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
Hey it's, good to see you again. Here are the calculations for each part: The given functions are: 1. Parabola: y = x^2 - 6x + 8 2. Line: y - 3x + 6 = 0 y = 3x - 6 a) Coordinates of A Point A is an intersection of the parabola and the line. To find the intersection points, set the y-values equal. Step 1: Set the equations equal to each other. x^2 - 6x + 8 = 3x - 6 Step 2: Rearrange the equation into a standard quadratic form. x^2 - 6x - 3x + 8 + 6 = 0 x^2 - 9x + 14 = 0 Step 3: Factor the quadratic equation. (x - 2)(x - 7) = 0 Step 4: Solve for x. x = 2 or x = 7 From the graph, point A is the intersection with the smaller x-coordinate. So, x_A = 2. Step 5: Substitute x_A = 2 into the linear equation to find y_A. y_A = 3(2) - 6 y_A = 6 - 6 y_A = 0 The coordinates of A are (2, 0). A(2, 0) b) X-intercepts of the graph of the function y = x^2 - 6x + 8 X-intercepts occur when y = 0. Step 1: Set the function equal to zero. x^2 - 6x + 8 = 0 Step 2: Factor the quadratic equation. (x - 2)(x - 4) = 0 Step 3: Solve for x. x = 2 or x = 4 The x-intercepts are (2, 0) and (4, 0). (2, 0) and (4, 0) c) Coordinates of point C Point C is the other intersection of the parabola and the line. From part a), the x-coordinates of the intersection points are x = 2 and x = 7. Since A has x=2, C must have x=7. Step 1: Use x_C = 7. Step 2: Substitute x_C = 7 into the linear equation to find y_C. y_C = 3(7) - 6 y_C = 21 - 6 y_C = 15 The coordinates of C are (7, 15). C(7, 15) d) Average rate of change for the function y = x^2 - 6x + 8 between points B and C 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). Step 1: Identify the coordinates of points B and C. From part b), point B is the other x-intercept, so B(4, 0). From part c), point C is (7, 15). Step 2: Apply the average rate of change formula. Average rate of change = (y_C - y_B)/(x_C - x_B) Average rate of change = (15 - 0)/(7 - 4) Average rate of change = (15)/(3) Average rate of change = 5 The average rate of change is 5. 5 Send me the next one 📸