining... Otherwise the certain people that might need to know this, which includes Trig. students (We have an exam on this tomorrow.) and some Algebra 2 students, who are just getting to this mess.
School was much better than expected today... The history test isn't until WEDNESDAY, so that's good... We just goofed around and did practically nothing in Physics... She said we were reviewing, but it was a circus... No one was listening... I just went to sleep. In Trig., we ... did... something, but I forgot exactly what. x.x; Ask Bridgett Nutt or Kathleen if you really care... maybe they remember. Trig. test tomorrow, though. I remember that. In Spanish we read La Catrina and went over the subjunctive tense again. In Anatomy, we dissected a sheep heart (lots of fat). Aaand in Chemistry, we did a lesson on the Kinetic-Energy Theorem and its relation to Solids and Liquids. No homework.
My library books are due... I should get them in sometime...
IN ENGLISH, Mrs. Davis started "EOC Boot Camp", which is a rigorous three-week-long training for the Arkansas End of Course, which before wouldn't matter to her, but due to new legislation, you must be remediated in English if you don't get Proficient or Advanced on this test. I'm not too worried... I know Mrs. Davis wants an advanced, and for her I'll try my hardest, but I know I can get a proficient, and after that, this test doesn't matter at all.
The first writing example she showed our class was a composition made by a sophomore (Her English 10 Honors classes are also training for the EOC.) over a book called The Monkey's Paw. I don't know how well you know this book... but... Oh well, it's irrelevent. She put the paper on the projector, read it, and spent about ten minutes talking about how amazing this writing excerpt is, and how Mrs. Davis keeps telling her she needs to be in AP next year. I want to say "During this long spill about the excellence of this sophomore, Ali looks at me, and I look back at him, and we each give a nod, signifying a mutual understanding.", but it was really after class, he asked me, and I said "I have no doubt in my mind." or something like that. Anyway, after class, we went to Mrs. Davis to find out. Ali asked something like "Who wrote that excerpt that you showed us at the beginning of class?" Mrs. Davis said something like "I'm not at liberty to discuss it." At which point I said something like "Would she happen to be in your seventh period class?" At which Mrs. Davis started laughing a little bit and said "Maybe," signifying an affirmative.
GOOD JOB, CRUST~! ♥ Your English skills are superior. I think that paper might even score a 7 or an 8 if the AP people gave you a prompt like that. And your work was used as an example of an Advanced writing style in front of both AP English classes. Good job! ♥♥♥
MARDI GRAS SALES START TOMORROW. BUY BEADS AND MASKS. I IMPLORE YOU.