What is symmetry of a function?

Symmetry of a function is associated with whether it is even, odd, both, or neither. Even functions have symmetry about the y-axis. Odd functions have symmetry about the origin. The only function that is both even and odd is the zero function: f(x)=0. Functions that are not symmetric about the y-axis or the origin are considered neither even nor odd.

How do I find, determine, or test for symmetry of a function?

There are a few properties you can use to find, determine, or test for symmetry of a function. If the function f is a constant, then it is an even function and therefore symmetrical with respect to the y-axis. If the function f is a monomial function with an even exponent, then it is an even function and therefore symmetrical with respect to the y-axis. If the function f is a monomial function with an odd exponent, then it is an odd function and therefore symmetrical with respect to the origin. If the function f is a polynomial function where every term is either a constant or has an even exponent, then it is an even function and therefore symmetrical with respect to the y-axis. If the function f is a polynomial function where every term has an odd exponent, then it is an odd function and therefore symmetrical with respect to the origin. If the function f is a polynomial function consisting of terms with even and odd exponents, and/or constants, then it is neither even nor odd and therefor not symmetrical to either the y-axis or the origin. You can also find the symmetry of a function algebraically. A function f is even if: f(-x)=f(x) for all possible values of x. A function f is odd if: f(-x)=-f(x) for all possible values of x.

How do I find, determine, or test for symmetry of a rational function?

You can find the symmetry of a rational function algebraically. A rational function f is even if: f(-x)=f(x) for all possible values of x. A rational function f is odd if: f(-x)=-f(x) for all possible values of x.

How do I identify the axis of symmetry of a function?

You can identify the axis of symmetry of a function either graphically or algebraically. To identify the axis of symmetry of a function graphically, you must sketch the graph and look for the line of symmetry. To identify the axis of symmetry of a function algebraically, there are a few methods. For a quadratic function (a parabola), the axis of symmetry can be found using the formula: x=-b/(2a). A function f is even if: f(-x)=f(x) for all possible values of x. Therefore, its axis of symmetry can be identified as the y-axis. A function f is odd if: f(-x)=-f(x) for all possible values of x. Therefore, its axis of symmetry can be identified as the origin.

Symmetry of Functions

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Certain functions display properties of symmetry that help us understand them and the shapes of their graphs. There are two types of symmetry when we talk about functions and their graphs: even and odd. If a function has even symmetry, that means it is symmetrical with respect to the y-axis. If a function has odd symmetry, that means it is symmetrical with respect to the origin.

In this article, we cover everything you need to know regarding symmetry of functions and their graphs, including what an axis of symmetry is and how to find it, an in-depth analysis of even and odd functions and their properties, how to find the axis of symmetry for a quadratic function, and the symmetry of trig functions.

What is the axis (or line) of symmetry in a Function?
Even and odd functions
Axis of symmetry for a quadratic function
Symmetry of trigonometric functions
Key takeaways

What is the Axis (or Line) of Symmetry in a Function?

A shape, or graph, has what we call reflective symmetry if it remains unchanged after we reflect it over a line.

Symmetry of Functions pentagon showing reflective symmetry StudySmarter A pentagon with reflective symmetry over a line - StudySmarter Originals

Notice how the purple line cuts the pentagon in half? That means the purple line is an axis of symmetry for the pentagon.

The Axis or Line of Symmetry is the line that passes through the center of a shape and divides it into identical halves, so that one half is the mirror-image of the other half.

In other words, an Axis or Line of Symmetry creates exact reflections of an object on both sides.

An axis of symmetry doesn't have to be a vertical line. It could also be horizontal or angled. As long as it is a straight line, any line can be an axis of symmetry!

In fact, different shapes have different axes of symmetry.

An equilateral triangle has 3 axes of symmetry.
A square has 4 axes of symmetry.
A regular pentagon has 5 axes of symmetry.
A regular hexagon has 6 axes of symmetry.
- If we notice the pattern here: any regular polygon with n number of sides has n number of axes of symmetry!
And a circle actually has an infinite number of axes of symmetry.

