Topic Review — Linear and Non-Linear Relationships — Solutions

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1. Graphing linear relationships

   1. y = −4x + 9: m = −4, b = 9
   2. y = ⅔x − 5: m = ⅔, b = −5
   3. y = 3x − 2 table: x=−2: y=−8; x=−1: y=−5; x=0: y=−2; x=1: y=1; x=2: y=4
   4. 2x + 5y = 20 intercepts: y-int: set x=0 → 5y=20, y=4, so (0,4). x-int: set y=0 → 2x=20, x=10, so (10,0).
   5. Equation m=−3, b=7: y = −3x + 7
   6. Parallel?: Yes. Both have gradient m = 5; they have different y-intercepts (−1 and 8) so they are parallel, not the same line.

2. Non-linear relationships

   1. y = x² − 3: x=−2: y=1; x=−1: y=−2; x=0: y=−3; x=1: y=−2; x=2: y=1
   2. y = 1/x: x=−4: y=−0.25; x=−2: y=−0.5; x=−1: y=−1; x=1: y=1; x=2: y=0.5; x=4: y=0.25
   3. y = x² + 1 features: (a) Vertex: (0, 1). (b) Axis of symmetry: x = 0. (c) y-intercept: (0, 1).
   4. y = 2^x values: x=−1: 0.5; x=0: 1; x=1: 2; x=2: 4; x=3: 8; x=4: 16

3. Classify and compare

   1. Classify: (a) Linear — x to power 1. (b) Non-linear — x³. (c) Linear — horizontal line. (d) Non-linear — x in denominator. (e) Non-linear — x is an exponent.
   2. First differences table (y: 6, 2, −2, −6, −10): Differences: −4, −4, −4, −4. Linear. m=−4, b=6.   y = −4x + 6
   3. First differences (y: 3, 6, 12, 24, 48): Differences: 3, 6, 12, 24. Not equal (doubling) ⇒ Non-linear (exponential y = 3 × 2^x).
   4. y=x vs y=x²: y=x is a straight line through the origin at 45°. y=x² is a U-shaped parabola through the origin; it curves symmetrically and grows much faster than y=x for large x values.

4. Finding equations and sketching

   1. Through (0,5) and (3,14): m = (14−5)/(3−0) = 3; b = 5.   y = 3x + 5
   2. Through (−2,6) and (4,−6): m = (−6−6)/(4−(−2)) = −12/6 = −2; sub (4,−6): −6=−2(4)+b → b=2.   y = −2x + 2
   3. Sketch y=−2x+4: y-int: (0,4); x-int: set y=0 → x=2, so (2,0); additional point e.g. (1,2). Line slopes downward steeply.
   4. Sketch y=x²−1: Vertex at (0,−1); x-ints at x=±1 (set 0=x²−1 → x=±1); y-int at (0,−1); U-shape opening upward.
   5. Parallel to y=3x−2, through (0,5): y = 3x + 5 (same gradient m=3, new y-int = 5).

5. Problem solving

   1. Pool draining:
      1. Equation: V = −800t + 24000 (or V = 24000 − 800t)
      2. Time to drain: Set V=0: 800t = 24000 → t = 30 hours
      3. Graph features: Straight line; V-intercept at (0, 24000); t-intercept at (30, 0); gradient = −800 (volume decreasing at constant rate); linear relationship.
   2. Profit plans:
      1. Linear/non-linear: Plan X (P=8x−40) is linear. Plan Y (P=x²−2x) is non-linear (quadratic).
      2. Profit at x=10: Plan X: P = 8(10)−40 = 40. Plan Y: P = 100−20 = 80. Plan Y gives higher profit at x=10.
      3. Equal profits: 8x−40 = x²−2x → 0 = x²−10x+40. Discriminant: 100−160 = −60 < 0. The plans never give equal profit for real values of x.
   3. y=x² vs y=x comparison: Correct for x=0 (both give 0) and x=1 (both give 1). For x=2: y=x gives 2, y=x² gives 4. For x=3: y=x gives 3, y=x² gives 9. For x=4: y=x gives 4, y=x² gives 16. This illustrates that two relationships can agree at isolated points but behave very differently overall — we must check multiple values before concluding they are the same.