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 the solution to the problem: 4.1.1 Determine the coordinates of points E and P. Points E and P are the x-intercepts of the graph f(x) = -x^2 - 4x + 5. To find them, set f(x) = 0: -x^2 - 4x + 5 = 0 Multiply by -1 to make the leading coefficient positive: x^2 + 4x - 5 = 0 Factor the quadratic equation: (x+5)(x-1) = 0 This gives two possible x-values: x = -5 or x = 1 From the graph, E is on the negative x-axis and P is on the positive x-axis. Therefore, the coordinates are: E = (-5, 0) P = (1, 0) 4.1.2 Calculate the value of m and c. The line g(x) = mx + c passes through points E and M. From 4.1.1, E is (-5, 0). Point M is the y-intercept of f(x). To find M, set x=0 in f(x): f(0) = -(0)^2 - 4(0) + 5 = 5 So, M is (0, 5). Now we have two points on the line g(x): E(-5, 0) and M(0, 5). Calculate the slope m: m = (y_2 - y_1)/(x_2 - x_1) = (5 - 0)/(0 - (-5)) = (5)/(5) = 1 The value of c is the y-intercept of the line, which is the y-coordinate of point M. c = 5 Therefore, the values are: m = 1 c = 5 4.1.3 Calculate the length of NT. N is the turning point of f(x) = -x^2 - 4x + 5. The x-coordinate of the turning point is given by x = (-b)/(2a). For f(x), a=-1 and b=-4. x_N = (-(-4))/(2(-1)) = (4)/(-2) = -2 Substitute x_N = -2 into f(x) to find the y-coordinate: y_N = f(-2) = -(-2)^2 - 4(-2) + 5 = -(4) + 8 + 5 = -4 + 8 + 5 = 9 So, the coordinates of N are (-2, 9). NTD is the axis of symmetry of f, which is the vertical line x = -2. Point T is on the graph of g(x) and on the axis of symmetry x = -2. From 4.1.2, g(x) = x + 5. Substitute x = -2 into g(x) to find the y-coordinate of T: y_T = g(-2) = -2 + 5 = 3 So, the coordinates of T are (-2, 3). To calculate the length of NT, since both points have the same x-coordinate, we find the absolute difference of their y-coordinates: Length NT = |y_N - y_T| = |9 - 3| = 6 The length of NT is 6 units. 4.1.4 Determine the coordinates of point S, the mirror image of point M with respect to the axis of symmetry. Point M is (0, 5). The axis of symmetry is the line x = -2. When a point is reflected across a vertical line, its y-coordinate remains the same. So, y_S = y_M = 5. The x-coordinate of the axis of symmetry is the midpoint of the x-coordinates of M and S. Let S be (x_S, y_S). (x_M + x_S)/(2) = -2 (0 + x_S)/(2) = -2 x_S = -4 Therefore, the coordinates of point S are (-4, 5).