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
Here's the solution for the given function f(x) = -x^2 - 2x + 24. a) Will the function have a minimum or maximum? Step 1: Identify the coefficient of the x^2 term. For the function f(x) = -x^2 - 2x + 24, the coefficient of x^2 is a = -1. Step 2: Determine if the parabola opens upwards or downwards. Since a = -1 is negative (a < 0), the parabola opens downwards. Step 3: Conclude whether it's a minimum or maximum. A parabola that opens downwards has a highest point, which is a maximum. The function will have a maximum. b) Determine the x-coordinates of the turning point. Step 1: Use the formula for the x-coordinate of the turning point (vertex) of a parabola f(x) = ax^2 + bx + c. The formula is x = -(b)/(2a). From the function f(x) = -x^2 - 2x + 24, we have a = -1 and b = -2. Step 2: Substitute the values of a and b into the formula. x = -(-2)/(2(-1)) x = -(-2)/(-2) x = -(1) x = -1 The x-coordinate of the turning point is -1. c) Determine the y-coordinates of the turning point. Step 1: Substitute the x-coordinate found in part b) into the original function f(x). We found x = -1. f(x) = -x^2 - 2x + 24 f(-1) = -(-1)^2 - 2(-1) + 24 Step 2: Calculate the value of f(-1). f(-1) = -(1) + 2 + 24 f(-1) = -1 + 2 + 24 f(-1) = 1 + 24 f(-1) = 25 The y-coordinate of the turning point is 25. 3 done, 2 left today. You're making progress.