The axes of symmetry of common shapes - StudySmarter Originals

Symmetry of Functions axes of symmetry of common regular shapes StudySmarter

This idea of reflective symmetry can be applied to the graphs of functions, too!

Even and Odd Functions

Whether a function is even or odd is a property that is related to its symmetry. This property makes dealing with functions a bit easier because it helps us graph them.

There are four states of classification for even and odd functions:

A function can be even,
A function can be odd,
A function can be neither even nor odd, or
A function can be both even and odd.

There are two ways we can check if a function is even, odd, or neither:

Algebraically: by replacing the input value (x) with a negative input (-x) and considering the output, and
Graphically: by checking for reflective symmetry with respect to the y-axis or for rotational symmetry with respect to the origin

Even and odd functions got their name from the fact that the power function, which is a monomial function of the form: $f (x) = x^{\pm n}$ is an even function if $n$ is even, and it is an odd function if $n$ is odd.

But, an exponent does not always mean a function is even or odd! For instance, the function f(x)=(x+1)^2 is not an even function, and the function f(x)=(x^3)+1 is not an odd function! We can verify this by using the definition for even and odd functions below.

In general, we consider functions to be even, odd, or neither. To determine whether a function, $f (x)$ , is even or odd, all we need to do is plug in $- x$ for $x$ and check the output value of $f (- x)$ .

Consider the real-valued function $f (x) = x^{2}$ .

Choose any positive number for x and plug it into the function. What is the output?
Now, plug the negative value of your number x into the function. What is the output?
- Hint: you should get the same answer for both!
This phenomenon can be expressed by writing f-x=fx for all x.
- In other words, if you plug in $x$ as an input to $f$ , you get the same answer as if you used $- x$ instead of $x$ .

This property is summarized in the definition of an even function below.

If a real-valued function $f$ satisfies $f (- x) = f (x)$ for every number $x$ in its domain, then it is an even function.

Put simply, if we replace every number $x$ with $- x$ , and we get the same result, the function is even.

Consider the real-valued function $f (x) = x^{3}$ .

Choose any positive number for x and plug it into the function. What is the output?
Now, plug the negative value of your number x into the function. What is the output?
- Hint: you should get the negative answer for what you chose first.
This phenomenon can be expressed by writing f-x=-fx for all x.
- In other words, if you plug in $- x$ as an input to $f$ , you get the negative of the answer for what value of $x$ you chose first.

This property is summarized in the definition of an odd function below.

If a real-valued function $f$ satisfies $f (- x) = - f (x)$ for every number $x$ in its domain, then it is an odd function.

Put simply, if we replace every number $x$ with $- x$ , and we get the negative version of the same result, the function is odd.

Graphs of Even and Odd Functions

The visual cue that a function is even is that its graph is symmetric with respect to the y-axis. What does that mean, exactly? It means that we can think of the y-axis as a mirror, or the axis of symmetry: what is on one side of the y-axis is reflected over to the other side. In other words, if we had the graph of an even function on a piece of paper, and we folded that paper along the y-axis of the graph, the two sides of the function would perfectly overlap each other.

The function $f (x) = x^{2}$ is even because:

$f (- x) = {(- x)}^{2} = x^{2} = f (x)$

Symmetry of Functions graph of an even function StudySmarter The graph an even function - StudySmarter Originals

The visual cue that a function is odd is that its graph is symmetric with respect to the origin. What does that mean, exactly? It means that if we reflect the graph of the function over both the x- and y-axes, it would look exactly the same as it did before we reflected it. In other words, if we had the graph of an odd function on a piece of paper, we could rotate that paper 180° (using the origin point of the graph as the point of rotation) and we would get the original graph.

