# Symmetry and intercepts (§2.2)

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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