Transformations and symmetry

This chapter only applies to AMC 10/12 test takers.

Transforming graphs is a useful skill on many AMC geometry problems. Here, you’ll work with three common ideas:

• symmetry (even and odd functions)
• reflections
• rotations

Symmetry

A graph can have symmetry in a few different ways. For functions, the two most common are:

• y-axis symmetry, which corresponds to an even function
• origin symmetry, which corresponds to an odd function

A graph is symmetric about the origin if it looks the same after a $180^{\circ }$ rotation around the origin.

You won’t see a nontrivial function that’s symmetric across the x-axis, because x-axis symmetry would require both $(x,y)$ and $(x,-y)$ to be on the graph. That gives two different outputs for the same input $x$, which violates the definition of a function.

Here are the definitions you’ll use.

Even functions

$$f(-x)=f(x)$$

For an even function, you can think of the y-axis as a mirror: whatever appears on the left side appears identically on the right.

For example, the graph of $f(x)=x^4$ is even. If you plug in inputs with the same magnitude but opposite signs (like $2$ and $-2$), you get the same output. That’s exactly what $f(-x)=f(x)$ means.

Odd functions

$$f(-x)=-f(x)$$

For an odd function, points come in pairs that are opposite across the origin: if $(x,y)$ is on the graph, then $(-x,-y)$ is also on the graph. That’s why odd symmetry matches a $180^{\circ }$ rotation about the origin.

All odd, continuous functions must pass through the origin. This isn’t the only requirement for odd symmetry, but it’s necessary: if the graph is unchanged by a $180^{\circ }$ rotation about the origin, then the origin must be on the graph.

For example, $f(x)=x^3$ is odd.

Rotation

Picture graph paper on a table. If you place a pin at a point and spin the paper, you rotate the graph around that point. On the AMC, rotations are usually about the origin, so the “pin” is at $(0,0)$.

To rotate a point, it helps to memorize what happens to coordinates.

• A $90^{\circ }$ clockwise rotation about the origin sends $(a,b)$ to $(b,-a)$.

  • You swap the coordinates, then multiply the new y-coordinate by $-1$.

• A $90^{\circ }$ counterclockwise rotation about the origin sends $(a,b)$ to $(-b,a)$.

  • You swap the coordinates, then multiply the new x-coordinate by $-1$.

Table showing the resulting coordinates after rotation

Rotation	Before rotation	After rotation
90° clockwise	$(x,y)$	$(y,-x)$
90° counterclockwise	$(x,y)$	$(-y,x)$
180°	$(x,y)$	$(-x,-y)$

Try this challenging AMC problem that uses both rotation and reflection.

• Any total rotation that’s a multiple of 360° ends back at the original position.
• Reflecting across both the $y$-axis and the $x$-axis (in either order) is equivalent to a 180° rotation.

Example: The question below is from 2020 AMC 10A

Let$T$ be the triangle in the coordinate plane with vertices $(0,0),(4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ },180^{\circ },$ and $270^{\circ }$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ }$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ }$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)
A. $12$
B. $15$
C. $17$
D. $20$
E. $25$

Common themes

• Do not be afraid to draw out diagrams and plot a few points to help visualize a graph.
• Functions do not have x-axis symmetry because a function must have exactly one y-value for every given x-value.
• Utilize symmetry to do half the work. If you need the area of a shape that has y-axis symmetry, just find the area of the right half and double it.
• Try practicing quickly visualizing what happens to graphs when you rotate or reflect them.
• Recall from the inverse functions section of the functions chapter that the inverse of a function is reflective of the original function across the line $y=x$.