Point $$A(5, -3)$$ is in Quadrant IV (positive $$x$$, negative $$y$$). Point $$C(-5, 3)$$ is in Quadrant II (negative $$x$$, positive $$y$$). Since $$A$$ and $$C$$ differ only by signs in both coordinates, they are reflections across both axes. Choice A is wrong because $$B$$ is in Quadrant III, not a reflection of $$C$$ across the $$x$$-axis. Choice B is wrong because $$A$$ is in Quadrant IV, not III. Choice D is wrong because $$B$$ is in Quadrant III, not II.

When you see coordinate points with variables and conditions like $$a > 0$$ and $$b < 0$$, you need to determine which quadrants the points fall into and identify any transformations between them.

Let's analyze each point systematically. Point $$R(a, b)$$ has coordinates where $$a > 0$$ (positive x-coordinate) and $$b < 0$$ (negative y-coordinate). This combination of positive x and negative y places point $$R$$ in Quadrant IV. Point $$S(-a, b)$$ has coordinates where $$-a < 0$$ (negative x-coordinate, since $$a > 0$$) and $$b < 0$$ (negative y-coordinate). This combination of negative x and negative y places point $$S$$ in Quadrant III.

To identify the relationship, notice that $$S(-a, b)$$ has the opposite sign for the x-coordinate compared to $$R(a, b)$$, while the y-coordinates remain identical. This indicates a reflection across the y-axis, which changes the sign of x-coordinates but preserves y-coordinates.

Looking at the wrong answers: Choice A incorrectly places $$S$$ in Quadrant II and suggests reflection across both axes. Choice B wrongly puts $$R$$ in Quadrant III and claims reflection across the x-axis. Choice C misplaces $$R$$ in Quadrant II and $$S$$ in Quadrant I, though it correctly identifies y-axis reflection.

Remember this pattern: when only the x-coordinate changes sign between two points, you have a reflection across the y-axis. Always check the signs of both coordinates against the given conditions to determine quadrants correctly.

Reflecting $$P(-4, 7)$$ across the $$x$$-axis gives $$(-4, -7)$$. Then reflecting across the $$y$$-axis gives $$(4, -7)$$, which is exactly point $$Q$$. When two points differ only by signs in both coordinates, reflecting across both axes maps one to the other. Choice B is wrong because the points are identical after the reflections. Choice C is wrong because there's no vertical separation. Choice D is wrong because the final positions coincide.

When working with coordinate points and quadrants, you need to remember the sign patterns: Quadrant I (+,+), Quadrant II (-,+), Quadrant III (-,-), and Quadrant IV (+,-).

Given the conditions $$m < 0$$ and $$n > 0$$, let's analyze each point. The first point $$(m, n)$$ has a negative x-coordinate and positive y-coordinate, placing it in Quadrant II. The second point $$(m, -n)$$ has a negative x-coordinate and negative y-coordinate (since $$-n < 0$$ when $$n > 0$$), placing it in Quadrant III.

Both points share the same x-coordinate $$m$$, so they lie on the same vertical line. Since one has y-coordinate $$n$$ and the other has y-coordinate $$-n$$, they are equidistant from the x-axis (the same distance above and below it).

Choice A incorrectly places the points in Quadrants IV and I, which would require different sign combinations. Choice B correctly identifies the quadrants but wrongly claims they share the same y-coordinate and are equidistant from the y-axis. Choice C places the first point in Quadrant I, which is impossible since $$m < 0$$, and incorrectly describes their relationship.

Choice D correctly identifies that the first point is in Quadrant II and the second is in Quadrant III, they share the same x-coordinate, and are equidistant from the x-axis.

Study tip: Always check the signs of coordinates against the given conditions first, then identify quadrants using the sign patterns. Points with the same x-coordinate form vertical lines, while points with opposite y-coordinates are reflections across the x-axis.

A point in Quadrant III has negative $$x$$ and negative $$y$$ coordinates. Reflecting across the $$x$$-axis changes the sign of $$y$$ only, giving negative $$x$$ and positive $$y$$ (Quadrant II). Reflecting across the $$y$$-axis changes the sign of $$x$$ only, giving positive $$x$$ and negative $$y$$ (Quadrant IV). Choice A is wrong because reflecting across one axis cannot reach Quadrant I. Choice C is wrong because reflection across the $$x$$-axis keeps $$x$$ negative. Choice D is wrong because reflection across the $$y$$-axis does change the $$x$$-coordinate sign.

Point $$W$$ in Quadrant I has positive $$x$$ and positive $$y$$ coordinates. Reflecting across the $$x$$-axis changes the $$y$$-coordinate sign, so $$X$$ has positive $$x$$ and negative $$y$$ (Quadrant IV). Reflecting across the $$y$$-axis changes the $$x$$-coordinate sign, so $$Y$$ has negative $$x$$ and positive $$y$$ (Quadrant II). Reflecting across both axes changes both coordinate signs, so $$Z$$ has negative $$x$$ and negative $$y$$ (Quadrant III). Choice A incorrectly places $$X$$ and $$Y$$. Choice C incorrectly places all three points. Choice D incorrectly places $$Y$$ and $$Z$$.

