Chapter 4 · Geometry and Equations · Concept 1
4-2 Points on the Coordinate Plane
Two number lines crossed at right angles, and suddenly every point on a flat surface has a name. Which quarter it lives in, and where it lands when the picture is folded, are the two things this lesson settles.
Guiding question
What does a point's pair of numbers tell you before you plot anything?
Learning outcome. Determine the quadrant of a point on the coordinate plane.
Four quarters, and the axes are in none of them
The two axes cut the plane into four quadrants, numbered anticlockwise from the top right — and the axes themselves belong to none of them.
1st
x > 0, y > 0
right and up
2nd
x < 0, y > 0
left and up
3rd
x < 0, y < 0
left and down
4th
x > 0, y < 0
right and down
And the three reflections.
· across the x-axis → change the sign of y
· across the y-axis → change the sign of x
· through the origin → change both
The rule that catches people is the first one. Reflecting in the x-axis changes y, not x — because folding over a flat line moves a point up or down, never sideways.
Pick a point, then press a mirror. Watch which way the dotted line goes — that is the direction the point moved, and the coordinate that did not move is the one that stays.
A(-3, 2) is in the 2nd quadrant
2. For A(−3, 2), find the point symmetric to it about the x-axis, the y-axis, and the origin.
- 1
Sketch it first. A(−3, 2) is left and up, so the 2nd quadrant.
x is negative, y is positiveThe picture is what makes the next three steps obvious instead of remembered.
Work with a partner. One of you says the answer from the signs alone; the other checks it on a sketch. Swap after every question — the point is to end up trusting both routes.
1. Which quadrant?
2. For A(−1, 2), find:
1. Which quadrant?
2–4. Find P (about the x-axis), Q (about the y-axis) and R (about the origin).
A quick check that costs nothing.
Each image must be the same distance from the origin as the point you started with — folding never moves a point closer in or further out. If A is 5 units out, all three images are too.
A robotic welding arm is programmed at point A(−3, 4) on a metal sheet. To weld identical support brackets, the arm must mirror this point about the x-axis, about the y-axis, and through the origin. Where does it weld?
Reflecting in the x-axis by changing x.
It changes y. Folding over a flat line moves a point up or down, so the sideways coordinate cannot change.
Numbering the quadrants clockwise.
They go anticlockwise from the top right: 1st, 2nd, 3rd, 4th.
Putting a point on an axis into a quadrant.
(0, 3) and (−4, 0) are in no quadrant at all. The book excludes the axes deliberately.
Changing only one sign for the origin.
Through the origin means BOTH signs. It is the two reflections done one after the other.
Not sketching.
Every question here takes ten seconds on a grid, and a sketch catches a flipped sign before it costs anything.
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