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 ThEe, good to see you again. Step 1: Show that g(x) = -x^2 + 4x + 5. The general form of a parabola with a turning point (p, q) is g(x) = a(x-p)^2 + q. Given the turning point C(2; 9), we substitute p=2 and q=9: g(x) = a(x-2)^2 + 9 Given the y-intercept A(0; 5), we substitute x=0 and g(x)=5 to find a: 5 &= a(0-2)^2 + 9 \\ 5 &= a(-2)^2 + 9 \\ 5 &= 4a + 9 \\ 4a &= 5 - 9 \\ 4a &= -4 \\ a &= -1 Now substitute a=-1 back into the equation: g(x) &= -1(x-2)^2 + 9 \\ g(x) &= -(x^2 - 4x + 4) + 9 \\ g(x) &= -x^2 + 4x - 4 + 9 \\ g(x) &= -x^2 + 4x + 5 This matches the required form. Step 2: Calculate the average gradient of g between A and C. The coordinates are A(0; 5) and C(2; 9). The formula for the average gradient (slope) between two points (x_1, y_1) and (x_2, y_2) is m = (y_2 - y_1)/(x_2 - x_1). m = (9 - 5)/(2 - 0) = (4)/(2) = 2 The average gradient is 2. Step 3: Calculate the length of BD. First, find the x-intercepts of f(x) = (1)/(2)x^2 - 8. Set f(x) = 0: (1)/(2)x^2 - 8 &= 0 \\ (1)/(2)x^2 &= 8 \\ x^2 &= 16 \\ x &= ± 4 From the graph, B is the negative x-intercept, so B(-4; 0). Next, find the x-intercepts of g(x) = -x^2 + 4x + 5. Set g(x) = 0: -x^2 + 4x + 5 &= 0 \\ x^2 - 4x - 5 &= 0 \\ (x-5)(x+1) &= 0 \\ x = 5 &or x = -1 From the graph, D is the positive x-intercept, so D(5; 0). The length of BD is the distance between B(-4; 0) and D(5; 0): Length BD = |5 - (-4)| = |5 + 4| = |9| = 9 The length of BD is 9 units. Step 4: Use the graphs to solve for x. Step 4.1: f(x) 0 The x-intercepts of f(x) are x = -4 and x = 4. Since f(x) = (1)/(2)x^2 - 8 is a parabola opening upwards, f(x) 0 when x is outside or on the x-intercepts. Therefore, x -4 or x 4. The solution is x -4 or x 4. Step 4.2: f and g are both strictly increasing For f(x) = (1)/(2)x^2 - 8: This parabola opens upwards and its turning point is at x=0. So, f(x) is strictly increasing for x > 0. For g(x) = -x^2 + 4x + 5: This parabola opens downwards and its turning point is at x = -(4)/(2(-1)) = 2. So, g(x) is strictly increasing for x < 2. For both functions to be strictly increasing, we need the intersection of these two intervals: x > 0 AND x < 2. The solution is 0 < x < 2. Send me the next one 📸