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## Algebra II (2018 edition)

### Unit 4: Lesson 14

Introduction to symmetry of functions# Function symmetry introduction

Learn what even and odd functions are, and how to recognize them in graphs.

#### What you will learn in this lesson

A shape has

**reflective symmetry**if it remains unchanged after a reflection across a line.For example, the pentagon above has reflective symmetry.

Notice how line l is a line of symmetry, and that the shape is a mirror image of itself across this line.

This idea of reflective symmetry can be applied to the shapes of graphs. Let's take a look.

## Even functions

A function is said to be an

**even function**if its graph is symmetric with respect to the y-axis.For example, the function f graphed below is an even function.

Verify this for yourself by dragging the point on the x-axis from right to left. Notice that the graph remains unchanged after a reflection across the y-axis!

#### Check your understanding

### An algebraic definition

Algebraically, a function f is even if f, left parenthesis, minus, x, right parenthesis, equals, f, left parenthesis, x, right parenthesis for all possible x values.

For example, for the even function below, notice how the y-axis symmetry ensures that f, left parenthesis, x, right parenthesis, equals, f, left parenthesis, minus, x, right parenthesis for all x.

## Odd functions

A function is said to be an

**odd function**if its graph is symmetric with respect to the origin.Visually, this means that you can rotate the figure 180, degrees about the origin, and it remains unchanged.

Another way to visualize origin symmetry is to imagine a reflection about the x-axis, followed by a reflection across the y-axis. If this leaves the graph of the function unchanged, the graph is symmetric with respect to the origin.

For example, the function g graphed below is an odd function.

Verify this for yourself by dragging the point on the y-axis from top to bottom (to reflect the function over the x-axis), and the point on the x-axis from right to left (to reflect the function over the y-axis). Notice that this is the original function!

#### Check your understanding

### An algebraic definition

Algebraically, a function f is odd if f, left parenthesis, minus, x, right parenthesis, equals, minus, f, left parenthesis, x, right parenthesis for all possible x values.

For example, for the odd function below, notice how the function's symmetry ensures that f, left parenthesis, minus, x, right parenthesis is always the

*opposite*of f, left parenthesis, x, right parenthesis.## Reflection question

## Want to join the conversation?

- What is the use of describing a function as "even" or "odd"?(51 votes)
- Even and odd functions have properties that can be useful in different contexts. The most basic one is that for an even function, if you know f(x), you know f(-x). Similarly for odd functions, if you know g(x), you know -g(x). Put more plainly, the functions have a symmetry that allows you to find any negative value if you know the positive value, or vice versa.(100 votes)

- Can an equation be both even and odd?(16 votes)
- The only function which is both even and odd is f(x) = 0, defined for all real numbers. This is just a line which sits on the x-axis. If you count equations which are not a function in terms of y, then x=0 would also be both even and odd, and is just a line on the y-axis.(53 votes)

- How can you prove definitively that a function is even or odd (or neither) just by its equation? Is there even a way?(12 votes)
- Mona's explanation works very well for polynomials. Two things to keep in mind:

1) Odd functions cannot have a constant term because then the symmetry wouldn't be based on the origin.

2) Functions that are not polynomials or that don't have exponents can still be even or odd. For example, f(x)=cos(x) is an even function.(10 votes)

- How can a function be neither even or odd?(7 votes)
- Even and odd describe 2 types of symmetry that a function might exhibit.

1) Functions do not have to be symmetrical. So, they would not be even or odd.

2) If a function is even, it has symmetry around the y-axis. What is a function has symmetry around y=5? It would not be even, because the symmetry is not around the Y-axis.

3) Similarly, odd functions have symmetry around the origin. Functions might have symmetry based on some point other than the origin. So, they would not be odd.

Hope this helps.(21 votes)

- Let's say the parent function y=x^2 gets translated to the left by 4. So now the equation is y=(x+4)^2. Is it still an even function? It is confusing because now the graph is not symmetric over the y-axis. So does this mean it is an odd function now? Or is it neither?(6 votes)
- Even function are strictly symmetrical about the y axis, so it's neither.(6 votes)

- How about symmetry with respect to x-axis only? Is it a thing?

Why did we define an even function to be symmetric with respect to y-axis and not the x one?(4 votes)- Remember the
**vertical line test**? A curve*cannot*be a function when a vertical line interesects it more than once.

And a curve that is symmetrical around the x-axis will*always fail*the vertical line test (unless that function is f(x) = 0). So, a function can*never*be symmetrical around the x-axis.

Just remember:*symmetry around x-axis*≠*function*

To answer your second question, "even" and "odd" functions are named for the exponent in this**power function**:

f(x) = xⁿ

-*if n is an even integer, then f(x) is an "even" function*

-*if n is an odd integer, then f(x) is an "odd" function*

Hope this helps!(7 votes)

- What is the name for a function that is neither even nor odd?(3 votes)
- There is no such terminology, it's just that a function that does not exhibit both the symmetry i.e. even or odd.(5 votes)

- I know that a function can be neither even or odd. The only way this is possible is if it's a line right?(2 votes)
- No, there are other ways that it can happen. You can have a functions that has multiple curves and the curves are not symmetrical according to the rules for even or odd symmetry.(4 votes)

- How do you know whether a function is even or odd if the functions consists of sines and cosines?(2 votes)
- Use the fact that sine is odd and cosine is even and observe the function's behavior when you plug in -𝑥. Just see if the function satisfies 𝑓(-𝑥) = 𝑓(𝑥) or 𝑓(-𝑥) = -𝑓(𝑥).(4 votes)

- can a function be even with respect to a line that is not the y axis?(3 votes)
- "Even" refers to a function that has line symmetry with respect to the y-axis.

Other functions can have line symmetry (which I have most often seen constrained to symmetry with respect to vertical lines only) with respect to other lines, but we would not call them even. We would simply say they have line symmetry.

For example, quadratic functions (parabolas) have line symmetry with respect to the line x = h, where h is the x-coordinate of the vertex.

For another example, transformations of the absolute value function y = |x| have line symmetry.

Transformations of the sine, cosine, secant, and cosecant functions have an infinite number of lines of symmetry, spaced every 2*pi unless they have been horizontally stretched or compressed.(1 vote)