Monday, October 15, 2007

Monday's Blog : The Quiz.

Bonjour, I am today's scribe. Alright, today Mr.K began our math class by reviewing blogging techniques. I'm sure all of us, or the majority of the class, had a chance to blog. He concentrated the group blogs we did last week, The Ferris Wheel Function. For those who weren't there on Thursday you can look over the blogs every group made. Okay, so finally we got into a bit of math...

We were surprised with a quiz from Mr.K. The quiz was a review of what we learned on transformations of functions. The first question was as follows...

1.) Given the graph of f(x) below, sketch the graph of f(x+1)+3.

... so this is f (x) but we are asked to find f(x+1)+3. When I worked on this question I found the four major points on this graph which are:

After choosing these points I changed them according to f(x+1)+3.

-we know when you take the 1 out of (x+1) it becomes a negative which makes all the X coordinates move to the left one unit.

-we know when you are adding a number outside the brackets it effects the Y coordinates and makes them move up 3 because it's positive.

Once you apply these facts to the individual points above you receive these coordinates:

Once you graph the new coordinates you can connect the lines to get f(x+1)+3, shown beneath this graph.

And now onto our next quiz question, question #2.

2.) Given the graphs of f(x) and g(x):

a) Express f(x) as a function of g(x).

b) Express g(x) as a function of f(x).

a) The question is asking us to express f(x) as a function of g(x). This means we look at g(x) and make it equal f(x). We have to find out what is done to g(x) to make it identical to f(x).

-there's no phase shift between the functions.
-to turn g(x) into f(x) you must squash the X coordinates.
-notice that the Y coordinates aren't changed in anyway.
-to change g(x) into f(x) you need to find a point on each function and compare their coordinates.
-take the coordinates on f(x) : (1,-3)
- and the coordinates on g(x) : (2,-3)

The X coordinates are being multiplied by 2. This squashes the function. Now to put this into a equation...


***Remember that you are multiplying the X coordinates by 2 so you MUST put the 2 in the brackets, NOT out of the brackets because that changes the Y coordinates.

b) For question #2 you just reverse the equation, this time you are finding the changes done to f(x) to get a result of g(x). Instead of multiplying by 2 you do the exact opposite, so you're dividing by two. BUT, remember we don't divide, we multiply. So take the reciprocal of 2, which is ... 1/2


***Remember that you are multiplying the X coordinates by 2 so you MUST put the 2 in the brackets, NOT out of the brackets because that changes the Y coordinates.

Why was there no shift? Simply because the X coordinates are being multiplied. The bottom point of the f(x) function was x=1 and once multiplied by 2 it was moved. BUT if the point was x=0 then it would stay the same because, zero multiplied by zero equals zero.

Okay, now onto the next question that was on the quiz, question # 3.

3.) Given the graph of f(x), sketch the graph of f^1(x) on the same cartesian plane.

Now we are trying to find the inverse of f(x). To find the inverse you let the function reflect over y=x.

Here's a graph with the dotted line of y=x.

Now you know that it must look identical on the other side of the dotted line, to accomplish this let all the X coordinates equal the Y coordinates and let all the Y coordinates equal the X coordinates. Once you do this to a couple points you gain an idea of how the curve will look and you can graph it.

Last but not least, question # 4.

4.) State whether each of the following is even, odd or neither.

a) f(x) = 3x^2 -when check whether it's odd or even, first you do the even f(-x)= 3(-x)^2 check. To check if the function is even you let the X coordinates f(-x)= 3x^2 equal negative (-). If the result from the equation isn't the same as the one you started with it's not even.

In question A it is even because the beginning and end result match.

b) f(x)= -sin(x)
f(-x)= -sin(-x)
f(-x)= sinx

- since the end result isn't the same as the beginning equation you know it is NOT even. Now you can check whether it is odd by letting Y coordinates equal negative (-). If the beginning is the same as the end result then it is odd, if not it is neither.

f(-x)= sinx
f-(x)= -((sin)(x))
f-(x)= sinx

Question B is an odd function.

c.) f(x)= |3x|
f(-x)= |3(-x)|
f(-x)= |-3x|
f(x)= |3x|

Question C is an even function because absolute values are always positive.

d.) f(x) = -4x^2+ 3x
f(-x) = -4(-x)^2+ 3(-x)
f(-x) = -4x^2-3x

Therefore, not even. Now check if it's odd..

f(-x) = -4x^2-3x
f-(x) = -(-4x^2-3x)
f-(x) = 4x^2+3x

Therefore, neither. Remember when you're check to see if it's odd you must check if it's even first.

Well this is it. I have answered all the quiz questions to my best ability. I hope my effort helps other students, whom may not have understood before, get a better understanding.

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