On this page, we cover symmetry of graphs in rectangular form. This other page covers symmetry of graphs in polar form.
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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.
y-axis Symmetry [Even Function]
There are several ways to describe even symmetry, listed below.
even symmetry |
symmetric about the y-axis |
even function |
Here are a couple of examples of even functions.
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.
1. | Find \(f(-x)\) |
2. | If \(f(-x) = f(x)\), then the function is even. |
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.
x-axis Symmetry
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.
Origin Symmetry [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 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.
odd symmetry |
symmetric about the origin |
odd function |
Here are a couple of examples.
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.
video by MIP4U |
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Okay, time for the 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 |
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Determine the symmetry of the graph of the equation \( f(x) = x^2 + 1 \) using algebraic techniques.
Final Answer |
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\(f(x)\) 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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\(y=x^3-2x\)
Problem Statement
Unless otherwise instructed, determine the symmetry of the graph of the equation \(y=x^3-2x\) 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
video by MIP4U |
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\(y^2-2y^4=6x\)
Problem Statement
Unless otherwise instructed, determine the symmetry of the graph of the equation \(y^2-2y^4=6x\) using algebraic techniques.
Solution
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
video by Freshmen Math Doctor |
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\( y = x(x-2)(x+3) \)
Problem Statement
Determine the symmetry of the graph of the equation \( y = x(x-2)(x+3) \) using algebraic techniques.
Solution
video by Freshmen Math Doctor |
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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
video by Freshmen Math Doctor |
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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
video by Freshmen Math Doctor |
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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
video by Freshmen Math Doctor |
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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
video by Freshmen Math Doctor |
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Topics You Need To Understand For This Page |
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[functions] - [vertical line test] |
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