# 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.

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This web page was written by Toby Bartels, last edited on 2021 August 26.
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