Curves in geometry are defined as lines or surfaces that are not straight or flat. There are many types of curves, from circular arcs to conic sections like parabolas and hyperbolas.
When graphing a curve, such as those in the parabola graph example, you must determine if the graph is symmetric about the x-axis, y-axis, or origin. This is an important step in graphing a curve because it determines where the intercepts will be.
When looking at a paraboloid graph, you would determine that it is symmetric about the y-axis. The value of y varies but x remains the same, so switching the x and y values would not change the output value. This makes it easy to identify where the y-intercepts are!
There are cases where a curve is not symmetric about any axis or origin, like a circle. These graphs must be approached differently.
Calculate the rotation about the axis of symmetry
Now that you know the graph is symmetric, you can determine the rotation about the axis of symmetry.
If the graph is symmetric about both the x-axis and y-axis, then the rotation is 2Cos(5θ) where θ is the angle of rotation. If the graph is only symmetric about the x-axis, then rotate 2Cos(5θ) about the x-axis. If it is only symmetric about the y-axis, then rotate 2Cos(5θ) about the y-axis.
Symmetry About Two Axes: Two possible cases exist for symmetry about two axes: one with a rotation around one axis, and one with no rotation. In both of these cases, there is only one axis of symmetry.
Compare the transformed graph to the original graph
Now that you have the transformed graph, you can determine if the graph is symmetric about the x-axis, y-axis, or origin.
If the graph is symmetric about the x-axis, then the median line of the graph will be a line where values on either side of it are equal. In other words, if you divided the graph into two halves using a line through the middle point of the x-axis, then the values on either side would be equal.
If the graph is symmetric about the y-axis, then the median line of the graph will be a vertical line. In other words, if you divided the graph into two halves using a line through the middle point of the y-axis, then values on either side would be equal.
If symmetry occurs around both axes, then there are two possibilities: one side could be reversed or mirrored. Which one occurs depends on which axis is larger in magnitude.
See if there is an angle that makes the graph symmetric
If the graph does not have an angle that makes it symmetric, then you can check whether it is symmetric about the x-axis, y-axis, or origin.
To check symmetry about the x-axis, determine whether the function is positive for values greater than zero and negative for values less than zero. If this is not the case, then the graph is not symmetric about the x-axis.
To check symmetry about the y-axis, determine whether the function is positive for values greater than zero and negative for values equal to zero. If this is not the case, then the graph is not symmetric about the y-axis.
To check symmetry about the origin, determine whether the function is positive for values less than zero and negative for values equal to zero. If this is not the case, then there is no symmetry about either axis.
Try different axes of symmetry to find which gives you a symmetric graph
The next step is to determine which axis of symmetry the graph has. Again, this is done by finding points where the curve crosses the axis and seeing whether the values are equal.
For example, if the curve crosses the y-axis at -1 and +1, then it is symmetric about the y-axis. If it crossed -1 and +1 at some other point on the curve, then it would not be symmetric about the y-axis.
Once you have determined that the graph is symmetric about an axis, you can determine whether it is symmetric about another axis by doing the same thing. Check if the graph crosses that axis at points where values are equal, and if it does not, then it is not symmetric about that axis.
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