To find the maximum value of the function \( y = 1 - 2x - 3x^2 \), we will follow a systematic approach using calculus. This function is a quadratic equation in the standard form \( y = ax^2 + bx + c \), where \( a = -3 \), \( b = -2 \), and \( c = 1 \). ### Step 1: Identify the Type of Quadratic Function The coefficient of \( x^2 \) (which is \( a = -3 \)) is negative, indicating that the parabola opens downwards. This means that the function has a maximum point. ### Step 2: Find the Vertex of the Parabola The maximum value of a quadratic function occurs at its vertex. The x-coordinate of the vertex can be found using the formula: \[ x = -\frac{b}{2a} \] Substituting the values of \( a \) and \( b \): \[ x = -\frac{-2}{2 \times -3} = \frac{2}{-6} = -\frac{1}{3} \] ### Step 3: Calculate the Maximum Value of \( y \) Now that we have the x-coordinate of the vertex, we can substitute \( x = -\frac{1}{3} \) back into the original equation to find the corresponding y-value: \[ y = 1 - 2\left(-\frac{1}{3}\right) - 3\left(-\frac{1}{3}\right)^2 \] Calculating each term: 1. \( -2\left(-\frac{1}{3}\right) = \frac{2}{3} \) 2. \( \left(-\frac{1}{3}\right)^2 = \frac{1}{9} \) and thus \( -3\left(-\frac{1}{3}\right)^2 = -3 \times \frac{1}{9} = -\frac{1}{3} \) Now substituting these values into the equation: \[ y = 1 + \frac{2}{3} - \frac{1}{3} \] Combining the fractions: \[ y = 1 + \frac{2}{3} - \frac{1}{3} = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \] ### Conclusion Thus, the maximum value of \( y \) is \( \frac{4}{3} \). ### Step 4: Evaluate the Options Now, let's evaluate the options provided: - **A. \( \frac{5}{4} \)**: This is less than \( \frac{4}{3} \) (which is approximately 1.33). - **B. \( \frac{5}{3} \)**: This is greater than \( \frac{4}{3} \) (which is approximately 1.33). - **C. \( \frac{3}{4} \)**: This is also less than \( \frac{4}{3} \). - **D. \( \frac{4}{3} \)**: This is exactly the maximum value we calculated. ### Why Other Options Are Incorrect - **A. \( \frac{5}{4} \)**: This value is approximately 1.25, which is less than the maximum value of \( \frac{4}{3} \). - **B. \( \frac{5}{3} \)**: This value is approximately 1.67, which exceeds the maximum value of \( \frac{4}{3} \). - **C. \( \frac{3}{4} \)**: This value is approximately 0.75, which is significantly less than the maximum value of \( \frac{4}{3} \). ### Revision Summary - The maximum value of the quadratic function \( y = 1 - 2x - 3x^2 \) occurs at the vertex. - The x-coordinate of the vertex is found using \( x = -\frac{b}{2a} \). - Substituting the vertex x-value back into the function gives the maximum y-value. - The correct maximum value is \( \frac{4}{3} \), corresponding to option D.