To write an equation for a quadratic function that has an absolute maximum at a given point, we can use the vertex form of a quadratic function. The vertex form is particularly useful when we know the vertex of the parabola (the highest or lowest point on the graph), which in this case is the point of the absolute maximum.
The vertex form of a quadratic function is given by:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
Here, [tex]\((h, k)\)[/tex] represents the vertex of the parabola, and [tex]\(a\)[/tex] is a coefficient that determines the width and the direction of the opening of the parabola:
- If [tex]\(a > 0\)[/tex], the parabola opens upwards, and the vertex is the absolute minimum.
- If [tex]\(a < 0\)[/tex], the parabola opens downwards, and the vertex is the absolute maximum.
Given that we want an absolute maximum at the point [tex]\((-7, -2)\)[/tex], this point must be the vertex of our parabola. Therefore, in our equation, [tex]\(h = -7\)[/tex] and [tex]\(k = -2\)[/tex].
Since the problem states that we have an absolute maximum, it implies that the parabola must open downwards, which means that [tex]\(a\)[/tex] must be negative. The value of [tex]\(a\)[/tex] also affects the "stretch" or "width" of the parabola. Without additional information, we cannot determine the exact value of [tex]\(a\)[/tex], as any negative value would suffice to make the parabola open downwards. For simplicity, let's choose [tex]\(a = -1\)[/tex], but keep in mind that any negative value of [tex]\(a\)[/tex] is valid.
Substitute [tex]\(a\)[/tex], [tex]\(h\)[/tex], and [tex]\(k\)[/tex] into the vertex form, we get:
[tex]\[ f(x) = -1(x - (-7))^2 - 2 \][/tex]
Which simplifies to:
[tex]\[ f(x) = -(x + 7)^2 - 2 \][/tex]
So the equation that represents the new function with an absolute maximum at (-7, -2) is:
[tex]\[ f(x) = -(x + 7)^2 - 2 \][/tex]
Remember that if you want the parabola to be wider or narrower than depicted by this equation, you can change the value of [tex]\(a\)[/tex] to any other negative value, which would stretch or compress the parabola accordingly.
