On this page, we cover symmetry of graphs in rectangular form. This other page covers symmetry of graphs in polar form.
We explain three types of graph symmetry on this page, even (yaxis), odd (origin) and xaxis symmetry. You will also come across symmetry described as symmetry about the xaxis, yaxis and y=x. Most graphs have no symmetry and some graphs can have more than one kind of symmetry. Knowing symmetry can sometimes simplify calculations in calculus.
Even Function
There are several ways to describe even symmetry, listed below.
even symmetry 
symmetric about the yaxis 
even function 
Here are a couple of examples of even functions.
\(y=x^2\)
\(y=\cos(x)\)
Looking at the graphs, notice that in each case, the graph on the left side of the yaxis is a mirror image of the graph on the right side. You can also think of it like this. If the point (a,b) is on the graph, then (a,b) is also on the graph of an even function.
Mathematically, you can show that a function, \(f(x)\) is even as follows.
1.  Find \(f(x)\) 
2.  If \(f(x) = f(x)\), then the function is even. 
Let's do an example. In the graph on the left, we have \(f(x)=x^2\).
\(f(x) = (x)^2 = x^2 = f(x)\)
Since \(f(x)=f(x)\), the function is even.
Odd Function
An odd function parallels the even function case, except that an odd function is mirrored or reflected about the origin. You can think of origin symmetry as a function reflected about the yaxis and then about the xaxis (or in reverse). Or you can think of the graph as rotating 180 degrees to get the same graph. Using points, you can notice that for a point (a,b) on the graph, the point (a,b) will also be on the graph of an odd function.
odd symmetry 
symmetric about the origin 
odd function 
Here are a couple of examples.
\(y=x^3\)
\(y=\sin(x)\)
We follow the same procedure with the equations that we did with an even function, except in this case, for a function to be odd, \(f(x) = f(x)\). Let's do an example.
\(g(x) = x^3\)
\(g(x) = (x)^3 = \) \( x^3 = (x^3) = g(x)\)
Since \(g(x) = g(x)\), the function \(g(x)=x^3\) is odd.
xaxis Symmetry
Here is a graph showing xaxis symmetry. Similar to an even function that is symmetric about the yaxis, this graph has a reflection across the xaxis. Notice that it is not a function since it does not pass the vertical line test. This is true of all graphs with xaxis symmetry. One way to think about xaxis symmetry is to notice if we have a point (a,b) on the graph, the point (a,b) is also on the graph.
\(y^2=x\)
Before jumping into some practice problems, let's watch this short video to make sure you are clear about the three kinds of symmetry.
video by MIP4U 

Okay, time for some practice problems.
Practice
Unless otherwise instructed, determine the symmetry of these graphs of these equations using algebraic techniques.
\( f(x) = x^2 + 1 \)
Problem Statement 

Determine the symmetry of the graph of the equation \( f(x) = x^2 + 1 \) using algebraic techniques.
Final Answer 

\(f(x)\) has even symmetry
Problem Statement 

Determine the symmetry of the graph of the equation \( f(x) = x^2 + 1 \) using algebraic techniques.
Solution 

First, let's test for even symmetry.
\(\begin{array}{rcl} f(x) & = & x^2+1 \\ f(x) & = & (x)^2+1 \\ & = & x^2+1 \\ & = & f(x) \end{array} \)
Since \( f(x) = f(x) \), this function has even symmetry.
If we look more closely at the equation, this is a parabola with vertex at the origin shifted up one unit.
Although the problem did not state that we needed a graph, we have provided one here to check if our answer makes sense.
Final Answer 

\(f(x)\) has even symmetry 
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\(y=x^4+4x^2+5\)
Problem Statement 

Unless otherwise instructed, determine the symmetry of the graph of the equation \(y=x^4+4x^2+5\) using algebraic techniques.
Solution 

video by MIP4U 

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\(y=x^32x\)
Problem Statement 

Unless otherwise instructed, determine the symmetry of the graph of the equation \(y=x^32x\) using algebraic techniques.
Solution 

video by MIP4U 

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\(x^2+4y^2=4\)
Problem Statement 

Unless otherwise instructed, determine the symmetry of the graph of the equation \(x^2+4y^2=4\) using algebraic techniques.
Solution 

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\(y^22y^4=6x\)
Problem Statement 

Unless otherwise instructed, determine the symmetry of the graph of the equation \(y^22y^4=6x\) using algebraic techniques.
Solution 

video by MIP4U 

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\( f(x) = x^4  29x^2 + 100 \)
Problem Statement 

Determine the symmetry of the graph of the equation \( f(x) = x^4  29x^2 + 100 \) using algebraic techniques.
Solution 

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\( y = x(x2)(x+3) \)
Problem Statement 

Determine the symmetry of the graph of the equation \( y = x(x2)(x+3) \) using algebraic techniques.
Solution 

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\( f(x) = x^3  9x \)
Problem Statement 

Determine the symmetry of the graph of the equation \( f(x) = x^3  9x \) using algebraic techniques.
Solution 

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\( f(x) = 4x^2  2x \)
Problem Statement 

Determine the symmetry of the graph of the equation \( f(x) = 4x^2  2x \) using algebraic techniques.
Solution 

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\( f(x) = 2x^2 + x^4 + 1 \)
Problem Statement 

Determine the symmetry of the graph of the equation \( f(x) = 2x^2 + x^4 + 1 \) using algebraic techniques.
Solution 

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\( f(x) = x^3  x \)
Problem Statement 

Determine the symmetry of the graph of the equation \( f(x) = x^3  x \) using algebraic techniques.
Solution 

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Really UNDERSTAND Precalculus
The Unit Circle
The Unit Circle [wikipedia]
Basic Trig Identities
Set 1  basic identities  

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\) 
\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\) 
\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\) 
\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\) 
Set 2  squared identities  

\( \sin^2t + \cos^2t = 1\) 
\( 1 + \tan^2t = \sec^2t\) 
\( 1 + \cot^2t = \csc^2t\) 
Set 3  doubleangle formulas  

\( \sin(2t) = 2\sin(t)\cos(t)\) 
\(\displaystyle{ \cos(2t) = \cos^2(t)  \sin^2(t) }\) 
Set 4  halfangle formulas  

\(\displaystyle{ \sin^2(t) = \frac{1\cos(2t)}{2} }\) 
\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\) 
Trig Derivatives
\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\) 
\(\displaystyle{ \frac{d[\cos(t)]}{dt} = \sin(t) }\)  
\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\) 
\(\displaystyle{ \frac{d[\cot(t)]}{dt} = \csc^2(t) }\)  
\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\) 
\(\displaystyle{ \frac{d[\csc(t)]}{dt} = \csc(t)\cot(t) }\) 
Inverse Trig Derivatives
\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\) 
\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = \frac{1}{\sqrt{1t^2}} }\)  
\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\) 
\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = \frac{1}{1+t^2} }\)  
\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 1}} }\) 
Trig Integrals
\(\int{\sin(x)~dx} = \cos(x)+C\) 
\(\int{\cos(x)~dx} = \sin(x)+C\)  
\(\int{\tan(x)~dx} = \ln\abs{\cos(x)}+C\) 
\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)  
\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\) 
\(\int{\csc(x)~dx} = \) \( \ln\abs{\csc(x)+\cot(x)}+C\) 
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