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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