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The graph of y = f(x) is shown below.

For each point (a,b) that is on the graph of y = f(x), the point $$\left( 3a - 1, \frac{b}{2} \right)$$ is plotted, forming the graph of another function y = g(x). As an example, the point (0,2) lies on the graph of y = f(x), so the point $$(3 \cdot 0 - 1, 2/2) = (-1,1)$$ lies on the graph of y = g(x).

(a) Plot the graph of y = g(x). Include the diagram as part of your solution.

(b) Express g(x) in terms of f(x)

(c) Describe the transformations that can be applied to the graph of y = f(x) to obtain the graph of y = g(x). For example, one transformation could be to stretch the graph vertically by a factor of 4.

Please, I really need help solving this and I have no idea how to do it.  Please include the steps on how to solve it, just the answer isn't enough to help me understand.  Thank you so much!!!

Oct 22, 2021
edited by sofsicle  Oct 22, 2021

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In my experience, this is a very unusual question and I found it difficult.

Five obvious points on the f(a)=b graph are

(-4,4)    (-1,0)   (0,2)   (2,-1)   (4,-4)

I have graphed these points and the new g function is in red.

I am not sure what the correct notation is to name it....

Now

$$f(a)=b\qquad g(3a+1)=\frac{b}{2}\\ g(3a-1)=\frac{b}{2}\\ g(3a-1)=\frac{f(a)}{2}\\ let\;\;\;x=3a-1\\ then\;\;\; a=\frac{x+1}{3}\\~\\ g(x)=\displaystyle \frac{f(\frac{x+1}{3})}{2}$$

Check

$$If \;\;x=-4 \;\;then\;\; g(-4)=\frac{f(-1)}{2}=0\quad true\\ If \;\;x=-1 \;\;then\;\; g(-1)=\frac{f(0)}{2}=\frac{2}{2}=1\quad true\\ etc$$

I do not know about describing the transformation.  That is all I have for now.

If you get an answer or work out how to add to this, please share it with me and with others that may also be interested.

LaTex:

g(3a-1)=\frac{b}{2}\\
g(3a-1)=\frac{f(a)}{2}\\
let\;\;\;x=3a-1\\
then\;\;\; a=\frac{x+1}{3}\\~\\
g(x)=\frac{f(\frac{x+1}{3})}{2}

Oct 22, 2021
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Attn:  sofsicle

A response from you would be appreciated.

Melody  Oct 23, 2021
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Yes i will post the answer very soon, thank you for answering it!

sofsicle  Oct 26, 2021
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Thanks Sofsicle

Melody  Oct 26, 2021
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NOTE: This is a homework problem and it would be helpful if such detailed answers were not posted online.

Oct 27, 2021
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I understand the problem of a teacher,  if I was you I would be equally frustrated.

But, people post questions for many reasons.

Sometimes they are trying to self-learn and people like me are almost essential tools.

It is one of the reasons why fully supervised exams, with new questions, are so vital for assessing students of maths.

Many dishonest students will always cheat.  What is the difference between a student coming here, or going to a knowledgeable parent?

And all conscientious, honest students will look for extra help if what they are offered is of poor quality.

On a slightly different vein:

What definitely annoys me is when I get asked to remove a question with answers, that is 3 or more years old, because it is an exam question.

How totally ridiculous is this.  Is the teacher or teaching facility too lazy to write new exams?  The question would be plastered across the internet in more than just this one site!  When I was teaching we had to write new exams every time

-------------------------

I would like to learn. Why don't you tell me how to finish this question.  You know I am genuinely interested!

I am not cheating.  I just want to know.   Or do you only teach when you are paid to do so?

Melody  Oct 27, 2021