The function $f (x) = x^{3}$ is odd because:

$f (- x) = {(- x)}^{3} = - x^{3} = - f (x)$

Symmetry of Functions graph of an odd function StudySmarter The graph of an odd function - StudySmarter Originals

Properties of Even and Odd Functions

There are several properties of even and odd functions that will help us out in AP Calculus.

Sum and difference properties:

$e v e n f u n c t i o n \pm e v e n f u n c t i o n = e v e n f u n c t i o n$
$o d d f u n c t i o n \pm o d d f u n c t i o n = o d d f u n c t i o n$
even function ± odd function = neither
- Unless one of the functions is the zero function, then it remains the same as the non-zero function.

Product and quotient properties:

$e v e n f u n c t i o n \times / \div e v e n f u n c t i o n = e v e n f u n c t i o n$
$o d d f u n c t i o n \times o d d f u n c t i o n = o d d f u n c t i o n$
$o d d f u n c t i o n \div o d d f u n c t i o n = e v e n f u n c t i o n$
$e v e n f u n c t i o n \times / \div o d d f u n c t i o n = o d d f u n c t i o n$

Composition properties:

$e v e n F u n c t i o n (e v e n F u n c t i o n (x)) = e v e n f u n c t i o n$
$o d d F u n c t i o n (o d d F u n c t i o n (x)) = o d d f u n c t i o n$
$o d d F u n c t i o n (e v e n F u n c t i o n (x)) = e v e n F u n c t i o n (o d d F u n c t i o n (x)) = e v e n f u n c t i o n$

Functions that are Neither Even nor Odd

Most functions we encounter in AP Calculus are neither even nor odd. A real-valued function that meets neither the even nor the odd requirements is considered to be neither even nor odd.

Consider the function:

$f (x) = 2 x^{5} + 3 x^{2} + 1$

If we plug in $- x$ for $x$ we get:

$f (- x) = 2 {(- x)}^{5} + 3 {(- x)}^{2} + 1 = - 2 x^{5} + 3 x^{2} + 1 \neq f (x) \Rightarrow f (x) = 2 x^{5} + 3 x^{2} + 1 \neq - f (x) \Rightarrow - f (x) = - 2 x^{5} - 3 x^{2} - 1$

Since the output of $f (- x)$ is neither $f (x)$ nor $- f (x)$ , the function is neither even nor odd.

The Zero Function: it's both Even and Odd

A real-valued function $f (x)$ is called both even and odd if it satisfies both $f (- x) = f (x)$ and $f (- x) = - f (x)$ for all values of $x$ in the domain of $f (x)$ .

There is only one function that is both even and odd, and that is the zero function: $f (x) = 0$ .

Why is this?

We know that for the zero function, $f (- x) = - f (x) = f (x) = 0$ , for all values of $x$ . This is because -0 is considered the same as 0.

Therefore, $f (x) = 0$ is both an even and odd function.

Even and Odd Functions: Examples

Determine whether the function $f (x) = {(x + 1)}^{2}$ is even, odd, or neither.

Solution:

f(-x)=-x+12=-x-12≠f(x)≠-f(x)
1. Therefore, $f (x) = {(x + 1)}^{2}$ is neither even nor odd.

Determine whether the function $f (x) = x^{3} + 1$ is even, odd, or neither.

f(-x)=-x3+1=-x3+1≠f(x)≠-f(x)
1. Therefore, $f (x) = x^{3} + 1$ is neither even nor odd.

Determine whether the rational function $f (x) = \frac{x}{x^{2} - 1}$ is even, odd, or neither.

Solution:

f(-x)=-x-x2-1=-xx2-1=-xx2-1=-f(x)
1. Therefore, $f (x) = \frac{x}{x^{2} - 1}$ is an odd function.

Axis of Symmetry for a Quadratic Function

The most common function used to describe an axis of symmetry is the parabola. The parabola is a quadratic function with a U-shaped graph whose axis of symmetry is a vertical line drawn through its vertex (the highest or lowest point on the graph).

