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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You're on a roll — 4.1 Determine the range of k. Step 1: Identify the type of graph and its vertex. The graph k(x) = ax^2 is a parabola with its vertex at the origin (0,0). From the diagram, the parabola opens downwards. Step 2: Determine the possible y-values. Since the parabola opens downwards from the vertex (0,0), the maximum y-value is 0, and all other y-values are less than or equal to 0. The range of k is y 0. 4.2 The domain of p. Step 1: Identify the type of graph. The graph p is a straight line, given by p(x) = mx + c. Step 2: Determine the possible x-values. A straight line extends infinitely in both the positive and negative x-directions. The domain of p is x R. 4.3 Given that the gradient of AD is 1, determine the equation of AD in the form of p(x) = mx + c. Step 1: Identify the gradient (m) and a point on the line. The gradient of line AD (which is line p) is given as m = 1. The line p passes through point C(0, -6). This is the y-intercept, so c = -6. Step 2: Write the equation of the line. Substitute m=1 and c=-6 into the equation p(x) = mx + c: p(x) = 1x - 6 p(x) = x - 6 The equation of AD is p(x) = x - 6. 4.4 Determine the equation of k in the form of k(x) = ax^2 if B is equidistant from A to C. Step 1: Use the equidistant property to find a relationship between the coordinates of B. Let B be (x_B, y_B). A is (6,0) and C is (0,-6). Since B is equidistant from A and C, BA^2 = BC^2. (x_B - 6)^2 + (y_B - 0)^2 = (x_B - 0)^2 + (y_B - (-6))^2 x_B^2 - 12x_B + 36 + y_B^2 = x_B^2 + y_B^2 + 12y_B + 36 -12x_B = 12y_B y_B = -x_B Step 2: Use the fact that B lies on line p to find its coordinates. From 4.3, the equation of line p is y = x - 6. Substitute y_B = -x_B into the line equation: -x_B = x_B - 6 -2x_B = -6 x_B = 3 Now find y_B: y_B = -x_B = -3 So, the coordinates of B are (3, -3). Step 3: Use the coordinates of B to find the value of 'a' for k(x) = ax^2. Point B(3, -3) lies on the parabola $k(