The axes divide the plane into four quadrants, each with one sign pattern. (3, 5) and (5, 3) are different points. Reflecting in an axis changes one sign and leaves every length unchanged.
The two axes divide the plane into four regions called quadrants. They are numbered I to IV, going anticlockwise, starting from the region where both coordinates are positive.
Each quadrant has one sign pattern. You can work it out from two facts: right of the y-axis means x > 0, and above the x-axis means y > 0.
Work the signs out from position. A point in Quadrant III is left of the y-axis and below the x-axis, so x < 0 and y < 0. The same reasoning gives every pattern.
A point has x-coordinate −5. Without knowing y, which quadrants could it be in?
x < 0 means the point is left of the y-axis. That rules out Quadrants I and IV.
If y > 0, it is in Quadrant II. If y < 0, it is in Quadrant III. If y = 0, it is on the x-axis and in no quadrant.
One coordinate settles left or right. The other settles above or below. Each one, alone, narrows the point down to half the plane.
The quadrants are the regions between the axes. The axes themselves are the boundaries, and a boundary is not part of either region it separates. So (0, 4), (−3, 0) and (0, 0) are in no quadrant.
In (x, y), the first number is always the x-coordinate and the second is always the y-coordinate. Swapping them usually gives a different point.
Q (−5, 3) is in Quadrant II: left and up. The swapped pair (3, −5) is in Quadrant IV: right and down.
If x = y, the swapped pair is identical: (4, 4) swapped is (4, 4). The exact rule is: (x, y) and (y, x) are the same point if and only if x = y.
When x ≠ y, the two points are reflections of each other in the line through the origin on which every point has x = y.
A reflection in an axis moves every point to the same distance on the other side of that axis.
Take the triangle A (3, 4), D (7, 1), M (9, 6). Reflected in the y-axis, it becomes A′ (−3, 4), D′ (−7, 1), M′ (−9, 6). Every x-coordinate has changed. The side lengths have not: AD = 5, DM = √29 and MA = √40, and the same three values hold for the image.
A reflection changes the position of a shape. It does not change its lengths.
Why are the side lengths unchanged when every x-coordinate has changed?
A length depends on the differences between coordinates, not on the coordinates themselves.
The horizontal difference between A (3, 4) and D (7, 1) is 7 − 3 = 4. After reflection, it is −3 − (−7) = 4. The vertical differences are not touched at all, because y does not change.
Same horizontal difference and same vertical difference give the same length. The next session turns this into the distance formula.
1Which quadrant does each point lie in?
2Which quadrant is the point (0, 4) in?
3Point W has x-coordinate −5. Which quadrants could W be in?
4For which points is (x, y) the same point as (y, x)?
5Reflecting the point (6, −2) in the y-axis gives the point ____.
6A triangle is reflected in the x-axis. What happens to its side lengths?
7The point (−4, 7) is reflected in the x-axis. In which quadrant is its image?
8Which statements are true?
More than one answer is correct.
Next: calculating the distance between any two points.
Noescio