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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Here's a step-by-step solution for question 4: a) To complete the table of values for the relation y = x^2 - 4x - 21, substitute each x value into the equation to find the corresponding y value. Step 1: Calculate y for each x value from -4 to 6. • For x = -4: y = (-4)^2 - 4(-4) - 21 = 16 + 16 - 21 = 11 • For x = -3: y = (-3)^2 - 4(-3) - 21 = 9 + 12 - 21 = 0 • For x = -2: y = (-2)^2 - 4(-2) - 21 = 4 + 8 - 21 = -9 • For x = -1: y = (-1)^2 - 4(-1) - 21 = 1 + 4 - 21 = -16 • For x = 0: y = (0)^2 - 4(0) - 21 = 0 + 0 - 21 = -21 • For x = 1: y = (1)^2 - 4(1) - 21 = 1 - 4 - 21 = -24 • For x = 2: y = (2)^2 - 4(2) - 21 = 4 - 8 - 21 = -25 • For x = 3: y = (3)^2 - 4(3) - 21 = 9 - 12 - 21 = -24 • For x = 4: y = (4)^2 - 4(4) - 21 = 16 - 16 - 21 = -21 • For x = 5: y = (5)^2 - 4(5) - 21 = 25 - 20 - 21 = -16 • For x = 6: y = (6)^2 - 4(6) - 21 = 36 - 24 - 21 = -9 Step 2: Complete the table with the calculated values. The completed table is: |c|c|c|c|c|c|c|c|c|c|c|c| x & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ y & 11 & 0 & -9 & -16 & -21 & -24 & -25 & -24 & -21 & -16 & -9 \\ b) To draw the graph of the relation y = x^2 - 4x - 21: Step 1: Set up the axes according to the given scale. • On the x-axis: 2cm represents 2 units (meaning 1cm = 1 unit). • On the y-axis: 2cm represents 5 units (meaning 1cm = 2.5 units). Step 2: Plot the points from the completed table. Plot the coordinates: (-4, 11), (-3, 0), (-2, -9), (-1, -16), (0, -21), (1, -24), (2, -25), (3, -24), (4, -21), (5, -16), (6, -9). Step 3: Draw a smooth curve. Connect the plotted points with a smooth, continuous curve to form a parabola. c) Use your graph to find: (i) The least value of the relation y = x^2 - 4x - 21. The least value of a parabola y = ax^2 + bx + c occurs at its vertex. The x-coordinate of the vertex is x = -(b)/(2a). For y = x^2 - 4x - 21, a=1 and b=-4. x = -(-4)/(2(1)) = (4)/(2) = 2 From the table, when x=2, y = -25. This is the lowest point on the graph. The least value of the relation is -25. (ii) The equation of the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex of the parabola. Its equation is x = -(b)/(2a). From the calculation in (i), the x-coordinate of the vertex is x=2. The equation of the axis of symmetry is x=2. (iii) The truth set of the equation x^2 - 4x - 16 = 0. To find the truth set using the graph of y = x^2 - 4x - 21, we can rewrite the equation x^2 - 4x - 16 = 0 as x^2 - 4x - 21 + 5 = 0. This means y + 5 = 0, or y = -5. On your graph, draw a horizontal line at y = -5. The truth set consists of the x-coordinates where this line intersects the parabola y = x^2 - 4x - 21. Reading from the graph, the approximate points of intersection are: x ≈ -2.5 and x ≈ 6.5. (For verification, using the quadratic formula x = -(-4) ± sqrt((-4)^2 - 4(1)(-16))2(1) = 4 ± sqrt(16+64)2 = 4 ± sqrt(80)2 = 4 ± 4sqrt(5)2 = 2 ± 2sqrt(5). 2 + 2sqrt(5) ≈ 2 + 2(2.236) = 2 + 4.472 = 6.472. 2 - 2sqrt(5) ≈ 2 - 2(2.236) = 2 - 4.472 = -2.472.) The truth set is approximately \-2.5, 6.5\. 3 done, 2 left today. You're making progress.