91Ó°ÊÓ

The vertex formula is a handy tool for finding the vertex of a quadratic function. A quadratic function is often written in the form of \( ax^2 + bx + c \), and the vertex represents a critical point of this function. By using the vertex formula, you can find the x-coordinate of the vertex, which is \( x = -\frac{b}{2a} \).
For our function \( f(x) = -\frac{1}{2}x^2 - x + 3 \), we identify:\

\( a = -\frac{1}{2} \) (coefficient of \( x^2 \))
\( b = -1 \) (coefficient of \( x \))
\( c = 3 \) (constant term)

Plugging these into the vertex formula gives \( x = -\frac{-1}{2(-\frac{1}{2})} \), simplifying to \( x = -1 \). Knowing this helps in finding both the nature of the graph and the function's maximum or minimum values.

Quadratic functions can show either a maximum or a minimum value because of their parabolic shape. For a parabola that opens downwards, like in our example, the vertex is where the function reaches its maximum value. This is determined by the sign of \( a \), the coefficient of \( x^2 \).
In our function, \( a = -\frac{1}{2} \), which means the parabola opens downwards. Therefore, it has a maximum value. Once you find the x-coordinate of the vertex, substitute it back into the function to find the corresponding y-value. For our function's vertex \( x = -1 \):\[\begin{align*} f(-1) &= 3 - (-1) - \frac{1}{2}(-1)^2 \&= 3 + 1 - \frac{1}{2} \&= 3.5 \end{align*}\] Thus, the maximum value of this function is \( 3.5 \), occurring at \( x = -1 \).

A quadratic function manifests as a parabolic graph when plotted. The equation \( f(x) = ax^2 + bx + c \) defines the shape and direction of this parabola. Key characteristics of the parabola include:

• It opens upwards if \( a > 0 \), showing a minimum point.
• It opens downwards if \( a < 0 \), showing a maximum point.

In our given function with \( a = -\frac{1}{2} \), \( b = -1 \), and \( c = 3 \), the parabola opens downwards, distinctly pointing to a peak point. The vertex coordinates \( (-1, 3.5) \) mark this highest point. Understanding this graphical representation provides intuitive insights into how changes in the function affect its shape and peaks.