Quadrant III is where x values are negative and y values are negative.
If x = -3 and y = x, then y = -3, giving us the point (-3,-3), which is in quadrant III.
Yes, y = x will have points in quadrant III.
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Quadrant IV is a more interesting question.
Quadrant IV is where x values are positive and y values are negative.
The function y = x will always give points with the same sign (+ or -). If x is -7, then y is -7, same sign. If x is +136 then y is +136, same sign.
The answer you're probably looking for is no, y = x will not have any points in quadrant IV.
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The interesting part of asking about y = x and quadrant IV is the origin at the point (0,0). That point fits the equation y = x, but does it lie within any of quadrants, all of the quadrants, or none of the quadrants?
The quadrants are defined in terms of having positive or negative x values and positive or negative y values, so the question of what quadrant (if any) the origin belongs to is similar to asking if 0 is positive or negative or both.
Positive and negative numbers are defined as numbers greater than or less than 0, but since 0 is neither greater than 0 nor less than 0, then 0 is neither positive nor negative. If 0 isn't positive or negative, then the point (0,0) doesn't fall within any of the quadrants.
Since (0,0) isn't within quadrant IV and no other point on the line y = x is in quadrant IV, it looks like y = x will never have any points in quadrant IV.