Symmetry of Functions axis of symmetry of a parabola StudySmarter The axis of symmetry of a parabola - StudySmarter Originals

A parabola has one axis of symmetry, and we can use that to determine the orientation of the parabola:

If the axis of symmetry is vertical, the parabola is vertical.
- Vertical parabolas open up or down.
If the axis of symmetry is horizontal, then the parabola is horizontal too.
- Horizontal parabolas open to the left or right.

A note on horizontal (or sideways) parabolas:

Symmetry of Functions horizontal parabola StudySmarter A horizontal parabola is not a function - StudySmarter Originals

If we remember from our Functions article: just because we can graph something, that doesn't mean it is a function. While a parabola that opens up or down is a function, a parabola that opens to the left or right is not a function.

Can you remember why this is the case?

It's because horizontal parabolas fail the vertical line test! This means a left/right opening parabola is a relation, not a function.

BUT...

Any shape (think squares, circles, triangles, stars, etc.) can still have an axis (or axes) of symmetry, even if they aren't functions!

So, even though horizontal parabolas aren't functions, they do still have an axis of symmetry. In this case, the axis of symmetry is the horizontal line through its vertex, instead of the vertical line through its vertex like the up/down opening parabola.

Axis of Symmetry Equation

If the vertex is the point where the axis of symmetry intersects a parabola, can we figure out how to come up with the equation for a parabola's axis of symmetry? Well, let's start with what we know:

We can use the axis of symmetry to determine a parabola's orientation, so we can also use a parabola's orientation to determine the axis of symmetry!
We know a vertical (up/down opening) parabola has a vertical axis of symmetry.
We know a horizontal (left/right opening) parabola has a horizontal axis of symmetry.

Based on this, we can infer:

For a vertical parabola whose vertex is $(h, k)$ , the axis of symmetry equation is $x = h$
For a horizontal parabola whose vertex is $(h, k)$ , the axis of symmetry equation is $y = k$

Axis of Symmetry Formula

Since a vertical parabola is a function, we can use a formula to find the axis of symmetry. This formula for an axis of symmetry can take two forms:

Standard Form

The standard form quadratic equation is:

$y = a x^{2} + b x + c$

where a, b, and c are real numbers.

We can find a parabola's axis of symmetry based on its quadratic equation by using this formula:

$x = - \frac{b}{2 a}$

Vertex Form

If we write the quadratic equation of a parabola in vertex form, we get:

$y = a {(x - h)}^{2} + k$

where $(h, k)$ is the vertex of the parabola.

This makes the formula for the axis of symmetry very simple:

$x = h$

since in vertex form, the vertex and axis of symmetry lie on the same line!

Factored Form

If we write the quadratic equation of a parabola in factored form, we get:

$y = a (x - p) (x - q)$

where p and q are the zeros (the points where the parabola crosses the x-axis) of the parabola.

Let's write the following in factored form:

$y = x^{2} - 5 x + 6$

Solution:

Multiply the leading coefficient: 1 by the constant: 6.
- $1 \times 6 = 6$
Factor the number we got in step 1 (the number 6) into two parts so that it meets two requirements:
1. The sum of those two parts is equal to the coefficient of $x$ , which is -5
2. The product of those two parts is equal to the number we got in step 1, which is 6.
  - With these in mind, can we think of two factors of 6 that when we add them we get -5 and when we multiply them we get 6?
    - We factor 6 as: $6 = (- 2) (- 3)$
Using these two factors: -2 and -3, factor the original quadratic equation:
- x2-5x+6=x2-2x-3x+6 =xx-2-3x-2 =x-2x-3
  - This means that the zeros (or roots) of this quadratic equation are $x = 2, x = 3$ . These are the points where the parabola crosses the x-axis. The graph of this quadratic is shown below.
  - The graph of a parabola showing the roots and axis of symmetry - StudySmarter Originals
  - Notice how the axis of symmetry is exactly in the middle of the roots of the equation?
    - This means that the factored form of the quadratic equation can be used to find the axis of symmetry. All we have to do is calculate the midpoint between the two roots, and we have it!
    - The midpoint can be calculated using the formula

$m i d p o i n t = \frac{p + q}{2}$

Axis of Symmetry: Derivation for a Parabola

Since the axis of symmetry always passes through the vertex of a parabola, finding the vertex is necessary for finding the axis of symmetry. For a vertical parabola, we know that the axis of symmetry formula is:

$x = - \frac{b}{2 a}$

Now, let's understand why this is the case.

