Mathematics • Year 8 • Unit 2 • Lesson 10
y-Intercept — Mixed Challenge
Pull together everything from Lesson 10: reading c from y = mx + c, substituting x = 0, rearranging equations, and building y = mx + c when given a gradient and a y-intercept. Six mixed problems, one "find the mistake", and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different combination of ideas from Lesson 10. Show all working. 3 marks each
1.1 Find the y-intercept of each: (a) y = 7x − 11, (b) y = −x + 9, (c) y = ½x, (d) y = −3.
1.2 Find the y-intercept of 3y = 12x − 18 by first rearranging to y = mx + c. State both m and c.
1.3 Write the equation of the line with gradient m = −2 and y-intercept (0, 7).
1.4 A line passes through (0, −4) and (3, 8). Find m and c, then write y = mx + c.
1.5 A line passes through (2, 9) and (5, 18). Find m, then use the equation y = mx + c with one of the points to find c. Write the equation.
1.6 Sketch (or describe) the line y = ½ x − 3, marking the y-intercept and one other point (use the gradient to step from the y-intercept).
2. Find the mistake
A student has tried to find the y-intercept of 2y = 4x + 6. Exactly one line of their reasoning is wrong. Spot it, explain why, then re-do correctly. 3 marks
Student's working:
Line 1: 2y = 4x + 6.
Line 2: It's already in the form y = mx + c.
Line 3: So m = 4 and c = 6.
Line 4: y-intercept = (0, 6).
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the corrected working — start by rearranging, then identify m, c, and the y-intercept.
Stuck? "y = mx + c" needs y alone on one side, not 2y. Divide everything by 2 first.3. Open-ended challenge — design three lines
This question has many valid answers. 4 marks
3.1 Find three different linear equations that all share the same y-intercept of (0, 4), but each has a different gradient (one positive, one negative, one zero).
For each line:
(i) Write its equation in the form y = mx + c.
(ii) State its m and c.
(iii) Substitute x = 0 to confirm it passes through (0, 4).
(iv) One sentence: describe how its direction differs from the other two.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Read off c
(a) y = 7x − 11 → c = −11; (0, −11). (b) y = −x + 9 → c = 9; (0, 9). (c) y = ½x → c = 0; (0, 0). (d) y = −3 → c = −3; (0, −3).
1.2 — Rearrange 3y = 12x − 18
Divide by 3: y = 4x − 6. So m = 4, c = −6; y-intercept (0, −6).
1.3 — Build line, m = −2, intercept (0, 7)
y = −2x + 7.
1.4 — Through (0, −4) and (3, 8)
(0, −4) has x = 0, so c = −4. m = (8 − (−4))/(3 − 0) = 12/3 = 4. Equation: y = 4x − 4.
1.5 — Through (2, 9) and (5, 18)
m = (18 − 9)/(5 − 2) = 9/3 = 3. Use (2, 9): 9 = 3(2) + c → c = 3. Equation: y = 3x + 3. (Check with (5,18): 3(5) + 3 = 18 ✓.)
1.6 — Sketch y = ½ x − 3
y-intercept at (0, −3). Gradient ½ means rise 1, run 2 → next plot point (2, −2). Draw the straight line through these — slopes gently uphill from below the x-axis through to above it.
2 — Find the mistake
(a) The mistake is on Line 2 (the wrong assumption is then carried into Line 3 and Line 4).
(b) The equation 2y = 4x + 6 is NOT in the form y = mx + c — y is not alone on the left side. You must divide everything by 2 first to get y on its own.
(c) Corrected working: 2y = 4x + 6 → y = 2x + 3. So m = 2, c = 3, and the y-intercept is (0, 3).
3 — Three lines through (0, 4) (sample solution)
Many valid sets. One example:
Line A (positive): y = 3x + 4. m = 3, c = 4. x = 0 → y = 4 ✓. Slopes uphill left-to-right.
Line B (negative): y = −2x + 4. m = −2, c = 4. x = 0 → y = 4 ✓. Slopes downhill left-to-right.
Line C (zero): y = 4. m = 0, c = 4. x = 0 → y = 4 ✓. Horizontal — doesn't slope at all.
All three pass through the y-axis at the same point (0, 4) but fan out in three different directions — like the spokes of a fan pinned at (0, 4).
Marking: 1 mark per valid line (positive / negative / zero gradient, all with c = 4) with correct check at x = 0 (up to 3 marks). 1 mark for a clear description of how they differ in direction.