Plotting Linear Relationships — Solutions

Click any answer to watch the solution video.

1. Tables of values (x = −2 to 2)

   1. y = x + 3: y: 1, 2, 3, 4, 5
   2. y = 2x: y: −4, −2, 0, 2, 4
   3. y = −x + 4: y: 6, 5, 4, 3, 2
   4. y = 3x − 2: y: −8, −5, −2, 1, 4
   5. y = −2x + 1: y: 5, 3, 1, −1, −3
   6. y = x − 5: y: −7, −6, −5, −4, −3
   7. y = 4x + 1: y: −7, −3, 1, 5, 9
   8. y = −3x − 1: y: 5, 2, −1, −4, −7

2. Gradient and y-intercept

   1. y = 3x + 5: gradient = 3, y-intercept = 5
   2. y = −2x + 7: gradient = −2, y-intercept = 7
   3. y = x − 4: gradient = 1, y-intercept = −4
   4. y = −x: gradient = −1, y-intercept = 0
   5. y = 5x: gradient = 5, y-intercept = 0
   6. y = −4x − 3: gradient = −4, y-intercept = −3
   7. y = 2x: gradient = 2, y-intercept = 0
   8. y = ½x + 2: gradient = 0.5, y-intercept = 2
   9. y = −3x + 6: gradient = −3, y-intercept = 6
   10. y = 10 − x: gradient = −1, y-intercept = 10

3. Calculate gradient between two points

   1. (0,0) to (3,6): 2
   2. (1,5) to (3,9): 2
   3. (0,4) to (2,0): −2
   4. (−1,3) to (2,9): 2
   5. (2,8) to (5,8): 0 (horizontal line)
   6. (0,0) to (4,−12): −3
   7. (−2,−5) to (3,10): 3
   8. (1,7) to (4,1): −2
   9. (−3,2) to (1,10): 2
   10. (5,−1) to (10,−6): −1

4. Identify linear from table & find rule

   1. x: 0,1,2,3 / y: 1,4,7,10: Linear; constant difference +3; y = 3x + 1
   2. x: 0,1,2,3 / y: 0,1,4,9: Not linear; differences are 1, 3, 5 (not constant) — this is y = x²
   3. x: −1,0,1,2 / y: 5,3,1,−1: Linear; constant difference −2; y = −2x + 3
   4. x: 0,2,4,6 / y: 3,7,11,15: Linear; gradient = (7−3)/(2−0) = 2; y = 2x + 3
   5. x: 1,2,3,4 / y: 3,6,10,15: Not linear; differences are 3, 4, 5 (not constant)
   6. x: 0,1,2,3 / y: −2,1,4,7: Linear; constant difference +3; y = 3x − 2

5. Describe each line

   1. y = 5x + 2: Steep positive slope (gradient 5), crosses y-axis at 2
   2. y = −x + 3: Negative slope (gradient −1), crosses y-axis at 3
   3. y = ¼x − 1: Gentle positive slope (gradient 0.25), crosses y-axis at −1
   4. y = −4x: Steep negative slope (gradient −4), passes through origin
   5. y = 3: Horizontal line (gradient = 0), crosses y-axis at 3
   6. y = 2x − 6: Positive slope (gradient 2), crosses y-axis at −6

6. Real-context linear relationships

   1. Cyclist: d = 25t; gradient = 25 means 25 km per hour (speed)
   2. Phone plan: C = 0.10n + 20; y-intercept = $20 (fixed monthly fee); gradient = 0.10 (cost per text)
   3. (0,−3) and (4,5): gradient = (5−(−3))/(4−0) = 2; equation: y = 2x − 3; when x=10, y=17
   4. y = 3x+1 and y = 3x−5: Parallel lines (same gradient of 3, different y-intercepts). They never intersect.

7. Complete table for y = 2x + 1 and check with graph

   1. Table of values (x = −2, −1, 0, 1, 2): y = −3, −1, 1, 3, 5
   2. y-intercept from table/graph: 1 — when x = 0, y = 2(0) + 1 = 1. Point (0, 1).
   3. Gradient from graph: 2 — for every 1 unit right, the line rises 2 units.
   4. Equation confirmed: y = 2x + 1 ✓

8. Read the graph — y = −x + 2

   1. y-intercept: 2 — line crosses y-axis at (0, 2)
   2. x-intercept: 2 — line crosses x-axis at (2, 0)
   3. Gradient (using (−2, 4) and (2, 0)): m = (0 − 4) ÷ (2 − (−2)) = −4 ÷ 4 = −1
   4. Equation: y = −x + 2 (gradient −1, y-intercept 2)

9. Swimming pool drain

   1. Linear relationship?: Yes — depth decreases by exactly 25 cm each hour (constant rate of change)
   2. Depth at t = 4: 100 cm (125 − 25 = 100)
   3. Gradient and meaning: −25 — the pool loses 25 cm of depth per hour
   4. Equation: D = −25t + 200 (or D = 200 − 25t)
   5. When pool empties (D = 0): 0 = 200 − 25t → t = 8 hours

10. Two lines — y = x + 3 and y = x − 3

    1. What you notice: The two lines are parallel — they have the same steepness and the same direction, but one is shifted up and the other is shifted down on the y-axis.
    2. Comparing the rules: Both rules have the same gradient (coefficient of x = 1). They differ only in the y-intercept: y = x + 3 crosses at (0, 3) and y = x − 3 crosses at (0, −3). Same slope, different starting heights.
    3. y-values when x = 2: y = x + 3: y = 2 + 3 = 5.   y = x − 3: y = 2 − 3 = −1.
    4. Will they ever cross?: No. Both lines have gradient 1, so they are parallel. Setting x + 3 = x − 3 gives 3 = −3, which is impossible — there is no x-value that satisfies both rules at the same time. Parallel lines never intersect.