This question tests understanding that the signs in ordered pairs indicate quadrant location (I: +,+; II: -,+; III: -,-; IV: +,-), and points differing only by signs are reflections across axes. Quadrants are defined as I (x>0, y>0 upper right both positive), II (x<0, y>0 upper left), III (x<0, y<0 lower left both negative), IV (x>0, y<0 lower right); signs indicate x-coordinate shows left/right of y-axis (positive right, negative left), y-coordinate shows above/below x-axis (positive above, negative below); reflections occur when pairs differ only in signs—(3,5) and (-3,5) differ in x-sign (across y-axis), (3,5) and (3,-5) in y-sign (across x-axis), (3,5) and (-3,-5) in both (across origin). For example, (3,5) has x=3>0 (right of y-axis), y=5>0 (above x-axis), both positive so Quadrant I; (-4,2) has x=-4<0 (left), y=2>0 (above) so Quadrant II; reflections include (3,5) across y-axis to (-3,5), across x-axis to (3,-5), across origin to (-3,-5). For point A(3,5), both coordinates are positive, placing it in Quadrant I, which is choice C. A common error is confusing quadrants, like thinking (+,+) is Quadrant II instead of I, or misreading signs and placing it in Quadrant IV. To determine the quadrant: (1) check x-sign (positive means right side, quadrants I or IV), (2) check y-sign (positive means upper, quadrants I or II), (3) combine to both positive for I. Remember quadrant order is counterclockwise from upper right: I to II to III to IV, and avoid mistaking axis points for quadrants.

This question tests understanding that the signs in ordered pairs indicate quadrant location (I: +,+; II: -,+; III: -,-; IV: +,-), and points differing only by signs are reflections across axes. Quadrants are defined as I (x>0, y>0 upper right both positive), II (x<0, y>0 upper left), III (x<0, y<0 lower left both negative), IV (x>0, y<0 lower right); signs indicate x-coordinate shows left/right of y-axis (positive right, negative left), y-coordinate shows above/below x-axis (positive above, negative below); reflections occur when pairs differ only in signs—(3,5) and (-3,5) differ in x-sign (across y-axis), (3,5) and (3,-5) in y-sign (across x-axis), (3,5) and (-3,-5) in both (across origin). For example, (3,5) has x=3>0 (right of y-axis), y=5>0 (above x-axis), both positive so Quadrant I; (-4,2) has x=-4<0 (left), y=2>0 (above) so Quadrant II; reflections include (3,5) across y-axis to (-3,5), across x-axis to (3,-5), across origin to (-3,-5). Points A(3,5) and E(-3,5) differ only in x-sign, so they are reflections across the y-axis, which is choice B. A common error is thinking they are across x-axis when y-signs are the same, or claiming no reflection. Reflections: identify which sign differs (only x for y-axis), understand symmetry as mirror image across axis. Mistakes include claiming origin reflection when only one sign differs or confusing the axes.

This question tests understanding that the signs in ordered pairs indicate quadrant location, with Quadrant I for (+,+), II for (-,+), III for (-,-), and IV for (+,-), and points differing only by signs are reflections across axes. Quadrants are defined as follows: I (x>0, y>0 upper right both positive), II (x<0, y>0 upper left), III (x<0, y<0 lower left both negative), IV (x>0, y<0 lower right); signs indicate x-coordinate for left/right of y-axis (positive right, negative left) and y-coordinate for above/below x-axis (positive above, negative below); reflections occur when pairs differ only in signs, like (3,5) and (-3,5) across y-axis (x-sign flip), (3,5) and (3,-5) across x-axis (y-sign flip), or (3,5) and (-3,-5) across origin (both flips). For example, (3,5) has x=3>0 (right of y-axis), y=5>0 (above x-axis), both positive so Quadrant I; (-4,2) has x=-4<0 (left), y=2>0 (above) so Quadrant II; reflections include (3,5) to (-3,5) across y-axis, (3,-5) across x-axis, (-3,-5) across origin. For point D(2,-3), x is positive and y is negative, placing it in Quadrant IV. A common error is identifying (+,-) as Quadrant II instead of IV, or confusing quadrant numbering like calling IV upper left. To determine the quadrant: (1) check x-sign (positive means right side, I or IV), (2) check y-sign (negative means lower, III or IV), (3) combine for x positive y negative as IV. Remember quadrant order is counterclockwise from upper right: I to II to III to IV, and avoid mistakes like confusing II and IV commonly.

This question tests understanding that the signs in ordered pairs indicate quadrant location, with Quadrant I for (+,+), II for (-,+), III for (-,-), and IV for (+,-), and points differing only by signs are reflections across axes. Quadrants are defined as follows: I (x>0, y>0 upper right both positive), II (x<0, y>0 upper left), III (x<0, y<0 lower left both negative), IV (x>0, y<0 lower right); signs indicate x-coordinate for left/right of y-axis (positive right, negative left) and y-coordinate for above/below x-axis (positive above, negative below); reflections occur when pairs differ only in signs, like (3,5) and (-3,5) across y-axis (x-sign flip), (3,5) and (3,-5) across x-axis (y-sign flip), or (3,5) and (-3,-5) across origin (both flips). For example, (3,5) has x=3>0 (right of y-axis), y=5>0 (above x-axis), both positive so Quadrant I; (-4,2) has x=-4<0 (left), y=2>0 (above) so Quadrant II; reflections include (3,5) to (-3,5) across y-axis, (3,-5) across x-axis, (-3,-5) across origin. For point C(-3,-5), both coordinates are negative, placing it in Quadrant III. A common error is confusing quadrants, like thinking (-,-) is Quadrant IV instead of III, or mixing up numbering where II is mistaken for lower left. To determine the quadrant: (1) check x-sign (negative means left side, II or III), (2) check y-sign (negative means lower, III or IV), (3) combine for both negative as III. Remember quadrant order is counterclockwise from upper right: I to II to III to IV, and avoid mistakes like quadrant identification wrong, especially confusing III and IV.