- Analyze the coefficient \(a = -2\) to determine that the parabola opens downward, indicating a maximum value.
- Calculate the x-coordinate of the vertex using \(h = -b/(2a)\), which results in \(x = 2\).
- Substitute \(x = 2\) into the original function \(g(x) = -2x^2 + 8x - 10\) to find the maximum value, which is \(-2\).
- Conclude that the function has a maximum value of \(-2\) occurring at \(x = \boxed{2}\).
Explanation
- Analyzing the nature of the extremum
The given quadratic function is \(g(x) = -2x^2 + 8x - 10\). A quadratic function of the form \(g(x) = ax^2 + bx + c\) defines a parabola. The direction in which the parabola opens is determined by the coefficient of the \(x^2\) term, \(a\). If \(a > 0\), the parabola opens upward, indicating a minimum value at its vertex. If \(a < 0\), the parabola opens downward, indicating a maximum value at its vertex. In this case, \(a = -2\), which is less than 0. Therefore, the function has a maximum value.
- Finding the x-coordinate of the maximum
The vertex of a parabola defined by \(g(x) = ax^2 + bx + c\) occurs at the \(x\)-coordinate \(h = -\frac{b}{2a}\). For the given function \(g(x) = -2x^2 + 8x - 10\), we have \(a = -2\) and \(b = 8\). Substituting these values into the formula gives:
\[h = -\frac{8}{2(-2)} = -\frac{8}{-4} = 2.\]
Thus, the maximum value occurs at \(x = 2\).
- Calculating the maximum value
To find the maximum value of the function, we substitute the \(x\)-coordinate of the vertex back into the original expression for \(g(x)\). Substituting \(x = 2\):
\[g(2) = -2(2)^2 + 8(2) - 10\]
\[g(2) = -2(4) + 16 - 10\]
\[g(2) = -8 + 16 - 10\]
\[g(2) = -2.\]
Therefore, the maximum value of the function is \(-2\).
- Conclusion
Based on the analysis, the function has a maximum value of \(-2\), which occurs at \(x = 2\).
| | |
| :--- | :--- |
| Does the function have a minimum or maximum value? | Maximum |
| What is the function's minimum or maximum value? | \(-2\) |
| Where does the minimum or maximum value occur? | \(x = \boxed{2}\) |
Examples
Quadratic functions are widely used in physics and economics to model phenomena. For example, if a company wants to maximize its profit, the profit function might be a quadratic equation that opens downward. By finding the vertex of this parabola, the company can determine the exact production level (x-coordinate) that results in the maximum profit (the y-coordinate). Similarly, a projectile's height over time can be modeled as a downward-opening parabola, where the vertex tells us the maximum height the object reaches and the time it takes to get there.