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.
ScanSolve AI Answer
Here are the calculations for each part of question 7. a) Coordinates of A Point A is an intersection of the parabola y = x^2 - 6x + 8 and the line y - 3x + 6 = 0. First, rewrite the equation of the line in the form y = mx + c: y = 3x - 6 Now, set the two equations equal to each other to find the x-coordinates of the intersection points: x^2 - 6x + 8 = 3x - 6 x^2 - 6x - 3x + 8 + 6 = 0 x^2 - 9x + 14 = 0 Factor the quadratic equation: (x - 2)(x - 7) = 0 This gives two possible x-coordinates: x = 2 or x = 7. From the graph, point A is the intersection with the smaller x-coordinate. So, x_A = 2. Substitute x_A = 2 into the line equation y = 3x - 6 to find y_A: y_A = 3(2) - 6 y_A = 6 - 6 y_A = 0 The coordinates of A are (2, 0). b) X-intercepts of the graph of the function y = x^2 - 6x + 8 X-intercepts occur when y = 0. Set the function equal to zero: x^2 - 6x + 8 = 0 Factor the quadratic equation: (x - 2)(x - 4) = 0 This gives two x-intercepts: x = 2 or x = 4. The x-intercepts are (2, 0) and (4, 0). c) Coordinates of point C Point C is the other intersection point of the parabola y = x^2 - 6x + 8 and the line y = 3x - 6. From part a), the x-coordinates of the intersection points are x = 2 and x = 7. Since point A has x=2, point C must have x_C = 7. Substitute x_C = 7 into the line equation y = 3x - 6 to find y_C: y_C = 3(7) - 6 y_C = 21 - 6 y_C = 15 The coordinates of C are (7, 15). d) Average rate of change for the function y = x^2 - 6x + 8 between points B and C. The function is f(x) = x^2 - 6x + 8. From part c), point C is (7, 15). Point B is the vertex of the parabola y = x^2 - 6x + 8. The x-coordinate of the vertex of a parabola y = ax^2 + bx + c is given by x = (-b)/(2a). For y = x^2 - 6x + 8, a=1 and b=-6. x_B = (-(-6))/(2(1)) x_B = (6)/(2) x_B = 3 Substitute x_B = 3 into the parabola equation to find y_B: y_B = (3)^2 - 6(3) + 8 y_B = 9 - 18 + 8 y_B = -1 So, point B is (3, -1). The average rate of change between two points (x_1, y_1) and (x_2, y_2) is given by (y_2 - y_1)/(x_2 - x_1). Using points B (3, -1) and C (7, 15): Average rate of change = (15 - (-1))/(7 - 3) Average rate of change = (15 + 1)/(4) Average rate of change = (16)/(4) Average rate of change = 4 Final Answers: a) The coordinates of A are (2, 0). b) The X-intercepts are (2, 0) and (4, 0). c) The coordinates of C are (7, 15). d) The average rate of change is 4.