We know the quadratic formula for a parabola is:

$y = a x^{2} + b x + c$

where a, b, and c are real numbers.

The term c is a constant that doesn't affect the parabola, so we can just consider the formula:

$y = a x^{2} + b x$

We know the axis of symmetry must pass through the vertex, but how do we find the vertex?

Well, since the axis of symmetry cuts the parabola into two equal halves, we need to do two things:

Find the x-intercepts of the parabola.
Then find the midpoint between the x-intercepts.

To find the x-intercepts, substitute $y = 0$ and solve for x:

$0 = a x^{2} + b x 0 = x (a x + b) x = 0 a n d a x + b = 0 x = 0 a n d x = - \frac{b}{a}$

To find the midpoint between the x-intercepts, plug both values of x into the midpoint formula:

$x = \frac{x_{1} + x_{2}}{2}$

and solve for x:

$x = \frac{0 + \frac{- b}{a}}{2} x = \frac{- \frac{b}{a}}{2} x = - \frac{b}{2 a}$

Axis of Symmetry for a Quadratic Function: Examples

Given the quadratic equation:

$y = x^{2} - 4 x + 3$

Find the axis of symmetry.

Solution:

This equation is in standard form, so we can gather that:

$a = 1, a n d b = - 4$

Using the axis of symmetry formula:

$x = - \frac{b}{2 a}$

So, plugging in the values for a and b, we get:

$x = - \frac{- 4}{2 (1)} = \frac{4}{2} = 2$

Therefore, the axis of symmetry is the vertical line: $x = 2$

Given the quadratic equation:

$y = - 4 x^{2}$

Find the axis of symmetry.

Solution:

We know that:

$a = - 4 a n d b = 0$

Using the axis of symmetry formula:

$x = - \frac{b}{2 a}$

So, plugging in the values for a and b, we get:

$x = - \frac{0}{2 (- 4)} = 0$

Therefore, the axis of symmetry is the vertical line: $x = 0$

Given the quadratic equation:

$y = 2 {(x - 3)}^{2} + 5$

Find the axis of symmetry.

Solution:

Since the quadratic equation is in vertex form, we know that:

$h = 3 a n d k = 5$

So, the vertex of the parabola is:

$(3, 5)$

Therefore, the axis of symmetry is the vertical line: $x = 3$

Symmetry of Trigonometric Functions

The six main trig functions all have properties of symmetry that are helpful for both understanding and evaluating them. All of them are either even or odd functions, and they have symmetry between angles.

Even and Odd Trig Functions

The 6 main trig functions:

sine and its reciprocal, cosecant
cosine and its reciprocal, secant
tangent and its reciprocal, cotangent

have even or odd properties that can be determined using the unit circle.

Symmetry of Functions unit circle StudySmarter The unit circle - StudySmarter Originals

To figure out whether a trig function is even or odd, we look at which of the trig functions have positive values in each quadrant of the unit circle:

In Quadrant I, all trig functions are positive.
In Quadrant II, only sine and cosecant are positive.
In Quadrant III, only tangent and cotangent are positive.
In Quadrant IV, only cosine and secant are positive.

Because $\cos (θ)$ and its reciprocal, $s e c (θ)$ are positive in both Quadrant I and Quadrant IV of the unit circle, we know that $\cos (- θ) = \cos (θ)$ and $s e c (- θ) = s e c (θ)$ , so they satisfy the requirement to be even functions.

