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...

f(x)=g(2x)

***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

g(x)=f(1/2x)

***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)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) = -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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