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17Calculus Precalculus - Graph Symmetry in Rectangular Coordinates

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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 (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.

\(y=x^2\)

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.

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.

\(y^2=x\)

Figure 3 [built with GeoGebra]

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.

\(y=x^3\)

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.

MIP4U - 3 kinds of symmetry [6min-45secs]

video by MIP4U

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

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

MIP4U - 1497 video 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

MIP4U - 1498 video 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

MIP4U - 1499 video 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

MIP4U - 1500 video 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

Freshmen Math Doctor - 2551 video solution

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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

Freshmen Math Doctor - 2552 video 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

Freshmen Math Doctor - 2553 video 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

Freshmen Math Doctor - 2554 video 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

Freshmen Math Doctor - 2555 video 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

Freshmen Math Doctor - 2556 video solution

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Really UNDERSTAND Precalculus

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Topics You Need To Understand For This Page

[functions] - [vertical line test]

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Practice Instructions

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

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