Determine whether the function y = -1/2 x^2 - 11x + 6 has a minimum or maximum value, and find that value.
Understand the Problem
The question asks to determine whether the quadratic function y = -1/2 x^2 - 11x + 6 has a minimum or maximum value, and then to find that value. Since the coefficient of x^2 is negative, the parabola opens downwards, indicating a maximum value. To find the maximum value, we need to find the vertex of the parabola.
Answer
The function has a maximum value of $\frac{133}{2}$ or $66.5$.
Answer for screen readers
Maximum value: $\frac{133}{2}$ or $66.5$
Steps to Solve
- Determine if the function has a minimum or maximum value
Since the coefficient of the $x^2$ term is negative ($-\frac{1}{2}$), the parabola opens downwards. Therefore, the function has a maximum value.
- Find the x-coordinate of the vertex
The x-coordinate of the vertex of a quadratic function in the form $y = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$. In this case, $a = -\frac{1}{2}$ and $b = -11$. Substituting these values, we get: $$ x = -\frac{-11}{2(-\frac{1}{2})} = -\frac{-11}{-1} = -11 $$
- Find the y-coordinate of the vertex
To find the maximum value (y-coordinate of the vertex), substitute the x-coordinate of the vertex ($x = -11$) into the original equation: $$ y = -\frac{1}{2}(-11)^2 - 11(-11) + 6 $$ $$ y = -\frac{1}{2}(121) + 121 + 6 $$ $$ y = -\frac{121}{2} + 127 $$ $$ y = -\frac{121}{2} + \frac{254}{2} $$ $$ y = \frac{133}{2} $$ $$ y = 66.5 $$
Maximum value: $\frac{133}{2}$ or $66.5$
More Information
The maximum value of the quadratic function $y = -\frac{1}{2}x^2 - 11x + 6$ is $\frac{133}{2}$, which occurs at $x = -11$.
Tips
A common mistake is to incorrectly calculate the x-coordinate of the vertex by making a mistake in applying the formula $x = -\frac{b}{2a}$. Another common mistake is making an arithmetic error when substituting the x-coordinate of the vertex back into the original equation to find the y-coordinate (maximum value). Carefully double-checking your calculations can help avoid these errors.
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