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We explain three types of graph symmetry on this page, even (y-axis), odd (origin) and x-axis symmetry. You will also come across symmetry described as symmetry about the x-axis, y-axis 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.

There are several ways to describe even symmetry, listed below.

Here are a couple of examples of even functions.

\(y=x^2\)

Figure 1 [built with GeoGebra]

\(y=\cos(x)\)

Figure 2 [built with GeoGebra]

Looking at the graphs, notice that in each case, the graph on the left side of the y-axis 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.

Let's do an example. For the graph in Figure 1, we have \(f(x)=x^2\).
\(f(-x) = (-x)^2 = x^2 = f(x)\)
Since \(f(x)=f(-x)\), the function is even.

Here is a graph showing x-axis symmetry. Similar to an even function that is symmetric about the y-axis, this graph has a reflection across the x-axis. Notice that it is not a function since it does not pass the vertical line test. This is true of all graphs with x-axis symmetry. One way to think about x-axis symmetry is to notice if we have a point \((a,b)\) on the graph, the point \((a,-b)\) is also on the graph.

\(y^2=x\)

Figure 3 [built with GeoGebra]

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 y-axis and then about the x-axis (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.

Here are a couple of examples.

\(y=x^3\)

Figure 4 [built with GeoGebra]

\(y=\sin(x)\)

Figure 5 [built with GeoGebra]

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.

Before jumping into some practice problems, let's watch this short video to make sure you are clear about the three kinds of symmetry.

Practice

Unless otherwise instructed, determine the symmetry of these graphs of these equations using algebraic techniques.

• Practice 2530
• Answer
• Solution

\( f(x) = x^2 + 1 \)

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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• Practice 1497
• Solution

\(y=-x^4+4x^2+5\)

Problem Statement

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

Solution

MIP4U - 1497 video solution

video by MIP4U

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• Practice 1498
• Solution

\(y=x^3-2x\)

Problem Statement

Unless otherwise instructed, determine the symmetry of the graph of the equation \(y=x^3-2x\) using algebraic techniques.

Solution

MIP4U - 1498 video solution

video by MIP4U

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• Practice 1499
• Solution

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

Problem Statement

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

Solution

MIP4U - 1499 video solution

video by MIP4U

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• Practice 1500
• Solution

\(y^2-2y^4=6x\)

Problem Statement

Unless otherwise instructed, determine the symmetry of the graph of the equation \(y^2-2y^4=6x\) using algebraic techniques.

Solution

MIP4U - 1500 video solution

video by MIP4U

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• Practice 2551
• Solution

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

Problem Statement

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

Solution

Freshmen Math Doctor - 2551 video solution

video by Freshmen Math Doctor

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• Practice 2552
• Solution

\( y = x(x-2)(x+3) \)

Problem Statement

Determine the symmetry of the graph of the equation \( y = x(x-2)(x+3) \) using algebraic techniques.

Solution

Freshmen Math Doctor - 2552 video solution

video by Freshmen Math Doctor

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• Practice 2553
• Solution

\( f(x) = x^3 - 9x \)

Problem Statement

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

Solution

Freshmen Math Doctor - 2553 video solution

video by Freshmen Math Doctor

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• Practice 2554
• Solution

\( f(x) = 4x^2 - 2x \)

Problem Statement

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

Solution

Freshmen Math Doctor - 2554 video solution

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• Practice 2555
• Solution

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

Problem Statement

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

Solution

Freshmen Math Doctor - 2555 video solution

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• Practice 2556
• Solution

\( f(x) = x^3 - x \)

Problem Statement

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

Solution

Freshmen Math Doctor - 2556 video solution

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