How to draw graph with transformed axis

When a graph is drawn on ( y + x ) axis and ( y - x) axis instead of original how to convert it into original Please help with a detailed approach ?

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Is this a homework problem? If so please tag it homework. –  Kazark Jul 3 '12 at 17:33
No this is not a homework problem. –  Arpit Bajpai Jul 4 '12 at 1:31
This appeared in an exam –  Arpit Bajpai Jul 4 '12 at 1:31
I wonder why it has been downvoted then. +1 –  Kazark Jul 4 '12 at 16:23

The $y+x$ axis is located at $+45^{\circ}$ from the $+x$ axis. So the $+x$ axis in your first picture is at $-45^{\circ}$. When you rotate the $+x$ axis by $+45^{\circ}$ everything moves with it, so you should rotate the picture by $45^{\circ}$ counterclockwise.

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Had the same graph being drawn on y/x and 1 /x axis then what will be your answer –  Arpit Bajpai Jul 3 '12 at 16:33
@ArpitBajpai: y/x vs. 1/x is not a simple rotation. There is not a simple graphic transformation that will take one to the other. You can certainly take points from one and replot it, but I have no geometric intuition like the above. –  Ross Millikan Jul 3 '12 at 16:37
How did you find out y + x axis is located at +45 from the x axis –  Arpit Bajpai Jul 3 '12 at 16:42
@ArpitBajpai: To do it mathematically, you use a rotation matrix: en.wikipedia.org/wiki/Rotation_matrix Intuitively, the perpendicular to the $y+x$ axis is the line where $y+x=0$, which is the line $y=-x$. –  Ross Millikan Jul 3 '12 at 16:58
I believe y + x is located at 135 from the x –  Arpit Bajpai Jul 3 '12 at 17:01

If this is a homework problem, one way to approach it is to simply look at the possible answers, and find which one would give you the correct y-x by y+x graph. So, for each graph, choose individual points, and determine which is larger, y-x or y+x.

If y and x are both positive, then y+x will give you a larger number than y-x. If both are negative, then y+x will give you a smaller number than y-x. You have to think a little bit more for points where x and y have different signs. This gives you an idea of what the converted graph will look like.

If you don't want to guess and check, and you want to start with the y-x by y+x graph, you can still think about the graph in the same way. Which is larger, y+x or y-x? Basically, is the slope of the line in your first graph greater than 1, or is it between 1 and 0?

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