Two numbers hide in every straight line — where it crosses, and how fast it climbs. Read both off the picture and the equation writes itself.
Read the graph
Which equation matches the graphed line?
The line crosses the y-axis at −2, so b = −2 — that alone kills A and D. From (0, −2) to (3, 4) it rises 6 while running 3, so m = 63 = 2: y = 2x − 2. B has the right crossing but a slope of 1 — it would pass through (3, 1), not (3, 4).
No computation. The crossing point IS the number.
crossing point: (0, −2)
read straight off the picture
b = −2
the y-intercept, sign included
In a multiple-choice question this single read often eliminates half the choices before you compute anything.
Pick lattice points — corners where the line crosses grid intersections exactly.
m = riserun = 4 − (−2)3 − 0
using the two marked points
m = 63 = 2
up 2 for every 1 across
Half of all slope errors are sign errors — and the sign is visible from across the room.
| The line… | The slope is… |
|---|---|
| climbs left → right | positive |
| falls left → right | negative |
| is horizontal | zero |
| is vertical | undefined — no m at all |
Our line climbs, so m must be positive — any choice with a negative slope is dead before you count a single square. Steeper than 45° means |m| > 1; flatter means a fraction.
m = 2, b = −2
both read from the picture
y = 2x − 2
the equation of the graphed line
The axes count differently. Across, each gridline is 1; up, each gridline is 5. The line crosses the y-axis at 10 and passes through the lattice point (2, 25).
squares: 2 across, 3 up
counting squares alone says 32
run = 2 × 1 = 2, rise = 3 × 5 = 15
turn squares into units first
m = 152, b = 10
b read where it crosses
y = 15x2 + 10
check (2, 25): 15 + 10 = 25 ✓
What is different: a square is not one unit on both axes. Read the labels, then count units, never squares.
A flat line. The graphed line passes through (−3, −4) and (2, −4).
rise = −4 − (−4) = 0
it never climbs
m = 0, b = −4
it meets the y-axis at −4
y = −4
the 0x term disappears
What is different: slope 0 leaves only the y-intercept, so the equation is y = a constant.
An upright line. The graphed line passes through (3, −2) and (3, 4).
run = 3 − 3 = 0
rise ÷ 0 has no value
no slope and no y-intercept
it never meets the y-axis
x = 3
every point has x = 3
What is different: there is no m and no b to read, so y = mx + b is the wrong template; the equation names the x it stands on.
Many graphs use 1 across and 2, 5 or 10 up. Rise over run in squares is only the slope when both axes count by the same step.
The exam rarely stops at y = mx + b. Suppose a graphed line passes through the lattice points (−1, 3) and (3, 1), and crosses the y-axis between two grid lines.
m = (1 − 3) ÷ (3 − (−1)) = −24 = −12
it falls, so m < 0 ✓
y − 3 = −12(x + 1) → y = −x2 + 52
b is off the grid: compute it, do not guess it
0 = −x2 + 52 → x = 5
x-intercept: set y = 0
2y = −x + 5 → x + 2y = 5
standard form: × 2, then move x across
⊥ through (5, 0): y = 2(x − 5) = 2x − 10
flip and switch −12 to 2
∥ through (0, 1): y = −x2 + 1
parallel keeps the slope; only b changes
(9, k) on the line: k = −92 + 52 = −2
a point further along: put its x in
where y = 4: 4 = −x2 + 52 → x = −3
given a y, solve for x
(k, k) on the line: k = −k2 + 52 → 3k = 5 → k = 53
a point written with k: substitute both, solve
y = 2x − 10 meets the axes at (5, 0) and (0, −10): area = ½ × 5 × 10 = 25
triangle with the axes: ½ × the two intercepts, as lengths
b is where the line meets the y-axis (here 52). The x-intercept comes from setting y = 0 (here 5). The two are always offered side by side.
Graph items ask for the matching equation (often in standard form), the x-intercept, a point further along the line, or a parallel/perpendicular line. Read m from two lattice points and b from the crossing (or compute it), then check your equation on a second marked point before choosing.
Read the graph
What is the slope of the graphed line?
From (0, 3) to (4, 1) the line falls 2 while running 4, so m = −24 = −12. The line falls left to right, so the slope must be negative — that kills C and D on sight. A flips rise and run: −2 would drop two squares for every one across, far steeper than this.
