Mathematics • Year 8 • Unit 2 • Lesson 14
Sketching Linear Graphs — Mixed Challenge
Pull together all three sketching methods, plus special lines (horizontal, vertical, through origin) and method-choosing. Six mixed problems, one "find the mistake", one open-ended challenge.
1. Mixed problems
For each, state your chosen method, two key points, and a one-line description of the sketch. 3 marks each
1.1 Sketch y = 3x + 2. (Which method is best?)
1.2 Sketch x + 2y = 4. State the two intercepts.
1.3 Sketch y = −2x + 6 using the gradient-intercept method. Step from (0, 6) using m = −2.
1.4 Sketch the two special lines x = −1 and y = 4 on the same axes. Where do they meet?
1.5 A point (3, k) lies on the line y = 2x − 3. Find k, then verify the line passes through (3, k) by sketching it through (0, −3) and your new point.
1.6 Sketch 2x − 3y = 6 by intercept method. Then state its gradient by computing m from the two intercepts.
2. Find the mistake
A student tried to sketch y = −½x + 4 using the gradient-intercept method. Their working is shown. Exactly one line contains a mistake. Spot it, explain why, and re-do correctly. 3 marks
Student's working — sketch y = −½x + 4:
Line 1: m = −½, c = 4.
Line 2: Plot (0, 4).
Line 3: m = −½ means run 2 right, rise 1 UP → from (0, 4) move to (2, 5).
Line 4: Draw a line through (0, 4) and (2, 5).
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the corrected step and the corrected second point.
Stuck? Negative gradient means rise is NEGATIVE — the line goes DOWN as it moves right, not up.3. Open-ended challenge — three lines, same x-intercept
This question has more than one valid answer. 4 marks
3.1 Design three different straight lines that all share the x-intercept (3, 0) but cross the y-axis at three different heights.
For each line you find:
(i) Write the equation in y = mx + c form.
(ii) State the gradient and the y-intercept.
(iii) Verify it passes through (3, 0).
Bonus: Which sketching method is fastest for the lines you've written? Explain your choice in one sentence.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — y = 3x + 2
Best method: gradient-intercept (equation is y = mx + c). Plot (0, 2); run 1, rise 3 → (1, 5). Steep uphill line.
1.2 — x + 2y = 4
Intercept method. y-int: 2y = 4 → (0, 2). x-int: x = 4 → (4, 0). Plot both, join.
1.3 — y = −2x + 6
Plot (0, 6). m = −2: run 1 right, rise 2 DOWN → (1, 4). Or step again: (2, 2). x-intercept at (3, 0). Downhill line.
1.4 — x = −1 and y = 4
x = −1 is a vertical line through (−1, 0). y = 4 is a horizontal line through (0, 4). They cross at the point (−1, 4).
1.5 — (3, k) on y = 2x − 3
k = 2(3) − 3 = 3. So the point is (3, 3). Sketch the line through (0, −3) and (3, 3). Gradient 2 checks out (rise 6 over run 3).
1.6 — 2x − 3y = 6
y-int (x = 0): −3y = 6 → y = −2 → (0, −2). x-int (y = 0): 2x = 6 → x = 3 → (3, 0). Plot both, join. Gradient m = (0 − (−2))/(3 − 0) = 2/3.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) m = −½ means the rise is NEGATIVE (−1), so from (0, 4) moving 2 right we should go 1 DOWN, not 1 up. The student treated the negative sign as if it didn't matter.
(c) Corrected: m = −1/2 → run 2 right, rise 1 DOWN → from (0, 4) to (2, 3). Draw the line through (0, 4) and (2, 3); the line goes downhill (as expected for negative gradient).
3 — Open-ended challenge (sample solution)
Substituting (3, 0) into y = mx + c gives 0 = 3m + c, so c = −3m. Pick three different m values:
Line 1: m = 1 → c = −3 → y = x − 3. Check (3, 0): 3 − 3 = 0 ✓. y-intercept (0, −3).
Line 2: m = 2 → c = −6 → y = 2x − 6. Check (3, 0): 6 − 6 = 0 ✓. y-intercept (0, −6).
Line 3: m = −1 → c = 3 → y = −x + 3. Check (3, 0): −3 + 3 = 0 ✓. y-intercept (0, 3).
Bonus: Gradient-intercept method — every equation is already in y = mx + c form, so we can read m and c straight off.
Marking: 1 mark per valid line with verification (3 marks). 1 bonus mark for naming the gradient-intercept method with a one-sentence reason.