Comparing Linear and Non-Linear Relationships — Solutions

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1. Classify equations as linear or non-linear

   1. y = 5x − 3: Linear — x appears only to power 1.
   2. y = x² + 1: Non-linear — x is squared (power 2).
   3. y = 3/x: Non-linear — x is in the denominator (equivalent to 3x⁻¹).
   4. y = −4x + 7: Linear — x to power 1; negative gradient does not make it non-linear.
   5. y = 2^x: Non-linear — x is an exponent (exponential relationship).
   6. y = x³ − 2: Non-linear — x is cubed (power 3).
   7. y = 9: Linear — a horizontal line; gradient = 0; x appears to power 0.
   8. y = ½x + 9: Linear — x to power 1 with fractional coefficient.

2. First difference test

   1. y: 2, 6, 10, 14, 18: First differences: 4, 4, 4, 4. All equal ⇒ Linear.
   2. y: 0, 1, 4, 9, 16: First differences: 1, 3, 5, 7. Not equal ⇒ Non-linear (quadratic, y = x²).
   3. y: −3, 1, 5, 9, 13: First differences: 4, 4, 4, 4. All equal ⇒ Linear.
   4. y: 1, 3, 9, 27, 81: First differences: 2, 6, 18, 54. Not equal (tripling) ⇒ Non-linear (exponential, y = 3^x).
   5. y: 5, 3, 1, −1, −3: First differences: −2, −2, −2, −2. All equal ⇒ Linear.

3. Match equation to description

   1. Straight line, positive gradient, negative y-intercept: A: y = 3x − 1 (gradient 3, y-intercept −1)
   2. U-shaped parabola, vertex (0, −2): B: y = x² − 2 (parabola shifted 2 down)
   3. Two-branch curve, asymptotes on both axes: C: y = 4/x (hyperbola)
   4. Always positive, passes through (0,1), rises rapidly: D: y = 2^x (exponential)

4. True or False

   1. Linear graph always passes through origin: False. A linear graph passes through the origin only if b = 0 (e.g. y = 3x). y = 2x + 5 is linear but does not pass through (0,0).
   2. Equal first differences ⇒ linear: True. A constant first difference means a constant rate of change, which is the defining feature of a linear relationship.
   3. y = −3x + 5 is non-linear because of negative gradient: False. A negative gradient does not make a relationship non-linear. y = −3x + 5 is linear — x appears to power 1.
   4. y = x² is a straight line: False. y = x² is a parabola (U-shaped curve), not a straight line.
   5. y = 2^x has constant rate of change: False. An exponential relationship has a changing rate of change — it grows faster and faster. First differences are not equal.
   6. Quadratic has constant second differences: True. For y = x²: first differences are 1, 3, 5, 7, … (not equal); second differences are 2, 2, 2, … (constant).

5. Real-world classification

   1. Earnings = $18/h: Linear. The earnings increase by $18 for every additional hour — a constant rate of change.
   2. Speed v = 10t: Linear. Speed increases by 10 m/s for each additional second — constant rate of change (straight line).
   3. Revenue triples each year: Non-linear (exponential). Revenue is multiplied by 3 each year, not added to by a fixed amount. First differences are not equal.
   4. Water drips at 2 mL/s: Linear. Volume increases by 2 mL every second — constant rate of change.

6. First differences and equation writing

   1. y: 4, 7, 10, 13: First differences: 3, 3, 3. Linear. m = 3, b = 4.   y = 3x + 4
   2. y: 0, 2, 8, 18: First differences: 2, 6, 10. Not equal ⇒ Non-linear (y = 2x²).
   3. y: −1, 2, 5, 8: First differences: 3, 3, 3. Linear. m = 3, b = −1.   y = 3x − 1
   4. y: 2, 4, 8, 16: First differences: 2, 4, 8. Not equal (doubling) ⇒ Non-linear (exponential, y = 2 × 2^x).

7. Interpreting graph differences

   1. y = 2x+1 vs y = 2x²+1: y = 2x+1 is a straight line (gradient 2, y-int 1). y = 2x²+1 is a U-shaped parabola with vertex at (0,1); symmetric, curves upward on both sides.
   2. y = x+3 vs y = 3^x: y = x+3 is a straight line rising at 45°, passing through (0,3). y = 3^x is an exponential curve passing through (0,1), which rises far more steeply for large x and approaches y = 0 for negative x.
   3. y = −x+4 vs y = −x²+4: y = −x+4 is a straight line sloping downward (gradient −1, y-int 4). y = −x²+4 is an inverted parabola with vertex (0,4) and x-intercepts at x = ±2.

8. Classify from description

   1. Straight line, (0,−2), positive slope: Linear. Possible equation: y = x − 2 (or any y = mx − 2, m > 0).
   2. Symmetric U-shape, lowest point (0,1): Non-linear (quadratic). Possible equation: y = x² + 1.
   3. First differences all −3: Linear with gradient m = −3. Possible equation: y = −3x + c for some constant c.
   4. First differences increasing by 2 each time (2, 4, 6, 8): Non-linear. The second differences are constant (all equal 2) ⇒ Quadratic relationship.

9. Mobile plans comparison

   1. Plan A equation: C = 0.10m + 30
   2. Plan A linear or non-linear: Linear — the cost increases by $0.10 for every additional minute (constant rate of change); m appears to power 1.
   3. Plan B first differences: Cost differences: 20−10=10, 40−20=20, 80−40=40, 160−80=80. Not equal (doubling) ⇒ Non-linear (exponential growth).
   4. Cheaper at 100 minutes: Plan A: C = 0.10(100)+30 = $40. Plan B: $40. They cost the same at 100 minutes.
   5. Very high minutes: Plan B becomes far more expensive because it is exponential — the cost doubles each 50 minutes. Plan A only grows linearly, so it stays much cheaper for large m values.

10. Square garden — area and perimeter

    1. Tables: A (s²): 1, 4, 9, 16, 25.   P (4s): 4, 8, 12, 16, 20.
    2. First differences: A first differences: 3, 5, 7, 9 — not equal ⇒ Non-linear. P first differences: 4, 4, 4, 4 — equal ⇒ Linear.
    3. Area increase 4→5 vs 1→2: 4→5: 25−16 = 9 m².   1→2: 4−1 = 3 m². Each extra metre of side adds more area as the garden gets bigger (non-linear).
    4. Perimeter increase 4→5 vs 1→2: Both: 4 m increase in perimeter. Perimeter increases by the same amount each time (linear).
    5. Student’s claim: Incorrect. y = −2x + 5 is linear (negative gradient, but x to power 1). y = −x² + 3 is non-linear. The power of x determines linearity, not the sign of coefficients.