Non-Linear Relationships — Solutions

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1. Table of values for y = x²

   y = 9, 4, 1, 0, 1, 4, 9 (for x = −3 to 3). Vertex: (0, 0). Axis of symmetry: x = 0.

2. Tables for y = x² + 2 and y = (x − 1)²

   1. y = x² + 2 (x = −3 to 3): y = 11, 6, 3, 2, 3, 6, 11. Shifted 2 units up from y = x². Vertex: (0, 2).
   2. y = (x − 1)² (x = −3 to 3): y = 16, 9, 4, 1, 0, 1, 4. Shifted 1 unit right from y = x². Vertex: (1, 0).

3. Identify graph type

   1. y = 3x + 1: Linear
   2. y = x² − 4: Parabola
   3. y = 6/x: Hyperbola
   4. y = 5^x: Exponential
   5. y = (x + 2)²: Parabola
   6. y = −4/x: Hyperbola
   7. y = 2^x: Exponential
   8. y = −x² + 5: Parabola (inverted)

4. True or False

   1. y = x² symmetric about y-axis: True. (−x)² = x² for all x, so the graph is symmetric about x = 0.
   2. y = 1/x passes through origin: False. The hyperbola never passes through the origin; x = 0 is an asymptote (undefined). Corrected: y = 1/x has asymptotes at x = 0 and y = 0.
   3. y = 3^x has y-intercept (0, 1): True. 3⁰ = 1, so the y-intercept is (0, 1) for any a^x.
   4. Vertex of y = x² + 5 is (0, 5): True. The parabola is shifted 5 units up; minimum at (0, 5).
   5. Vertex of y = (x − 3)² is (3, 0): True. The parabola is shifted 3 units right; minimum at (3, 0).
   6. Hyperbola y = k/x has horizontal asymptote at x = 0: False. x = 0 is the vertical asymptote. Corrected: The horizontal asymptote is y = 0.

5. Transformations of y = x²

   1. y = x² + 4: Shifted 4 units up. Vertex (0, 4). Opens upward. Same width as y = x².
   2. y = x² − 3: Shifted 3 units down. Vertex (0, −3). Opens upward. Same width.
   3. y = (x − 2)²: Shifted 2 units right. Vertex (2, 0). Opens upward. Same width.
   4. y = −x²: Reflected in x-axis (flipped upside down). Vertex (0, 0). Opens downward. Same width.
   5. y = 3x²: Stretched vertically by factor 3 — narrower than y = x². Vertex (0, 0). Opens upward.
   6. y = −x² + 6: Reflected in x-axis and shifted 6 units up. Vertex (0, 6). Opens downward. Maximum at (0, 6).

6. Finding y-values by substitution

   1. y = x² − 5, x = 4: y = 4² − 5 = 16 − 5 = 11
   2. y = (x − 3)², x = 1: y = (1 − 3)² = (−2)² = 4
   3. y = 8/x, x = −4: y = 8 ÷ (−4) = −2
   4. y = 2^x, x = 5: y = 2⁵ = 32
   5. y = −x² + 10, x = −3: y = −(−3)² + 10 = −9 + 10 = 1
   6. y = (x + 1)² − 4, x = 2: y = (2 + 1)² − 4 = 9 − 4 = 5

7. Ball trajectory h = −t² + 6t

   1. Table: t=0: h=0; t=1: h=5; t=2: h=8; t=3: h=9; t=4: h=8; t=5: h=5; t=6: h=0
   2. Maximum height: 9 metres at t = 3 seconds (the vertex of the parabola).
   3. Ball on ground: h = 0 at t = 0 and t = 6 (the ball starts and lands at the ground).
   4. Axis of symmetry: t = 3 (midpoint between t = 0 and t = 6).
   5. Direction: Opens downward. The coefficient of t² is −1 (negative), so the parabola is inverted with a maximum.

8. Comparing graph types

   1. y = x² vs y = 2^x: y = x² is a symmetric U-shaped parabola with vertex at origin; it decreases then increases. y = 2^x is always increasing and always positive, passing through (0,1) with a horizontal asymptote at y = 0; it has no axis of symmetry.
   2. y = x² + 1 vs y = −x² + 1: Both are parabolas with vertex (0, 1). y = x² + 1 opens upward (minimum at vertex); y = −x² + 1 opens downward (maximum at vertex). They are reflections of each other in the line y = 1.
   3. y = 4/x vs y = 4x: y = 4x is a straight line through the origin with gradient 4. y = 4/x is a hyperbola with two branches in quadrants 1 and 3; it never touches the axes; it is non-linear with no intercepts.

9. Projectile motion h = −x² + 4x

   1. Table: x=0: h=0; x=1: h=3; x=2: h=4; x=3: h=3; x=4: h=0
   2. Maximum height: 4 metres at x = 2 metres horizontal distance (the vertex).
   3. Ball on ground: −x² + 4x = 0 ⇒ x(−x + 4) = 0 ⇒ x = 0 or x = 4. The ball is on the ground at x = 0 m and x = 4 m.
   4. Axis of symmetry and vertex: Axis of symmetry: x = 2. Vertex: (2, 4).
   5. Above fence of 3 m: −x² + 4x > 3 ⇒ −x² + 4x − 3 > 0 ⇒ x² − 4x + 3 < 0 ⇒ (x−1)(x−3) < 0. The ball is above 3 m for 1 < x < 3.

10. Comparing N = 2^t and A = s²

    1. N = 2^t table: t=0: N=1; t=1: N=2; t=2: N=4; t=3: N=8; t=4: N=16; t=5: N=32
    2. A = s² table: s=0: A=0; s=1: A=1; s=2: A=4; s=3: A=9; s=4: A=16; s=5: A=25
    3. Key difference: N = 2^t is an exponential curve that starts slowly then rises very steeply; it always stays above 0 and never has a vertex. A = s² is a parabola that starts at the origin and also curves upward, but it is symmetric; it grows more slowly than the exponential for large values.
    4. Growing faster at value 5: At t=5: N = 32. At s=5: A = 25. The bacteria count (32) is higher, so 2^t is growing faster than s² at this point.
    5. Hours until N > 100: 2^t > 100. At t=6: 2⁶ = 64 (not enough). At t=7: 2⁷ = 128 > 100. After 7 hours there will be more than 100 bacteria.