Answer:
Maximum value: [tex](-1, \frac{7}{2})[/tex]
Step-by-step explanation:
Given the quadratic function, f(x) = -½x² - x + 3
where a = -½, b = -1, and c = 3
Since the value of a is negative, then it means that the graph of the parabola opens downward. Therefore, its vertex is the maximum point.
The vertex of the parabola occurs at the point (h, k).
Axis of symmetry: x = h.
The axis of symmetry is the imaginary straight line that passes through the x-intercept which divides a parabola into two symmetrical parts. To find the value of the h coordinate of the vertex (h, k), use the following formula:
[tex]x = \frac{-b}{2a}[/tex]
[tex]x = \frac{-(-1)}{2(-1/2)} = -1[/tex]
Now that we have the value for h, substitute its value into the original quadratic function:
h = -½(-1)² - (-1) + 3
h = 7/2 or 3.5
Vertex (h, k) = [tex](-1, \frac{7}{2})[/tex]
Therefore, the maximum value of the function is [tex](-1, \frac{7}{2})[/tex].
