Horizontal Reflection

Flipping a Graph across the Vertical Axis

Horizontal reflection flips a function graph across the $y$ -axis. As with a vertical mirror, each point appears at the same height and the same distance on the opposite side.

If we have a function $f(x)$ , then horizontal reflection produces a new function $g(x) = f(-x)$ which is the reflection of the original function across the $y$ -axis.

Rules of Horizontal Reflection

For any function $f(x)$ , horizontal reflection is defined as:

g(x) = f(-x)

This transformation changes every point $(x, y)$ on the original graph to $(-x, y)$ on the reflected graph.

Visualization of Horizontal Reflection

For the quadratic function $f(x) = (x - 2)^2$ , compare each original point with its reflected point:

Horizontal Reflection of Quadratic Function

f(x) = (x - 2)^2

Notice how the graph reflects across the

y

-axis, forming a symmetric reflection.

The graph shows:

The original function $f(x) = (x - 2)^2$ has its vertex at $(2, 0)$
The reflected function $g(x) = (-x - 2)^2$ has its vertex at $(-2, 0)$
The two graphs are mirror images of one another across the $y$ -axis

Horizontal Reflection on Linear Functions

The same reflection rule applies to the linear function $f(x) = 2x + 3$ .

Horizontal Reflection of Linear Function

f(x) = 2x + 3

The reflected line has opposite slope but the same

y

-intercept.

The two lines show:

The original function $f(x) = 2x + 3$ has a positive slope of $2$
The reflected function $g(x) = -2x + 3$ has a slope of $-2$
Both lines intersect the $y$ -axis at the same point $(0, 3)$

Which Values the Horizontal Flip Moves

Horizontal reflection flips a graph across the $y$ -axis, so the $y$ -axis acts as the mirror line. Each point on the original graph keeps its distance to the mirror line, but moves to the other side of it.

Take a point $(a, b)$ on the graph of $f(x)$ . The matching point on the graph of $f(-x)$ is $(-a, b)$ . The $y$ value stays at $b$ , and only the sign of the $x$ coordinate flips.

Part of the function	Before the reflection	After the reflection
Point $(a, b)$	On the graph of $f(x)$	Moves to $(-a, b)$
Domain $[c, d]$	The original interval	Becomes $[-d, -c]$
Range	The original output values	The same output values as before
$y$ -intercept	At $x = 0$	Stays at $x = 0$

The domain flips to the opposite side because it lists the allowed values of $x$ , and those values change sign. The range keeps its values because the height of the graph at each point stays the same. A vertical reflection flips those heights instead.

Turning Exponential Growth into Decay

For the exponential function $f(x) = 2^x$ , reflection turns growth from left to right into decay from left to right.

Horizontal Reflection of Exponential Function

f(x) = 2^x

The reflected graph decreases but keeps the same horizontal asymptote and

y

-intercept.

For exponential functions:

The horizontal asymptote remains at $y = 0$ for both functions
The $y$ -intercept remains the same at $(0, 1)$
The function that was originally increasing becomes decreasing after reflection

Horizontal Reflection on Square Root Functions

The square root function shows how reflection also moves a restricted domain.

Horizontal Reflection of Square Root Function

f(x) = \sqrt{x}

The reflected square root curve produces a curve that opens in the opposite direction.

The graph confirms:

The domain of the original function $f(x) = \sqrt{x}$ is $[0, \infty)$
The domain of the reflected function $g(x) = \sqrt{-x}$ is $(-\infty, 0]$
Both curves meet at the origin $(0, 0)$

Exercises

The three problems below use the mirror rule you have just learned. Each one replaces the input $x$ with $-x$ , so every result appears on the opposite side of the $y$ -axis. Try them without the answer key first, then compare your steps with the worked answers.

Given the function $f(x) = x^2 + 3x + 2$ . Determine the equation of the function resulting from horizontal reflection.
The graph of the function $g(x) = 3^x + 1$ is reflected across the $y$ -axis. Determine the equation of the resulting function, then determine the domain of that function.
Function $h(x) = |x + 2|$ undergoes horizontal reflection. Determine the vertex of the resulting reflected function.

