Graphing Absolute Values

We didn't do much today...
In the morning, we just worked on our exercises then a REALLY short lesson in the afternoon.

The lesson was about graphing absolute values.

As you can see, the x-intercepts (zeroes or roots) for the absolute graph (green graph) are -3 and 3.

This is because if you set the y to zero

y = x - 3
0 = x- 3
x = -3

y = x + 3
0 = x + 3
x = 3



It is important to remember that: ABSOLUTE VALUE ALWAYS HAS TO BE POSITIVE. (in the graph)

Since you can't have a negative value... the graph will just bounce and go the opposite direction.








the absolute value of -1 is 1 therefore the graph will just keep on bouncing and it will never have a negative value.


That is it for now... if i missed anything please inform me.

The next scribe is JAMILYN G.

Scribe # ??

Hello. This is Regine, the scribe for today. Formally known as..well, I don't know what I'm formally known as. Point is, I don't like math. But that doesn't matter.haha. I like lavender..

Anyways, on with the lesson for today...

Graphing Reciprocal Functions.


There are five simple rules on how to graph reciprocal functions. As Mr.Ingram states, if you follow these rules, you'll have no problem with the graphing.

1. Where the value of the original function is Zero, the value of the reciprocal is undefined, so
you have a vertical asymptote.

2. Where the value of the orginal function is 1, the reciprocal will be 1.
Where the value of the original function is -1, the reciprocal will be -1.

3. Where the value of the original is positive, the reciprocal will be positive.
Where the value of the original is negative, the reciprocal will be negative.

4. If the value of the original is increasing over an interval, the reciprocal will decrease over that interval.

5. As the absolute value of the original function approaches zero, the absolute value of the reciprocal function gets larger.

Examples


f(x) = x - 2 graph f^-1(x)





f(x)=x^2+2

Reciprocal:








Graph f^-1(x)








Graph sin(x) and f^1(x)






Anyways. That's the end of it. Sorry guys. I'm too lazy to go indepth. If I do, this blog will be about six pages long.haha. So to save you time and..time. Just stay in school and listen in class.


THE NEXT SCRIBE IS









KAT L.

UNTITLED

Let's start with solving for: SIN² ø + COS²ø + 2COSø = 0

Think about the equation for the unit circle: x² + y² = 1

COS²ø + SIN²ø = 1, and it just happens that COS²ø and SIN²ø are in the equation.
So ...

SIN² ø + COS²ø + 2COSø = 0 can now be written as

1 + 2COSø = 0
COSø = - 1/2

*SOLVE FOR ALL REAL NUMBERS and your answers are:
2pi/3 + 2Kpi, K E I and 4pi/3 + 2Kpi, K E I




SKETCHING GRAPHS WITHOUT A CALCULATOR

EX. 1 Sketch f(x) = x (x-4)

* find where y = 0

x - 4 = 0
x = 4

x value is 4 so the axis of symmetry = 2

* substitute the value of the axis of symmetry for the x's in the equation

y = 2 (2 - 4)
y = 4 - 8
y = -4

vertex = (2, -4)

EX. 2 Sketch f(2x) = 2x (2x - 4)

* find where y = 0
b > 1
2x - 4 = 0
2x = 4
x = 2

x value is 2 so the axis of symmetry = 1

* substitute the value of the axis of symmetry for the x's in the equation

y = 2 (1) (2(1) -4)
y = 2 (-2)
y = -4

vertex = (1, -4)

* we horizontally compressed the original graph.


EX. 3 Sketch y f(x) = 1/2x (1/2x - 4)

* find where y = 0
0
1/2x - 4 = 0
1/2 x = 4
x = 8

x value is 8 so the axis of symmetry = 4

* substitute the value of the axis of symmetry for the x's in the equation

y = 1/2 (4) (1/2(4) - 4)
y = -4

_ _ _ _ _

SO ...

EX. 1 Start with y = x²

* Now graph y = 2f(x) by referring to the y = x² graph as a guideline.

The parabola should stretch:

* Now graph y = 2f (x) + 3 still referring to the y = x² graph as a guideline.

The parabola not only stretches, but also moves 3 units up due to the + 3 in the equation.

EX.2 Start with y = x²

* Now graph y = f (2x)²

This will make the parabola a compressed horizontal:

* Now graph y = f(2(x -8))²

This will take the original parabola 8 units to the right:

* Now graph y = f(2(x -8))² + 3

This will move the parabola 3 units upward.

* Sketch the graph of y = 1/2 (x + 4)² - 5

start with the y = x² graph to help you.

