X-Axis Reflections

Simple overview of $x$-axis reflections.

The Main Concept

$x$-axis reflections simply boil down to

\[(x,y)\mapsto (x,-y)\]

To flip something over the $x$-axis, you keep the same value of $x$ but change the sign of $y$ (or whatever represents your other variable).

Watch this section: x-axis reflection rule at 0:00.

Reflecting A Point

Consider a point

\[(a,b)\]

After reflecting over the x-axis, the new point is

\[(a,-b)\]

Only the second value in the coordinate changes. A point like $(3,5)$ becomes $(3,-5)$, and a point like $(-2,-7)$ becomes $(-2,7)$.

Watch this section: reflecting a point at 0:15.

Reflecting A Triangle

Consider a triangle with the vertices

\[A(x_1,y_1),\quad B(x_2,y_2),\quad C(x_3,y_3),\]

then the reflected vertices are

\[A'(x_1,-y_1),\quad B'(x_2,-y_2),\quad C'(x_3,-y_3).\]

The triangle is now flipped across the x-axis, but its side lengths and shape stay the same.

Watch this section: reflecting a triangle at 0:40.

Reflecting A Function

The pattern continues for functions, if a point $(x,y)$ is on the original graph, then $(x,-y)$ is on the reflected graph. Since $y=f(x)$, the reflected function is

\[g(x)=-f(x).\]

That negative sign outside the function flips every output.

Consider the function

\[f(x)=x^2\]

then its reflection over the x-axis is

\[g(x)=-f(x)=-x^2.\]

Watch this section: reflecting a function at 1:24.

Timestamp Guide

Section	What is shown	Video
Reflection rule	Use $(x,y)\mapsto(x,-y)$ for an x-axis reflection.	0:00
Point reflection	Change the sign of the y-coordinate for one point.	0:15
Triangle reflection	Apply the point rule to each vertex.	0:40
Function reflection	Write the reflected function as $g(x)=-f(x)$.	1:24
Summary	Review the rule across the examples.	2:47
Support	Subscribe or join to support the channel.	3:05

Related Algebra Work

For another place where the x-axis matters, review the polynomial sign chart notes.

You can also browse the Algebra topic page for more JoeCMath algebra videos.