Let's go through the process of finding the maximum and minimum values of a quadratic function based on the sign of the leading coefficient (a) in three different forms: standard form, factored form, and vertex form.
Standard Form: \(f(x)=ax^2+bx+c\)

When \(a>0\):
 The graph opens upwards, and the vertex represents the minimum value.
 To find the vertex, use the formula: Vertex = \((\frac{b}{2a},f(\frac{b}{2a}))\).
Example: Consider \(f(x)=2x^2−4x+3\).
 Find the vertex: Vertex=\((\frac{4}{(2)(2)},f(\frac{4}{2(2)}))=(1,1)\)
 So, the minimum value is 1, and it occurs at \(x=1\).

When \(a<0\):
 The graph opens downwards, and the vertex represents the maximum value.
 To find the vertex, use the same formula as above.
Example: Consider \(f(x)=−x^2+4x−3\).
 Find the vertex: Vertex=\((\frac{4}{2(1)},f(\frac{4}{2(1)}))=(2,5)\).
 So, the maximum value is 5, and it occurs at \(x=2\).
Factored Form: \(f(x)=a(x−p)(x−q)\)
To find the maximum and minimum values from the factored form, you can use the same principles as in the standard form.
Vertex Form: \(f(x)=a(x−h)^2+k\)

When \(a>0\):
 The graph opens upwards, and the vertex represents the minimum value.
 The vertex form provides the vertex directly as \((h,k)\).
Example: Consider \(f(x)=2(x−1)^2+3\).
 The vertex is \((1,3)\), so the minimum value is 3, and it occurs at \(x=1\).

When \(a<0\):
 The graph opens downwards, and the vertex represents the maximum value.
Example: Consider \(f(x)=−3(x−2)^2+4\).
 The vertex is \((2,4)\), so the maximum value is 4, and it occurs at \(x=2\).
Finding Vertex from Standard Form Algebraically:

Completing the Square:
 Start with the standard form \(ax^2+bx+c\).
 Complete the square by adding and subtracting \((\frac{b}{2a})^2\).
 Factor the squared expression, and you'll have the vertex form.
Example: \(f(x)=x^2−4x+7\)
 Complete the square: \(f(x)=(x^2−4x+4)+3=(x−2)^2+3\).
 The vertex form is \(f(x)=(x−2)^2+3\), and the vertex is \((2,3)\).

Factoring:
 Factor out the leading coefficient from \(ax^2+bx+c\).
 Write it as \(a(x^2+\frac{b}{a}x)+c\).
 Complete the square inside the parentheses, and factor it into \((x+\frac{b}{2a})^2\).
Example: \(f(x)=2x^2−4x+5\)
 Factor out 2: \(2(x^2−2x)+5\).
 Complete the square inside the parentheses: \(2(x^2−2x+1)+5−2\).
 Factor it: \(2(x−1)^2+3\).
 The vertex form is \(f(x)=2(x−1)^2+3\), and the vertex is \((1,3)\).
In summary, understanding the sign of \(a\) helps determine whether the vertex represents a minimum or maximum value. Completing the square and factoring are useful techniques for finding the vertex form and subsequently identifying the vertex.