How to test for symmetry for the x-axis
To test for symmetry about the x-axis, we need to check if a graph is unchanged when reflected across the x-axis
To test for symmetry about the x-axis, we need to check if a graph is unchanged when reflected across the x-axis.
The symmetry test can be done by checking if the function remains the same when we change the sign of the y-coordinate. Here are the steps to test for symmetry about the x-axis:
1. Start with the given equation or graph that represents the function.
2. Replace every instance of y with -y in the equation or graph.
3. Simplify the equation or redraw the graph using the new y-values.
4. Compare the original equation or graph with the new equation or graph after the sign change.
If the original equation and the equation obtained after changing the sign of y remain the same, then the function is symmetric about the x-axis. If they are different, then the function is not symmetric about the x-axis.
Let’s take an example to understand this better.
Example: Test the symmetry of the function f(x) = x^2 about the x-axis.
1. Start with the equation f(x) = x^2.
2. Replace y with -y in the equation: -y = x^2.
3. Simplify the equation: y = -x^2.
4. Compare the original equation f(x) = x^2 with the new equation y = -x^2.
The original equation f(x) = x^2 represents a U-shaped graph, opening upwards. The new equation y = -x^2 represents an inverted U-shaped graph, opening downwards. As the two graphs are different, we can say that the function f(x) = x^2 is not symmetric about the x-axis.
Remember, a function is symmetric about the x-axis if replacing y with -y in the equation or graph gives the same equation or graph.
More Answers:
Understanding Graph Symmetry with Respect to the X-Axis: Explained with Examples and EquationsUnderstanding Symmetry with Respect to the Y-Axis in Mathematics: Definition, Characteristics, and Testing
Understanding Symmetry: Graph and Equation Symmetry with the Origin