Mastering Maximum and Minimum Values of Quadratic Expressions

Understanding the maximum and minimum values of quadratic expressions is a fundamental skill for success in competitive exams like JEE. This topic bridges algebra and calculus, providing insights into the behavior of parabolic functions. We'll explore how to find these extreme values using both algebraic and graphical methods.

What is a Quadratic Expression?

A quadratic expression is a polynomial of degree two. It generally takes the form: $ax^2 + bx + c$ , where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The graph of a quadratic expression is a parabola.

The Shape of the Parabola: 'a' is Key

The sign of the leading coefficient 'a' determines if the parabola opens upwards or downwards.

If 'a' is positive, the parabola opens upwards, forming a 'U' shape. If 'a' is negative, it opens downwards, forming an inverted 'U' shape.

When $a > 0$ , the parabola opens upwards. This means the vertex of the parabola represents the lowest point on the graph. Consequently, the quadratic expression will have a minimum value at its vertex. When $a < 0$ , the parabola opens downwards. In this case, the vertex represents the highest point on the graph, and the quadratic expression will have a maximum value at its vertex. The value of 'b' and 'c' influence the position of the vertex but not the direction of opening.

Finding the Vertex: The Core of the Solution

The vertex of a parabola $y = ax^2 + bx + c$ is the point where the maximum or minimum value occurs. The x-coordinate of the vertex can be found using the formula $x = -b / (2a)$ . Once you have the x-coordinate, you can substitute it back into the expression to find the corresponding y-coordinate, which is the maximum or minimum value.

What is the formula for the x-coordinate of the vertex of a quadratic expression

ax^2 + bx + c

$x = -b / (2a)$

Method 1: Completing the Square (Algebraic Approach)

Completing the square is a powerful algebraic technique to rewrite a quadratic expression in vertex form: $a(x-h)^2 + k$ . In this form, $(h, k)$ is the vertex. The value of $k$ directly gives the maximum or minimum value of the expression. If $a > 0$ , $k$ is the minimum value. If $a < 0$ , $k$ is the maximum value. The term $a(x-h)^2$ is always non-negative. Therefore, when $a > 0$ , the minimum occurs when $(x-h)^2 = 0$ (i.e., $x=h$ ), making the minimum value $k$ . When $a < 0$ , the term $a(x-h)^2$ is always non-positive, and its maximum value (closest to zero) occurs when $x=h$ , making the expression $k$ , which is the maximum value.

Consider the quadratic expression $f(x) = 2x^2 - 8x + 5$ . To find its minimum value, we complete the square. First, factor out the leading coefficient (2) from the terms involving x: $f(x) = 2(x^2 - 4x) + 5$ . Now, complete the square inside the parenthesis for $x^2 - 4x$ . We need to add and subtract $(-4/2)^2 = (-2)^2 = 4$ : $f(x) = 2(x^2 - 4x + 4 - 4) + 5$ . Rearrange to form the perfect square: $f(x) = 2((x-2)^2 - 4) + 5$ . Distribute the 2: $f(x) = 2(x-2)^2 - 8 + 5$ . Simplify: $f(x) = 2(x-2)^2 - 3$ . This is in the vertex form $a(x-h)^2 + k$ , where $a=2$ , $h=2$ , and $k=-3$ . Since $a=2$ is positive, the parabola opens upwards, and the minimum value is $k = -3$ , occurring at $x = h = 2$ .

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Method 2: Using the Vertex Formula (Direct Approach)

This method is often quicker. For $f(x) = ax^2 + bx + c$ :

Identify $a$ , $b$ , and $c$ .
Calculate the x-coordinate of the vertex: $x_{vertex} = -b / (2a)$ .
Substitute $x_{vertex}$ back into the expression to find the maximum or minimum value: $f(x_{vertex})$ .

Example: For $f(x) = -x^2 + 6x - 10$ , we have $a=-1$ , $b=6$ , $c=-10$ . The x-coordinate of the vertex is $x_{vertex} = -6 / (2 * -1) = -6 / -2 = 3$ . The maximum value is $f(3) = -(3)^2 + 6(3) - 10 = -9 + 18 - 10 = -1$ . Since $a=-1$ is negative, this is indeed a maximum value.

For

f(x) = 3x^2 - 12x + 7

, what is the x-coordinate of the vertex?

$x = -(-12) / (2 * 3) = 12 / 6 = 2$

Maximum and Minimum Values on a Closed Interval

When a quadratic expression is considered over a specific closed interval $[p, q]$ , the maximum and minimum values can occur either at the vertex (if it lies within the interval) or at the endpoints of the interval ( $p$ or $q$ ). To find these values:

Calculate the vertex's x-coordinate. If it's within $[p, q]$ , evaluate the expression at the vertex.
Evaluate the expression at the endpoints $p$ and $q$ .
The largest of these values is the maximum, and the smallest is the minimum on the interval.

Remember: For parabolas opening upwards, the vertex is a minimum. For parabolas opening downwards, the vertex is a maximum. Always check the sign of 'a'!

Summary of Key Concepts

Feature	Parabola Opens Upwards ( $a > 0$ )	Parabola Opens Downwards ( $a < 0$ )
Vertex Type	Minimum Value	Maximum Value
Vertex x-coordinate	$x = -b / (2a)$	$x = -b / (2a)$
Value at Vertex	Minimum Value ( $k$ )	Maximum Value ( $k$ )
Completing the Square Form	$a(x-h)^2 + k$	$a(x-h)^2 + k$

Learning Resources

Quadratic Functions - Vertex Form(video)

Learn how to convert quadratic functions into vertex form and understand its relationship to the parabola's vertex and extreme values.

Finding the Maximum or Minimum Value of a Quadratic Function(documentation)

This page explains how to find the roots of a quadratic equation, which is closely related to finding the vertex and extreme values.

Quadratic Functions: Maximum and Minimum Values(blog)

A comprehensive explanation of finding maximum and minimum values of quadratic expressions with examples.

Quadratic Functions - Maximum and Minimum Values(blog)

Provides detailed methods and examples for determining the maximum and minimum values of quadratic expressions.

JEE Mathematics: Quadratic Equations(video)

A video tutorial focusing on quadratic equations, often covering vertex and extreme value concepts relevant to JEE.

Completing the Square - Math is Fun(documentation)

A clear explanation of the completing the square method, essential for rewriting quadratic expressions.

Quadratic Function(wikipedia)

Wikipedia's detailed article on quadratic functions, covering their properties, graphs, and applications, including vertex and extrema.

Maximum and Minimum Values of Quadratic Expressions - Toppr(blog)

Explains the concept of maximum and minimum values of quadratic expressions with solved examples for competitive exams.

Calculus: Finding Maxima and Minima(video)

While this is a calculus topic, understanding optimization is key. This video introduces the concept of finding maximum and minimum values, which is the calculus approach to this algebraic problem.

Practice Problems: Quadratic Functions(tutorial)

Practice applying your knowledge of quadratic functions, including finding maximum and minimum values in word problem contexts.