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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 ?Please help in solving Graph questio

3 Answers 3

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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 answer2012-07-03
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    @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.2012-07-03
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    How did you find out y + x axis is located at +45 from the x axis2012-07-03
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    @ArpitBajpai: To do it mathematically, you use a rotation matrix: http://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 $y=-x$.2012-07-03
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    I believe y + x is located at 135 from the x2012-07-03
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    @ArpitBajpai: No, the value of $y+x$ increases when you increase either $y$ or $x$, so has to be between them. The axis at 135 from +x is the $y-x$ axis, as it increases when $y$ is increased and when $x$ is decreased.2012-07-03
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    let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/3977/discussion-between-arpit-bajpai-and-ross-millikan)2012-07-03
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    @ArpitBajpai: Fine. I have loaded it2012-07-03
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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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You could use rotations to solve this, but a simpler approach is as follows:

y-x = y'; y+x = x'; where x' and y' represent the coordinate axes as drawn in the problem figure.

The presented graph is linear of the form: y' = m'.x' or y-x = m'.(y+x) , with m' > 1 (since the angle of the line with respect to the +x' axis is positive and between 45 deg and 90 deg).

Solving this for y as a function of x yields:

y = [(1+m')/(1-m')].x , which is a line with a slope m = (1+m')/(1-m').

Since m' is greater than 1, m will be less than -1 (that is, the slope will be negative but with an absolute value greater than 1). Hence (4) is the answer.