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 Lwaziih, good to see you again. Here are the calculations for each quadratic function: a) f(x) = x^2 Step 1: Determine the shape. The coefficient of x^2 is a=1. Since a > 0, the parabola opens upwards. Shape: Happy face Step 2: Determine the turning point. For f(x) = x^2, we have a=1, b=0, c=0. The x-coordinate of the turning point is x = -(b)/(2a) = -(0)/(2(1)) = 0. The y-coordinate is f(0) = (0)^2 = 0. Turning point: (0, 0) Step 3: Determine the x-intercepts. Set f(x) = 0: x^2 = 0 x = 0 x-intercept: (0, 0) b) y = 2x^2 - 2 Step 1: Determine the shape. The coefficient of x^2 is a=2. Since a > 0, the parabola opens upwards. Shape: Happy face Step 2: Determine the turning point. For y = 2x^2 - 2, we have a=2, b=0, c=-2. The x-coordinate of the turning point is x = -(b)/(2a) = -(0)/(2(2)) = 0. The y-coordinate is y = 2(0)^2 - 2 = -2. Turning point: (0, -2) Step 3: Determine the x-intercepts. Set y = 0: 2x^2 - 2 = 0 2x^2 = 2 x^2 = 1 x = ±sqrt(1) x = ± 1 x-intercepts: (-1, 0) and (1, 0) c) 10x^2 + 5y = -20 Step 1: Rewrite the equation in standard form y = ax^2 + bx + c. 5y = -10x^2 - 20 y = -2x^2 - 4 The coefficient of x^2 is a=-2. Since a < 0, the parabola opens downwards. Shape: Sad face Step 2: Determine the turning point. For y = -2x^2 - 4, we have a=-2, b=0, c=-4. The x-coordinate of the turning point is x = -(b)/(2a) = -(0)/(2(-2)) = 0. The y-coordinate is y = -2(0)^2 - 4 = -4. Turning point: (0, -4) Step 3: Determine the x-intercepts. Set y = 0: -2x^2 - 4 = 0 -2x^2 = 4 x^2 = -2 Since x^2 cannot be negative for real numbers, there are no real x-intercepts. x-intercepts: No real x-intercepts d) f(x) = -4x^2 + 16 Step 1: Determine the shape. The coefficient of x^2 is a=-4. Since a < 0, the parabola opens downwards. Shape: Sad face Step 2: Determine the turning point. For f(x) = -4x^2 + 16, we have a=-4, b=0, c=16. The x-coordinate of the turning point is x = -(b)/(2a) = -(0)/(2(-4)) = 0. The y-coordinate is f(0) = -4(0)^2 + 16 = 16. Turning point: (0, 16) Step 3: Determine the x-intercepts. Set f(x) = 0: -4x^2 + 16 = 0 -4x^2 = -16 x^2 = 4 x = ±sqrt(4) x = ± 2 x-intercepts: (-2, 0) and (2, 0) e) f(x) = 4 - x^2 Step 1: Rewrite the equation in standard form f(x) = ax^2 + bx + c. f(x) = -x^2 + 4 The coefficient of x^2 is a=-1. Since a < 0, the parabola opens downwards. Shape: Sad face Step 2: Determine the turning point. For f(x) = -x^2 + 4, we have a=-1, b=0, c=4. The x-coordinate of the turning point is x = -(b)/(2a) = -(0)/(2(-1)) = 0. The y-coordinate is f(0) = -(0)^2 + 4 = 4. Turning point: (0, 4) Step 3: Determine the x-intercepts. Set f(x) = 0: -x^2 + 4 = 0 -x^2 = -4 x^2 = 4 x = ±sqrt(4) x = ± 2 x-intercepts: (-2, 0) and (2, 0) f) 2y - 8x^2 = 2 Step 1: Rewrite the equation in standard form y = ax^2 + bx + c. 2y = 8x^2 + 2 y = 4x^2 + 1 The coefficient of x^2 is a=4. Since a > 0, the parabola opens upwards. Shape: Happy face Step 2: Determine the turning point. For y = 4x^2 + 1, we have a=4, b=0, c=1. The x-coordinate of the turning point is x = -(b)/(2a) = -(0)/(2(4)) = 0. The y-coordinate is y = 4(0)^2 + 1 = 1. Turning point: (0, 1) Step 3: Determine the x-intercepts. Set y = 0: 4x^2 + 1 = 0 4x^2 = -1 x^2 = -(1)/(4) Since x^2 cannot be negative for real numbers, there are no real x-intercepts. x-intercepts: No real x-intercepts Drop the next question! 📸