Symmetry and intercepts (§1.2)
The pattern here is that you always mess with the other variable.
Symmetry
You can test symmetry using either the equation or the graph.
- Symmetry with respect to the x-axis:
- change y to −y in the equation,
and see if this is equivalent to the original equation;
- change each point (x, y) on the graph
to (x, −y),
and see if this is also a point on the graph.
- Symmetry with respect to the y-axis:
- change x to −x in the equation,
and see if this is equivalent to the original equation;
- change each point (x, y) on the graph
to (−x, y),
and see if this is also a point on the graph.
- Symmetry with respect to the origin:
- change both x to −x and y to −y
in the equation,
and see if this is equivalent to the original equation;
- change each point (x, y) on the graph
to (−x, −y),
and see if this is also a point on the graph.
Each kind of symmetry is a separate Yes/No question.
The possible answers are:
- No, No, No: None of these three symmetries apply;
- Yes, No, No: Symmetric with respect to the x-axis only;
- No, Yes, No: Symmetric with respect to the y-axis only;
- No, No, Yes: Symmetric with respect to the origin only;
- Yes, Yes, Yes: All three Symmetries apply.
However, it is not possible to have two Yes and one No,
because each of these symmetries is a combination of the other two.
Intercepts
An intercept is a point on a graph that is also on one of the axes.
- x-intercepts:
- in the equation, set y to 0 and solve for x;
- any point (x, 0) on the graph is an x-intercept.
- y-intercepts:
- in the equation, set x to 0 and solve for y;
- any point (0, y) on the graph
is a y-intercept.
Every intercept is a point with two coordinates.
If the origin is on the graph,
then it is both an x-intercept and a y-intercept.
Go back to the course homepage.
This web page was written by Toby Bartels, last edited on 2023 September 18.
Toby reserves no legal rights to it.
The permanent URI of this web page
is
http://tobybartels.name/MATH-1300/2023FA/symmetry/.
