Symmetry - Algebra - Pauls Online Math Notes

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PaulFebruary 18, 2026

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Section 4.7 : Symmetry

In this section we are going to take a look at something that we used back when we were graphing parabolas. However, we’re going to take a more general view of it this section. Many graphs have symmetry to them.

Symmetry can be useful in graphing an equation since it says that if we know one portion of the graph then we will also know the remaining (and symmetric) portion of the graph as well. We used this fact when we were graphing parabolas to get an extra point of some of the graphs.

In this section we want to look at three types of symmetry.

  1. A graph is said to be symmetric about the \(x\)-axis if whenever \(\left( {a,b} \right)\) is on the graph then so is \(\left( {a, - b} \right)\). Here is a sketch of a graph that is symmetric about the \(x\)-axis. There are no tick marks on the x or y-axis in this graph and the 1st and 4th quadrants only are shown. The function graphed looks to be a parabola with a vertex at the origin and it opens off to the right into the 1st and 4th quadrants. In the portion of the graph in the 1st quadrant a point is marked on the graph with coordinates (a,b). The portion of the graph in the 4th quadrant has a point marked in it with coordinates (a,-b). This is to illustrate that the portion of the graph in the 4th quadrant is basically a reflection of the portion in the 1st quadrant and for any x value, say x=a, there will be one point above it at y=b and a point below it at y=-b.
  2. A graph is said to be symmetric about the \(y\)-axis if whenever \(\left( {a,b} \right)\) is on the graph then so is \(\left( { - a,b} \right)\). Here is a sketch of a graph that is symmetric about the \(y\)-axis. There are no tick marks on the x or y-axis in this graph and the 1st and 2nd quadrants only are shown. The function graphed looks to be a parabola with a vertex at the origin and it opens off up into the 1st and 2nd quadrants. In the portion of the graph in the 1st quadrant a point is marked on the graph with coordinates (a,b). The portion of the graph in the 2nd quadrant has a point marked in it with coordinates (-a,b). This is to illustrate that the portion of the graph in the 2nd quadrant is basically a reflection of the portion in the 1st quadrant and for any y value, say y=b, there will be one point to the right at x=a and another point to the left at x=-a.
  3. A graph is said to be symmetric about the origin if whenever \(\left( {a,b} \right)\) is on the graph then so is \(\left( { - a, - b} \right)\). Here is a sketch of a graph that is symmetric about the origin. In the 1st quadrant there is a graph that contains the point (a,b) and as we increase x away from this point the graph slopes down towards the x-axis and flattens out as it gets closer to the x-axis but never crosses the x-axis. As we decrease x from this point towards the y-axis the graph increases rapidly getting closer and closer to the y-axis becoming almost vertical near the y-axis but it never crosses the y-axis. In the 3rd quadrant there is a graph that contains the point (-a,-b) and as we increase x away from this point in the negative direction the graph slopes up towards the x-axis and flattens out as it gets closer to the x-axis but never crosses the x-axis. As we decrease x from this point towards the y-axis the graph decreases rapidly getting closer and closer to the y-axis becoming almost vertical near the y-axis but it never crosses the y-axis. The point of this graph is to show that for any point in the 1st quadrant there will be a point in the 3rd quadrant with the same coordinates except opposite signs.

Note that most graphs don’t have any kind of symmetry. Also, it is possible for a graph to have more than one kind of symmetry. For example, the graph of a circle centered at the origin exhibits all three symmetries.

Tests for Symmetry

We’ve some fairly simply tests for each of the different types of symmetry.

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