Finding the Maximum and Minimum Values of a Quadratic Function

🧑Zayn Alazawi 📅August 20, 2020 📊Grade 11 Math

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: \(f(x)=ax^2+bx+c\)

  1. 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 \(x=1\).

  2. 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: \(f(x)=a(x−h)^2+k\)

  1. 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 \(x=1\).

  2. 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:

  1. 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 \((2,3)\).

  2. 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.

 

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