Because $\sin (θ)$ and $\tan (θ)$ and their reciprocals, $c s c (θ)$ and $c o t (θ)$ are positive in Quadrant I but negative in Quadrant IV of the unit circle, we know that $\sin (- θ) = - \sin (θ), c s c (- θ) = - c s c (θ), \tan (- θ) = - \tan (θ), a n d c o t (- θ) = - c o t (θ)$ , so they satisfy the requirement to be odd functions.

Symmetry with Angles in Trig Functions

Remember the periodic and cofunction trig identities from our Manipulating Functions article? These identities show how the trig functions have symmetry based on what angle for $θ$ we use with the function. They also can be quite helpful when simplifying problems in AP Calculus.

Now let's take this idea and expand on it a little bit. If we examine the unit circle again, we can see that more properties of symmetry for the trigonometric functions can be established.

Symmetry of Functions unit circle showing trig functions symmetry StudySmarter The unit circle for trig functions symmetry - StudySmarter Originals

What this is showing is that when we use certain angles for $θ$ with the trig functions, the result is often one of the other trig functions! This leads us to find the following identities:

Reflected in θ = 0	Reflected in θ = π/4 (cofunction identities)	Reflected in θ = π/2
$\sin (- θ) = - \sin (θ)$	$\sin (\frac{π}{2} - θ) = + \cos (θ)$	$\sin (π - θ) = + \sin (θ)$
$\cos (- θ) = + c o s (θ)$	$\cos (\frac{π}{2} - θ) = + \sin (θ)$	$\cos (π - θ) = - \cos (θ)$
$\tan (- θ) = - \tan (θ)$	$\tan (\frac{π}{2} - θ) = + co t (θ)$	$\tan (π - θ) = - \tan (θ)$
$c s c (- θ) = - c s c (θ)$	$c s c (\frac{π}{2} - θ) = + s e c (θ)$	$c s c (π - θ) = + c s c (θ)$
$s e c (- θ) = + s e c (θ)$	$s e c (\frac{π}{2} - θ) = + c s c (θ)$	$s e c (π - θ) = - s e c (θ)$
$c o t (- θ) = - c o t (θ)$	$c o t (\frac{π}{2} - θ) = + \tan (θ)$	$c o t (π - θ) = - c o t (θ)$

The sign in front of the trig function does not necessarily indicate the sign of the value of the function. For instance, +cos(θ) doesn't always mean that cos(θ) is positive. In fact, if θ = π, +cos(θ) = -1.

Shifting by Angles and Periodicity of Trig Functions

It is also helpful to know that if we shift a trig function by certain angles, it is very likely that we can find a different trig function to express the result more simply. Common examples of this are shifting by $π / 2, π, and 2 π$ radians. Because the periods of the trig functions are either $π or 2 π$ , there are several cases where the new function is the same as the old function! Let's take a look at these shifts:

Shift of π/2 radians	Shift of π radians	Shift of 2π radians
$\sin (θ + \frac{π}{2}) = + \cos (θ)$	$\sin (θ + π) = - \sin (θ)$	$\sin (θ + 2 π) = + \sin (θ)$
$\cos (θ + \frac{π}{2}) = - \sin (θ)$	$\cos (θ + π) = - \cos (θ)$	$\cos (θ + 2 π) = + \cos (θ)$
$\tan (θ + \frac{π}{2}) = - c o t (θ)$	$\tan (θ + π) = + \tan (θ)$	$\tan (θ + 2 π) = + \tan (θ)$
$c s c (θ + \frac{π}{2}) = + s e c (θ)$	$c s c (θ + π) = - c s c (θ)$	$c s c (θ + 2 π) = + c s c (θ)$
$s e c (θ + \frac{π}{2}) = - c s c (θ)$	$s e c (θ + π) = - s e c (θ)$	$s e c (θ + 2 π) = + s e c (θ)$
$c o t (θ + \frac{π}{2}) = - \tan (θ)$	$c o t (θ + π) = + c o t (θ)$	$c o t (θ + 2 π) = + c o t (θ)$

Another way of looking at the symmetries of trig functions is summarized in the table below.

