Graphing Rational Functions

This morning we continued talking about Graphing Rational Functions. We went back to the example from yesterday's class and finish graphing the rational function.

Example:

Step 1: Find the y-intercept by letting x=0.



Step 2: Factor everything, which in this case it's already factored.

Step 3: Find the roots of the function by finding the roots of the numerator.
This one has NO ROOTS.

Step 4: Find the vertical asymptotes by finding the roots of the denominator.
V.A. → x=2

Step 5: Find the horizontal asymptotes by dividing each term in the function by highest power of x, and take the limit as x goes infinity.

H.A. → y=0

Step 6: Determine the sign of the function over the intervals defined by the roots (step 3) and vertical asymptotes (step 4).

Step 7: Sketch the graph.

This is not the the exact graph, but it gives us idea on how the graph would look like.

We did another example on graphing rational functions, but it's pretty much the exact same thing on how we solve it. The only thing that changed was the graph (we can see this on the slides that Mr. K post everyday).

Homework: Go back to the Exercise that was assigned yesterday and and do the graphing rational functions part of it. Also, don't forget to start on doing the last exercise in the book that covers all the units that we did in Pre-Cal.

Next scribe is Larlyn

- Jeamille

Reflection .

Hi everyone! The Logic Unit was tolerable :) I was pretty much okay with the Converse, Inverse, Contrapositive, Inductive Reasoning and Deductive Reasoning. I should say that the only thing I'm worried for tomorrow's test is the Proof by Contradiction (Direct and Indirect Reasoning). That's about it.

Good Luck!

Proof by Contradiction

Hi everyone! Although this is the last day of school for this year, we still had lessons in class that we talked about for the whole period...

Before we went into groups and answer group questions, Mr. K. recalled the sentence that we talked about the other day:

This sentence is false.

This sentence is a Paradox which means it can't be true and false at the same time, but it should be true and false at the same time. A Paradox is one kind of Contradiction that goes round and round.

When we went into our groups, we were given logic questions that we needed to answer. Here's the first question.

On of Isle of View, a knight always tell the truth and a knave always lie. A person who just came met Mr. A and Mr. B. Mr. A said "one of us is a knave". Which one is the knave, and which one is the night?

We need to answer this question by Proof by Contradiction, which means we have to assume the opposite of the statement and prove that it's right. This was how we tackled this question.

First, we choose the Direct Reasoning of the statement, which in this case was
1) Assume Mr. A is a knight. We can then tell that
2) His statement is true, therefore our conclusion is that
3) Mr. B is a knave and Mr. A is a knight.

Using the Indirect Reasoning, we take the opposite statement and check if it leads to contradiction...

1) Assume Mr. A is a knave
2) His statement is false
3) They are both knights

This part of the reasoning leads to a contradiction, which was what we were looking for. Therefore, Mr. A is a knight.

We looked at two more questions that deals with logic, obviously :) It was pretty much straight forward... We do have homework, however that we need to do. It's Exercise 50 all questions.

Next scribe is Charizze

Pre-test on Monday - January 5, 2009

HAPPY HOLIDAYS!

see you all next year...

reflection.

This unit was a little fun, all because of drawing circles and other stuff. Anyway, Circle Geometry was pretty good. I could understand most of it, especially the theorems. It's just kind of confusing or hard when we need to visualize the question and look at it carefully and think of which theorems to use for solving the problem. My weakness for this unit was the Statement and Reason part, it's just hard for me to state my reason on how I got the answer for a question in mathematical ways. I hope we'll all do well on the test tomorrow :) Good Luck !

What I'm Thinking.

Hello :)

Here are the things that I either need more practice on or get confused with:

☺ Absolute Values in Algebra
☺ Circles and Radius Problems in Analytic Geometry
☺ Adding-Subtracting Method in Systems of Linear Equations in Three Variables

Circle Hyperbola

Hello ...
Sorry for the late post, but here's what we did on Friday.

First off, we went through the homework that were given on Thursday.

w + l + h = 80 cm
Length=10cm less than twice the sum of width and height and twice the width exceeds the height by 6 cm. Find the width, length and height of the box.

~We didn't go over the procedure on how to solve this problem, but the answers were:
h = 18cm, w = 12 cm, l = 50 cm

Then there was the question from the book (Ex. 25 #1) Letter "c" was done by Ramina on the white board. For those who didn't get to copy the procedure, here it is...

1. a) 3x-4y+5z=2
4x+5y-3z=-5
5x-3y+2z=-11

(3) 3x-4y+5z=2 (3)
(5) 4x+5y-3z=-5 (5)

9x-12y-15z=6
+ 20x+25y-15z=-25
ˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉ
29x+13y=-19


(2) 4x+5y-3z=-5 (2)
(3) 5x-3y+2z=-11 (3)

8x+10y-6z=-10
+ 15x-9y+6z=-33
ˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉ
23x+y=-43


29x+13y=-19
(13) 23x+y=-43 (13)

29x+13y=-19
+ 229x+13y=-559
ˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉˉ
-270x = 540
ˉˉˉˉˉˉ ˉˉˉˉˉ
-270 -270

x = -2


29x+13y=-19
29(-2)+13y=-19
-58+13y=-19
13y=58-19
13y=39
ˉˉˉ ˉˉˉ
13 13

y = 3


3x-4y+5z=2
(3)(-2)-4(3)+5z=2
-6-12+5z=2
-18+5z=2
5z=18+2
5z = 20
ˉˉˉ ˉˉˉ
5 5

z = 4


After this, we talked about a little something new called Circle Hyperbola.
It looks like this ...



A Circle Hyperbola can only have 0, 1, or 2 intercepts and no more than that. We can find the intercept by putting the equation on our calculator OR solve it algebraically. We did some examples as a practice on finding the intercepts of Circle Hyperbola.

► y=x-1
y=x²-6x+9

x-1=x²-6x+9
-10=x²-7x
x²-7x+10=0
(x-5) (x-2)
x=5 x=2
y=4 y=1

another example was...

►x²-y²=5
xy+6=0

y²=x²-5 → y= √x²-5 (#1)

y=-6/x (#2)

-6/x=√x²-5
(-6/x)² = (√x²-5)²
36/x² = x²-5
x^4-5x²=36 (substitute x^4 to u=x²)
u²-5u-36=0
(u-9) (u=4)
u=9, -4
x²=9, -4
x=±3
x=3 , y=-2
x=-3 , y=2

After doing these examples, there were homework that were assigned for us to do. Exercise 26 (# 1-5) Also, don't forget to hand in your bonus mark homework sheets before Dr. Eviatar leaves!

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Next Scribe ... Jamie

September 8, 2008

For today's class, we finally talked about Quadratic Functions. But before that, I have to inform all of you that Mrs. Ingram will be the one teaching us for the rest of the semester. It is because Mr. K is on maternal leave and have to spend time with his family. :)

Okay, Mrs. Ingram talked about Quadratic Functions and how often we'll use it from now on. Quadratic Function is a function of second degree. This is how a Quadratic Function look like:

* y=ax²+bx+c * Quadratic Function cannot be completed if ax² is not there.

We did a few examples on this. We made table of values and then put it on a graph indicating the Domain and Range...

Table of Values


1.) y=x²


2.) y=2x²
↓
This (2) coefficient affects our parabola. The higher your "a" coefficient, the skinnier your parabola will be.







Mrs. Ingram gave us 2 books to use for the semester (Cumulative Exercises & Answers to Cumulative Exercises). Don't forget to do your homework! It's Exercise 1 Numbers 1-4.


Next Scribe: Randall :))