# 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.