91Ó°ÊÓ

In quadratic functions, the maximum or minimum value is a crucial point, typically located at the vertex of the parabola. These values hint at the highest or lowest points that a function can achieve. Quadratic functions have standard forms of \( y = ax^2 + bx + c \). When nestled in the graph, these parabolas can either open upwards or downwards, dictated by the coefficient \( a \):

• If \( a > 0 \), the parabola opens upwards, and the vertex represents the minimum value of the function, because all values of \( y \) must be greater than this point.
• If \( a < 0 \), the parabola opens downwards, and the vertex represents the maximum value because all values of \( y \) must be less than this point.

For the given function \( y=-2x^2-3x+2 \), \( a = -2 \), showing a downward-facing parabola. Consequently, the vertex will hold the maximum value. When calculating these values, make sure to compare the vertex result with other parts of the graph to ensure you've found the true extreme point in context.

The vertex of a parabola is a fundamental point that serves as a peak or a trough of the function. It gives us both the x- and y-coordinates. The x-coordinate of a vertex in a quadratic function \( y = ax^2 + bx + c \) is found with the formula:\[x = -\frac{b}{2a}\]Substituting \( a = -2 \) and \( b = -3 \) into this formula yields:\[x = -\frac{-3}{2(-2)} = \frac{3}{4}\]This computation shows that the vertex's x-coordinate is \( \frac{3}{4} \). To find the corresponding y-value, substitute this x back into the original function:\[y = -2\left( \frac{3}{4} \right)^2 - 3\left( \frac{3}{4} \right) + 2\]Solving this gives us \( y = -\frac{11}{8} \). Thus, the vertex is at \( \left( \frac{3}{4}, -\frac{11}{8} \right) \), combining both coordinates into one pivotal point on the graph.

A downward-opening parabola is noted by its concave shape, which means it curves down like a hill. This shape contrasts with the upward-opening parabolas that resemble a valley. In quadratic terms, a downward-opening parabola arises when the coefficient \( a \) in \( y = ax^2 + bx + c \) is less than zero (\( a < 0 \)). This characteristic ensures that the parabola will never rise above the vertex, making the vertex strategically important as the maximum point of the function.Key features of a downward-opening parabola include:

• The extremum, or peak, is the highest point on the graph—this is your maximum for the function.
• The sides of the parabola will continue indefinitely towards negative infinity, never reaching or surpassing the y-coordinate of the vertex.
• Understanding the vertex's role and the direction of opening helps predict and analyze the behavior of such functions efficiently.

In the equation \( y = -2x^2 - 3x + 2 \), the negative leading coefficient confirms the downward opening, ensuring that the vertex we calculated is indeed the maximum value point.