Case	Relationship between angles A and B	Diagram	Conclusion
1	The angles differ by a multiple of 360°. $B - A = 360 k ° o r B = A + 360 k °$		Since $A$ and $A + 360 k °$ are coterminal, they share the same value of cosine and sine.
2	The angles differ by an odd multiple of 180°. $B - A = 180 ° (2 k - 1) o r B = A + 180 ° (2 k - 1)$		Since $A$ and $A + 180 ° (2 k - 1)$ have terminal sides in diagonally opposite quadrants, both cosine and sine change sign.
3	The sum of the angles is a multiple of 360°. $A + B = 360 k ° o r B = 360 k ° - A$		Since $A$ and $360 k ° - A$ lie in vertically adjacent quadrants, the cosine values are the same, but the sine values are opposite.
4	The sum of the angles is an odd multiple of 180°. $A + B = 180 ° (2 k - 1) o r B = 180 ° (2 k - 1) - A$		Since $A$ and $180 ° (2 k - 1) - A$ lie in horizontally adjacent quadrants, the cosine values are opposite but the sine values are the same.

Symmetry of Trigonometric Functions: Examples

Determine whether the function $f (x) = \sin (x) \cos (x)$ is even, odd, or neither.

Solution:

We know that $\sin (x)$ is an odd function, and that $\cos (x)$ is an even function.
We know that multiplying an even function and an odd function gives us an odd function.
1. Therefore, $f (x) = \sin (x) \cos (x)$ is an odd function.

Verification:

$f (- x) = \sin (- x) \cos (- x) = - \sin (x) \cos (x) = - f (x)$

Determine whether the function $f (x) = \cos h (x)$ is even, odd, or neither.

Solution:

$f (x) = \cos h (x) = \frac{e^{x} + e^{- x}}{2}$ , so to determine if the function is even, odd, or neither, replace x with -x and simplify.
f(-x)=cosh-x=e-x+e--x2=e-x+ex2=ex+e-x2=f(x)
1. Therefore, $f (x) = \cos h (x)$ is an even function.

What is the relationship between $\tan (θ + π) a n d \tan (θ)$ ?

Solution:

$t a n (θ + π) = \frac{\sin (θ + π)}{\cos (θ + π)} = \frac{- \sin (θ)}{- \cos (θ)} = t a n (θ)$

Therefore, $t a n (θ + π) = t a n (θ)$

This shows that the period of the tangent function is $π$ .

Symmetry of Functions - Key takeaways

There are two main types of symmetry in functions:
- Even functions - are symmetrical with respect to the y-axis.
- Odd functions - are symmetrical with respect to the origin.
Though some functions can be either even or odd, most functions are neither.
- Only the zero function is both even and odd.
An axis (or line) of symmetry is a line that divides an object into two equal halves that are mirror images of each other.
The quadratic function has two formulas for finding the axis of symmetry for a parabola:
- Standard form
- Vertex form
The 6 main trig functions are even or odd functions.
- Cosine and Secant are even functions.
- Sine, Cosecant, Tangent, and Cotangent are odd functions.

Frequently Asked Questions about Symmetry of Functions

Symmetry of a function is associated with whether it is even, odd, both, or neither.

Even functions have symmetry about the y-axis.
Odd functions have symmetry about the origin.
The only function that is both even and odd is the zero function: f(x)=0.
Functions that are not symmetric about the y-axis or the origin are considered neither even nor odd.

There are a few properties you can use to find, determine, or test for symmetry of a function.

