Mathematics • Year 8 • Unit 2 • Lesson 12
Finding Equation from a Graph
Build fluency with the two-step method: read c from where the line crosses the y-axis, then calculate m from a gradient triangle. One worked example, one guided fill-in, then eight independent problems.
1. I do — fully worked example
Read every line. The two-step method always works the same way: read c, then calculate m.
Problem. A straight line passes through (0, 2) and (3, 8). Find its equation in y = mx + c form.
Step 1 — Read c off the y-axis.
The line crosses the y-axis at (0, 2), so c = 2.
Reason: c is the y-value when x = 0 — exactly where the line meets the y-axis.
Step 2 — Calculate m using the gradient formula.
m = (y₂ − y₁) / (x₂ − x₁) = (8 − 2) / (3 − 0) = 6/3 = 2
Reason: gradient = rise / run. Between the two points the line rises 6 units while running 3 units across.
Step 3 — Substitute into y = mx + c.
y = 2x + 2
Step 4 — Verify with the second point.
At x = 3: y = 2(3) + 2 = 8 ✓ — matches (3, 8).
Answer: y = 2x + 2.
2. We do — fill in the missing steps
A line crosses the y-axis at (0, 5) and passes through (4, 1). Fill in the blanks. 4 marks
Step 1 — Read c off the y-axis:
Line crosses at (0, ____ ), so c = ______ .
Step 2 — Calculate m using points (0, 5) and (4, 1):
m = ( ______ − ______ ) / ( ______ − ______ ) = ______ / ______ = ______
Step 3 — Substitute into y = mx + c:
y = ______ x + ______
Step 4 — Verify with (4, 1):
y = ______ × 4 + ______ = ______ . Matches y = 1 ? ______
3. You do — independent practice
Show your working in the space under each problem. Foundation (read directly), Standard (compute from two points), Extension (special cases).
Foundation — read m and c from given info
3.1 A line crosses the y-axis at (0, 4) and has gradient 1. Write its equation. 1 mark
3.2 A line crosses the y-axis at (0, −3) and has gradient 2. Write its equation. 1 mark
3.3 A graph shows a line with y-intercept 1 and a gradient triangle with rise 3, run 1. Find m, then state the equation. 1 mark
3.4 A line passes through the origin (0, 0) and the point (1, 4). Find m and c, then write the equation. 1 mark
Standard — compute from two points
3.5 A line passes through (0, 1) and (2, 7). Find its equation. Show your gradient calculation. 2 marks
3.6 A line passes through (0, 6) and (3, 0). Find its equation and verify with the second point. 2 marks
Extension — special cases
3.7 A line passes through (0, 3) and (2, 3). What is its gradient? Write the equation and explain in one sentence what kind of line this is. 2 marks
3.8 A line passes through (−2, 1) and (2, 5). (Hint: c is NOT directly visible — you need to find it from y = mx + c after computing m.) 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (downhill line)
Step 1: crosses at (0, 5), c = 5.
Step 2: m = (1 − 5)/(4 − 0) = −4/4 = −1.
Step 3: y = −1x + 5, i.e. y = −x + 5.
Step 4: y = −1 × 4 + 5 = 1. Matches y = 1 ✓.
3.1 — c = 4, m = 1
y = (1)x + 4 → y = x + 4.
3.2 — c = −3, m = 2
y = (2)x + (−3) → y = 2x − 3.
3.3 — c = 1, rise 3, run 1
m = rise/run = 3/1 = 3. So y = 3x + 1.
3.4 — Through (0,0) and (1, 4)
c = 0 (passes through origin). m = (4 − 0)/(1 − 0) = 4. So y = 4x.
3.5 — Through (0, 1) and (2, 7)
c = 1. m = (7 − 1)/(2 − 0) = 6/2 = 3. So y = 3x + 1. Check at (2, 7): 3(2) + 1 = 7 ✓.
3.6 — Through (0, 6) and (3, 0)
c = 6. m = (0 − 6)/(3 − 0) = −6/3 = −2. So y = −2x + 6. Check at (3, 0): −2(3) + 6 = 0 ✓.
3.7 — Through (0, 3) and (2, 3)
m = (3 − 3)/(2 − 0) = 0/2 = 0. So y = 3. This is a horizontal line — flat, with y = 3 at every point.
3.8 — Through (−2, 1) and (2, 5)
m = (5 − 1)/(2 − (−2)) = 4/4 = 1. Now find c using (2, 5): 5 = 1(2) + c → c = 3. So y = x + 3. Check at (−2, 1): (−2) + 3 = 1 ✓.