1. Because of the 1/2 the parabola will vertically compress.
2. Because of the 4 the parabols will move 4 units left.
3. Because of the - 5 the parabola will move 5 unit down.

For the next blog, I choose m0nkeh ! (I don't know who that is).

2nd Scribe

Okay, so i'm the second scribe. We didn't do that much today since the lesson was yesterday afternoon. We just did the question #8 in the exercises as well as #11.

Question 8:

Given the sketch of g(x) = f(x + 2) + 1, sketch f(x):



The question simply asks that in the given graph, sketch the graph where it originally came from.

So we can start in the inside bracket. It says f(x + 2) + 1. To go to the original point, instead of shifting to the left by 2 units, it will be 2 units to the right.




After shifting to the right, the next is the f(x + 2) + 1. Instead of shifting it up by 1 unit, the points are going to be shifted down by 1 unit.


And this is f(x).

Question #11.
Find the approximate values of the equation 3sin² ϑ + 5sin ϑ - 2 = 0 for 0< ϑ<2 π.

First, it is easy for me to just substitute 'x' to 'sin ϑ'. It is going to be like factoring an equation.
3x² + 5x - 2 = 0.

Factor:
(3x-1)(x+2)

Substitute:
(3sin ϑ - 1)(sin ϑ + 2)

Solve:

3sin ϑ = 1
sin ϑ = 1/3
ϑ = 19.5^o ; 160.5^o

sin ϑ = 2
no solution


I hope i did it right. Considering it should be easy. Ok, well the next scribe would be Bale! (Ms. Ingram said. =P)

Bloooggg eett . . . .

Yuup. I'm the scribe today. Ms. Ingram picked bale but I wanted to do it so I'm doing it. Moving on, today's lesson was pretty short. There was no twist to it. Though I know its going to be harder and its going to be much more complicated next time. Hope you like it and actually understand it. If you don't then uh ... just ask Ms. Ingram, haha. Well, here goes ...




Its the same as x^3 but its just shifted two units down.


Its the same as x^3 but shifted THREE units to the LEFT.



It shifted ONE unit to the RIGHT.

y = f ( x + k )
K shifts the graph horizontally
If it's POSITIVE it shifts to the LEFT.
If it's NEGATIVE it shifts to the RIGHT.

y = f ( x )+ k
K shifts the graph vertically
If it's POSITIVE it goes UP.
If it's NEGATIVE it goes DOWN.

Does it make sense so far...? yea...? Ok great! no...? Well here are some more examples.


Do you see how it shifted three units to the left ...?


This one shifted five units to the right.

K. You get the point. Its pretty straight forward. Homework is EXERCISE 7: #1-9.. That's all for now. Next scribe is ..... CARLA. Good luck and have fun :))

First Blog ...

Precal is hard. no doubt. I have to look over my notes everyday and do some questions to actually get it. I don't have really much to say. Time is passing by so fast. We finished unit one already. Well that's all. Theee end.

Good? Great? Awesome!

So, this is my first blogging for such a reason. It's new, but I can get used to it. Anyway, this Grade 12 Pre-cal course is pretty easy in my case. I actually already have taken it last year. I'm just upgrading so I can't say this stuff is hard because it shouldn't be. And as for the work? Honestly, it's not as many as I did before. But it still makes you think, which is actually good.

And for Mrs. Ingram? She's awesome. Haha. But really, I like the way she teaches. Especially when she tries to make things easier for us to understand and such.

Yeah... The End.

Blog Mark ^_^

I feel the same way as Carla... I'm used to a different way of teaching. I love how Mrs. Ingram teach. She's makes it fun and all but I'm not used to it yet. I guess I will be after couple more days. The first week was alright but then this week isn't that great. I guess it's because I've been talking too much -_-" .I need to keep up with the exercises too! Hm, I hope I pass the test tomorrow... I'll end this blog now `cause I still need to study. Goodluck guys!

ENTRY 1

Grade 12 Precal is starting off pretty difficult, but I'm guessing it's my own fault for not really keeping up with the exercises. I think it's better to do the exercises for marks because it pressures me into doing the homework and also contributes to my grade. To be honest, I don't know if I'll do that great on the test...but I will study.

First Blog Entry

Kk, so this is my first entry. I don't know about the other students but i feel dumb now with this course. Lol. I'm used to a different teaching style. Or i guess i'm still on vacation mode. =P. My brain doesn't wanna work. But im going to try my best to pass this course. Lol. I hope i pass the test this Thursday.

Welcome

Welcome to your blog for semester 1, grade 12 pre calculus.
This blog has been created as a learning tool to assist you with your studies in the grade 12 pre calculus course. I hope you make good use of this blog, and find it very helpful. Enjoy.