If the function f is a constant, then it is an even function and therefore symmetrical with respect to the y-axis.
If the function f is a monomial function with an even exponent, then it is an even function and therefore symmetrical with respect to the y-axis.
If the function f is a monomial function with an odd exponent, then it is an odd function and therefore symmetrical with respect to the origin.
If the function f is a polynomial function where every term is either a constant or has an even exponent, then it is an even function and therefore symmetrical with respect to the y-axis.
If the function f is a polynomial function where every term has an odd exponent, then it is an odd function and therefore symmetrical with respect to the origin.
If the function f is a polynomial function consisting of terms with even and odd exponents, and/or constants, then it is neither even nor odd and therefor not symmetrical to either the y-axis or the origin.

You can also find the symmetry of a function algebraically.

A function f is even if: f(-x)=f(x) for all possible values of x.
A function f is odd if: f(-x)=-f(x) for all possible values of x.

You can find the symmetry of a rational function algebraically.

A rational function f is even if: f(-x)=f(x) for all possible values of x.
A rational function f is odd if: f(-x)=-f(x) for all possible values of x.

You can identify the axis of symmetry of a function either graphically or algebraically.
To identify the axis of symmetry of a function graphically, you must sketch the graph and look for the line of symmetry.
To identify the axis of symmetry of a function algebraically, there are a few methods.

For a quadratic function (a parabola), the axis of symmetry can be found using the formula: x=-b/(2a).
A function f is even if: f(-x)=f(x) for all possible values of x. Therefore, its axis of symmetry can be identified as the y-axis.
A function f is odd if: f(-x)=-f(x) for all possible values of x. Therefore, its axis of symmetry can be identified as the origin.

Flashcards in Symmetry of Functions13 Start learning

What is the Axis (or Line) of Symmetry?

The Axis (or Line) of Symmetry is the line that passes through the center of a shape and divides it into identical halves, so that one half is the mirror-image of the other half.

What is an even function?

If a function has even symmetry, that means it is symmetrical with respect to the y-axis.

What is an odd function?

If a function has odd symmetry, that means it is symmetrical with respect to the origin.

What are the two ways to check if a function is even or odd?

Graphically and Algebraically

True or False?

Most functions are neither even nor odd.

True

Which of the following is the only function that is both even and odd?

The zero function

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Symmetry of Functions

What is the Axis (or Line) of Symmetry in a Function?

Even and Odd Functions

Graphs of Even and Odd Functions

Properties of Even and Odd Functions

Functions that are Neither Even nor Odd

Even and Odd Functions: Examples

Axis of Symmetry for a Quadratic Function

Axis of Symmetry Equation

Axis of Symmetry Formula

Standard Form

Vertex Form

Factored Form

Axis of Symmetry for a Quadratic Function: Examples

Symmetry of Trigonometric Functions

Even and Odd Trig Functions

Symmetry with Angles in Trig Functions

Shifting by Angles and Periodicity of Trig Functions

Symmetry of Trigonometric Functions: Examples

Symmetry of Functions - Key takeaways

Frequently Asked Questions about Symmetry of Functions

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Symmetry of Functions

What is the Axis (or Line) of Symmetry in a Function?

Even and Odd Functions

Graphs of Even and Odd Functions

Properties of Even and Odd Functions

Functions that are Neither Even nor Odd

Even and Odd Functions: Examples

Axis of Symmetry for a Quadratic Function

Axis of Symmetry Equation

Axis of Symmetry Formula

Standard Form

Vertex Form

Factored Form

Axis of Symmetry for a Quadratic Function: Examples

Symmetry of Trigonometric Functions

Even and Odd Trig Functions

Symmetry with Angles in Trig Functions

Shifting by Angles and Periodicity of Trig Functions

Symmetry of Trigonometric Functions: Examples

Symmetry of Functions - Key takeaways

Frequently Asked Questions about Symmetry of Functions

What is symmetry of a function?

How do I find, determine, or test for symmetry of a function?

How do I find, determine, or test for symmetry of a rational function?

How do I identify the axis of symmetry of a function?

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Learn with 13 Symmetry of Functions flashcards in the free StudySmarter app

Learn with 13 Symmetry of Functions flashcards in the free StudySmarter app

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