Function symmetry introduction | Transformations of functions | Algebra 2 | Khan Academy

TL;DR

Even functions look the same when flipped over the y-axis, while odd functions look the same when flipped over both the y- and x-axes.

Transcript

• [Instructor] You've likely heard the concept of even and odd numbers, and what we're going to do in this video is think about even and odd functions. And as you can see or as you will see, there's a little bit of a parallel between the two, but there's also some differences. So let's first think about what an even function is. One way to think ab... Read More

Key Insights

• 👀 Even functions look the same when flipped over the y-axis, while odd functions look the same when flipped over the y- and x-axes.
• 🦕 Mathematically, even functions are represented as f(x) = f(-x), and odd functions are represented as f(x) = -f(-x).
• 🦕 Polynomial functions with even exponents are even functions, while those with odd exponents are odd functions.
• 😑 The structure of an expression in addition to the exponent determines whether a function is even or odd.
• 🦕 Functions that are neither even nor odd do not have symmetrical properties.
• 🖤 The concept of even and odd functions relates to the symmetry or lack thereof in mathematical functions.
• 🦕 Understanding even and odd functions can help analyze and identify patterns in various mathematical equations.

Questions & Answers

Q: What is the difference between even and odd functions?

Even functions remain the same when flipped over the y-axis, while odd functions look the same when flipped over both the y- and x-axes.

Q: How can we mathematically determine if a function is even or odd?

An even function is represented by f(x) = f(-x), meaning the function is equal to its reflection over the y-axis. An odd function is represented by f(x) = -f(-x), indicating that the function is equal to its reflection over both the y- and x-axes.

Q: Are all polynomial functions either even or odd?

No, not all polynomial functions are either even or odd. Polynomial functions with even exponents are even functions, while polynomial functions with odd exponents are odd functions. Other polynomial functions that do not follow this pattern are neither even nor odd.

Q: Is it possible for a function to be both even and odd?

No, a function cannot be both even and odd. This is because an even function remains unchanged when flipped over the y-axis and an odd function remains unchanged when flipped over both the y- and x-axes. However, a function that is constantly zero is both even and odd.

Summary & Key Takeaways

• Even functions are those that remain unchanged when flipped over the y-axis, while odd functions look the same when flipped over both the y- and x-axes.

• Mathematically, an even function is represented by f(x) = f(-x), while an odd function is represented by f(x) = -f(-x).

• Polynomial functions with even exponents are even functions, while polynomial functions with odd exponents are odd functions.