This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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a) (i) Use the table to find the values of p and q. Step 1: Use the point (0, -12) from the table and substitute x=0 and y=-12 into the relation y = px^2 - 5x + q. -12 = p(0)^2 - 5(0) + q -12 = 0 - 0 + q q = -12 Step 2: Use another point from the table, for example (4, 0), and substitute x=4, y=0, and q=-12 into the relation y = px^2 - 5x + q. 0 = p(4)^2 - 5(4) + (-12) 0 = 16p - 20 - 12 0 = 16p - 32 16p = 32 p = (32)/(16) p = 2 The values are p=2 and q=-12. The relation is y = 2x^2 - 5x - 12. (ii) Copy and complete the table. Using the relation y = 2x^2 - 5x - 12, we calculate the missing y-values: For x=-1: y = 2(-1)^2 - 5(-1) - 12 y = 2(1) + 5 - 12 y = 2 + 5 - 12 y = 7 - 12 = -5 For x=1: y = 2(1)^2 - 5(1) - 12 y = 2(1) - 5 - 12 y = 2 - 5 - 12 y = -3 - 12 = -15 For x=2: y = 2(2)^2 - 5(2) - 12 y = 2(4) - 10 - 12 y = 8 - 10 - 12 y = -2 - 12 = -14 For x=3: y = 2(3)^2 - 5(3) - 12 y = 2(9) - 15 - 12 y = 18 - 15 - 12 y = 3 - 12 = -9 The completed table is: | x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | |---|----|----|----|---|---|---|---|---|---| | y | 21 | 6 | -5 | -12 | -15 | -14 | -9 | 0 | 13 | b) Using scales of 2cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of the relation for -3 x 5. Step 1: Draw the x-axis and y-axis on graph paper. Step 2: Mark the x-axis from -3 to 5, with each 2 cm representing 1 unit. Step 3: Mark the y-axis to cover the range of y-values (from -15 to 21), with each 2 cm representing 5 units. Step 4: Plot the points from the completed table: (-3, 21), (-2, 6), (-1, -5), (0, -12), (1, -15), (2, -14), (3, -9), (4, 0), (5, 13). Step 5: Draw a smooth curve connecting all the plotted points. This curve represents the graph of y = 2x^2 - 5x - 12. c) Use the graph to find: (i) y when x = 1.8 To find y when x=1.8 from the graph, locate x=1.8 on the x-axis. Move vertically from x=1.8 to intersect the curve. From the intersection point, move horizontally to the y-axis to read the corresponding y-value. Using the equation y = 2x^2 - 5x - 12: y = 2(1.8)^2 - 5(1.8) - 12 y = 2(3.24) - 9 - 12 y = 6.48 - 9 - 12 y = 6.48 - 21 y = -14.52 From the graph, y would be approximately -14.5. (ii) x when y = -8 To find x when y=-8 from the graph, locate y=-8 on the y-axis. Move horizontally from y=-8 to intersect the curve. From the intersection points, move vertically to the x-axis to read the corresponding x-values. Using the equation y = 2x^2 - 5x - 12: -8 = 2x^2 - 5x - 12 0 = 2x^2 - 5x - 4 Using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a: x = -(-5) ± sqrt((-5)^2 - 4(2)(-4))2(2) x = 5 ± sqrt(25 + 32)4 x = 5 ± sqrt(57)4 x_1 = 5 + sqrt(57)4 ≈ (5 + 7.5498)/(4) ≈ (12.5498)/(4) ≈ 3.137 x_2 = 5 - sqrt(57)4 ≈ (5 - 7.5498)/(4) ≈ (-2.5498)/(4) ≈ -0.637 From the graph, x would be approximately 3.1 or -0.6. That's 2 down. 3 left today — send the next one.