How to Find Maxima and Minima of a Function?

Every function that curves up or down has maximum and minimum points — the peaks and valleys of its graph. Finding these points tells you the greatest or least output value a function can produce, whether over its entire domain (absolute extrema) or just in a local region (relative extrema). Algebra 1 focuses especially on quadratic functions, where the vertex gives the maximum or minimum directly.

What Are Maxima and Minima?

• An absolute (global) maximum is the highest output value the function ever reaches.
• An absolute (global) minimum is the lowest output value the function ever reaches.
• A relative (local) maximum is a peak that is higher than all nearby outputs (but not necessarily the overall highest).
• A relative (local) minimum is a valley that is lower than all nearby outputs.

How to Find Maxima and Minima of a Quadratic Function

Using the vertex formula

For \(\color{blue}{f(x) = \text{ ax }}\)² + \(\color{blue}{\text{ bx } + c}\), the vertex x-coordinate is:

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x = −\(\color{blue}{\frac{b}{(2a)}}\)

Substitute back into f(x) to find the y-coordinate of the vertex.

• If a > 0 (parabola opens upward), the vertex is a minimum.
• If a < 0 (parabola opens downward), the vertex is a maximum.

Example: absolute maximum

f(x) = −x² + \(\color{blue}{6x – 5}\).
a = −1 (opens downward, so there is a maximum).
x = −\(\color{blue}{\frac{6}{(2 \cdot -1)} = 3}\)
f(3) = −\(\color{blue}{9 + 18 – 5}\) = 4
Absolute maximum: 4 at \(\color{blue}{x = 3}\).

Example: absolute minimum

\(\color{blue}{g(x) = x}\)² − \(\color{blue}{4x + 3}\).
\(\color{blue}{a = 1}\) (opens upward, so there is a minimum).
x = −\(\color{blue}{\frac{(-4)}{(2 \cdot 1)} = 2}\)
\(\color{blue}{g(2) = 4 – 8 + 3}\) = −1
Absolute minimum: −1 at \(\color{blue}{x = 2}\).

Reading Maxima and Minima from a Graph

On a graph, a relative maximum is a point that is higher than all points immediately to its left and right. A relative minimum is lower than all points immediately to its left and right. Read the y-value at each peak or valley.

Step-by-Step Summary

1. Identify the type of function (quadratic, etc.).
2. For a quadratic \(\color{blue}{f(x) = \text{ ax }}\)² + \(\color{blue}{\text{ bx } + c}\), find x = −\(\color{blue}{\frac{b}{(2a)}}\).
3. Evaluate f at that x to get the extreme value.
4. Determine whether it is a max (a < 0) or min (a > 0).
5. For non-quadratics, locate peaks and valleys on the graph and read their coordinates.

Watch: Introduction to Maximum and Minimum Points (Video Lesson)

Khan Academy introduces the concept of maximum and minimum points on functions with visual explanations:

Maxima and Minima – Worked Examples

Example 1: Find the maximum of f(x) = −x² + \(\color{blue}{6x – 5}\).

a = −1 < 0 (maximum). x = −\(\color{blue}{\frac{6}{(2 \cdot -1)} = 3}\).
f(3) = −\(\color{blue}{9 + 18 – 5}\) = 4. Maximum value: 4 at \(\color{blue}{x = 3}\).
(Note: \(\color{blue}{f(1) = 0}\) and \(\color{blue}{f(5) = 0}\) are the roots, confirming \(\color{blue}{x = 3}\) is the midpoint/vertex.)

Example 2: Find the minimum of \(\color{blue}{g(x) = x}\)² − \(\color{blue}{4x + 3}\).

\(\color{blue}{a = 1}\) > 0 (minimum). x = −\(\color{blue}{\frac{(-4)}{(2 \cdot 1)} = 2}\).
\(\color{blue}{g(2) = 4 – 8 + 3}\) = −1. Minimum value: −1 at \(\color{blue}{x = 2}\).

Example 3: Find the minimum of \(\color{blue}{f(x) = x}\)² − \(\color{blue}{6x + 8}\).

\(\color{blue}{a = 1}\) > 0. \(\color{blue}{x = \frac{6}{2} = 3}\). \(\color{blue}{f(3) = 9 – 18 + 8}\) = −1. Minimum: −1 at \(\color{blue}{x = 3}\).

Example 4: Find the maximum of h(x) = −2x² + \(\color{blue}{4x + 1}\).

a = −2 < 0. x = −\(\color{blue}{\frac{4}{(2 \cdot -2)} = 1}\). h(1) = −\(\color{blue}{2 + 4 + 1}\) = 3. Maximum: 3 at \(\color{blue}{x = 1}\).

More Practice: Recognizing Relative and Absolute Extrema (Video Lesson)

This second Khan Academy video focuses on distinguishing between relative and absolute maxima and minima from graphs:

Exercises for Maxima and Minima

Find the maximum or minimum value of each function and state the x-value at which it occurs.

1. \(\color{blue}{f(x) = x}\)² − \(\color{blue}{6x + 8}\)
2. f(x) = −x² + \(\color{blue}{4x – 1}\)
3. \(\color{blue}{f(x) = 2x}\)² − \(\color{blue}{8x + 6}\)
4. f(x) = −3x² + \(\color{blue}{6x + 2}\)
5. \(\color{blue}{f(x) = x}\)² + \(\color{blue}{10x + 25}\)

Answers

1. \(\color{blue}{a = 1}\) (min). \(\color{blue}{x = 3}\). \(\color{blue}{f(3) = 9 – 18 + 8}\) = −1 (minimum at \(\color{blue}{x = 3}\))
2. a = −1 (max). \(\color{blue}{x = 2}\). f(2) = −\(\color{blue}{4 + 8 – 1}\) = 3 (maximum at \(\color{blue}{x = 2}\))
3. \(\color{blue}{a = 2}\) (min). \(\color{blue}{x = 2}\). \(\color{blue}{f(2) = 8 – 16 + 6}\) = −2 (minimum at \(\color{blue}{x = 2}\))
4. a = −3 (max). \(\color{blue}{x = 1}\). f(1) = −\(\color{blue}{3 + 6 + 2}\) = 5 (maximum at \(\color{blue}{x = 1}\))
5. \(\color{blue}{a = 1}\) (min). x = −5. \(\color{blue}{f(-5) = 25 – 50 + 25}\) = 0 (minimum at x = −5)

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Want More Practice?

We haven’t published a worksheet built specifically for Maxima and Minima of a Function just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Characteristics of Quadratic Functions Worksheet
• Download Graphing Quadratic Functions Worksheet

Frequently Asked Questions

What is the difference between a relative and an absolute maximum?

A relative (local) maximum is just a peak that is higher than all nearby points. An absolute (global) maximum is the highest output value the function ever reaches across its entire domain. A function can have multiple relative maxima but at most one absolute maximum.

Can a linear function have a maximum or minimum?

No. A linear function (\(\color{blue}{y = \text{ mx } + b}\) with m ≠ 0) is always increasing or always decreasing, so it has no peaks or valleys. Its range is all real numbers, with no maximum or minimum.

What if the parabola is written in vertex form?

Vertex form is \(\color{blue}{f(x) = a(x – h)}\)\(\color{blue}{^{2} + k}\). The vertex is (h, k) directly. If a > 0, k is the minimum; if a < 0, k is the maximum. No formula needed; just read h and k from the equation.

Related Topics

• Increasing and Decreasing Functions
• Finding Domain and Range of a Function
• Average Rate of Change of a Function
• Function Notation
• Graphing Rational Functions