Across first, then up.
(x, y) — the x always comes first.
(3, 2) and (2, 3) are different points
Order matters. Read it as "3 across, 2 up". A point on an axis has a zero in it: (5, 0) sits on the x-axis, (0, 5) on the y-axis.
- Home
- Grade 7-12 Math Review
- The Coordinate Plane and Graphing Lines
The Coordinate Plane and Graphing Lines
Two numbers name a point. Two points draw a line.
Find the intercepts
Where does the line 2x + 3y = 12 cross the x-axis?
On the x-axis, y is 0. Substituting: 2x = 12, so x = 6, giving (6, 0). A is the y-intercept — that comes from setting x = 0, which gives 3y = 12 and y = 4. The two are easy to swap: the x-intercept is where y disappears, not x.
A table of values turns an equation into points.
y = 2x − 1
choose easy x values
| x | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| y | −3 | −1 | 1 | 3 | 5 |
Three points are enough for a straight line, and the third is your check: if it does not sit on the line through the other two, something is wrong.
The intercept method is faster for Ax + By = C.
Set x = 0 for one intercept, y = 0 for the other.
2x + 3y = 12
standard form
x = 0 → 3y = 12 → y = 4
(0, 4)
y = 0 → 2x = 12 → x = 6
(6, 0)
plot both, join, extend
two points fix a line
Standard form hides the shape of the line but makes the intercepts trivial. Two substitutions and the graph is drawn.
Horizontal and vertical lines.
y = 3
horizontal
y is 3 everywhere
x = −2
vertical
x is −2 everywhere
y = 0
the x-axis itself
and x = 0 is the y-axis
These look backwards to almost everyone at first: y = 3 is a horizontal line, because it is the y-value that stays fixed while x roams.
Is a point on the line? Substitute and see.
Is (3, 2) on 2x + 3y = 12?
test it
2(3) + 3(2) = 6 + 6 = 12 ✓
yes, it is on the line
Is (1, 4)? 2 + 12 = 14 ≠ 12
no
A line is the complete set of points that make the equation true. Testing a point is not a trick — it is the definition.
Reflecting a point across an axis.
Across the x-axis, y changes sign. Across the y-axis, x changes sign.
Reflect the point (−6, 2) across each axis, and through the origin.
| Reflect across | Rule | Image of (−6, 2) |
|---|---|---|
| x-axis | (x, y) → (x, −y) | (−6, −2) |
| y-axis | (x, y) → (−x, y) | (6, 2) |
| the origin | (x, y) → (−x, −y) | (6, −2) |
The mirror line keeps its own coordinate. The x-axis is horizontal, so a point flipped over it moves straight up or down: its x stays −6 and only the y changes sign. The point moves from quadrant II to quadrant III.
Changing the wrong coordinate
Reflecting (−6, 2) across the x-axis is not (6, 2) — that point is the reflection across the y-axis. Picture it: a flip over the horizontal axis can only move a point up or down.
Variations: a missing coordinate, fraction intercepts, a line through the origin.
Substitution and intercepts, in three less obvious settings.
The point (k, 11) lies on y = 4x − 5. Find k.
11 = 4k − 5
substitute x = k, y = 11
16 = 4k, k = 4
solve for the unknown coordinate
Check: 4(4) − 5 = 11 ✓. The unknown can be either coordinate; substitute what you know and solve.
Find the intercepts of 4x + 6y = 9.
x = 0 → 6y = 9 → y = 32
(0, 32)
y = 0 → 4x = 9 → x = 94
(94, 0)
Intercepts need not be whole numbers. Plot 32 as 1.5 and 94 as 2.25.
Graph y = −3x.
x = 0 → y = 0
both intercepts are (0, 0)
x = 1 → y = −3
a second point: (1, −3)
When the line passes through the origin, the two intercepts are the same point, so they cannot fix the line. Choose any other x for the second point.
On the exam
Exams ask for the quadrant of a point, an intercept, whether a point lies on a line, or the image of a point after a reflection. The fastest routes: x = 0 for the y-intercept, y = 0 for the x-intercept, and substitute to test a point. For a 10-second check, put your point back into the equation, or sketch it to see that the quadrant matches the signs.
Now you try
Practice problem
Find the y-intercept
Where does 5x − 2y = 20 cross the y-axis?
On the y-axis x = 0, so −2y = 20 and y = −10, giving (0, −10). B drops the minus — dividing 20 by −2 gives −10, not 10. C is the x-intercept, found by setting y = 0 instead.
Hint: set x = 0 first, then watch the sign when you divide by −2.