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.