Answer Key

Every worked answer starts by replacing $x$ with $-x$ , because that is where the mirror rule lives. The equation is then simplified and the new graph is checked to confirm that it sits on the opposite side of the $y$ -axis.

Horizontal reflection: $f'(x) = f(-x) = (-x)^2 + 3(-x) + 2 = x^2 - 3x + 2$

Function $f(x) = x^2 + 3x + 2$ and Its Reflection Result
Reflection across the $y$ -axis moves the vertex to the opposite side while the parabola still opens upward.
Equation of the resulting reflected function:
- Horizontal reflection: $g'(x) = g(-x) = 3^{-x} + 1$
- Domain after reflection: Remains $(-\infty, \infty)$ because exponential functions are defined for all real numbers
Visualization:

Function $g(x) = 3^x + 1$ and Its Reflection Result
The increasing exponential curve is reflected to become a decreasing curve with the same asymptote.
The original function $h(x) = |x + 2|$ has its vertex at $(-2, 0)$ . After horizontal reflection: $h'(x) = |-x + 2| = |-(x - 2)| = |x - 2|$ , the vertex becomes $(2, 0)$ .

Function $h(x) = |x + 2|$ and Its Reflection Result
The absolute value function is reflected across the $y$ -axis producing a function with opposite vertex position.

Flipping a Graph across the Vertical Axis

Horizontal reflection flips a function graph across the $y$ -axis. As with a vertical mirror, each point appears at the same height and the same distance on the opposite side.

If we have a function $f(x)$ , then horizontal reflection produces a new function $g(x) = f(-x)$ which is the reflection of the original function across the $y$ -axis.

Rules of Horizontal Reflection

For any function $f(x)$ , horizontal reflection is defined as:

g(x) = f(-x)

This transformation changes every point $(x, y)$ on the original graph to $(-x, y)$ on the reflected graph.

Visualization of Horizontal Reflection

For the quadratic function $f(x) = (x - 2)^2$ , compare each original point with its reflected point:

Horizontal Reflection of Quadratic Function

f(x) = (x - 2)^2

Notice how the graph reflects across the

y

-axis, forming a symmetric reflection.

The graph shows:

The original function $f(x) = (x - 2)^2$ has its vertex at $(2, 0)$
The reflected function $g(x) = (-x - 2)^2$ has its vertex at $(-2, 0)$
The two graphs are mirror images of one another across the $y$ -axis

Horizontal Reflection on Linear Functions

The same reflection rule applies to the linear function $f(x) = 2x + 3$ .

Horizontal Reflection of Linear Function

f(x) = 2x + 3

The reflected line has opposite slope but the same

y

-intercept.

The two lines show:

The original function $f(x) = 2x + 3$ has a positive slope of $2$
The reflected function $g(x) = -2x + 3$ has a slope of $-2$
Both lines intersect the $y$ -axis at the same point $(0, 3)$

Which Values the Horizontal Flip Moves

Take a point $(a, b)$ on the graph of $f(x)$ . The matching point on the graph of $f(-x)$ is $(-a, b)$ . The $y$ value stays at $b$ , and only the sign of the $x$ coordinate flips.

Part of the function	Before the reflection	After the reflection
Point $(a, b)$	On the graph of $f(x)$	Moves to $(-a, b)$
Domain $[c, d]$	The original interval	Becomes $[-d, -c]$
Range	The original output values	The same output values as before
$y$ -intercept	At $x = 0$	Stays at $x = 0$

Turning Exponential Growth into Decay

For the exponential function $f(x) = 2^x$ , reflection turns growth from left to right into decay from left to right.

Horizontal Reflection of Exponential Function

f(x) = 2^x

The reflected graph decreases but keeps the same horizontal asymptote and

y

-intercept.

For exponential functions:

The horizontal asymptote remains at $y = 0$ for both functions
The $y$ -intercept remains the same at $(0, 1)$
The function that was originally increasing becomes decreasing after reflection

Horizontal Reflection on Square Root Functions

The square root function shows how reflection also moves a restricted domain.

Horizontal Reflection of Square Root Function

f(x) = \sqrt{x}

The reflected square root curve produces a curve that opens in the opposite direction.

The graph confirms:

The domain of the original function $f(x) = \sqrt{x}$ is $[0, \infty)$
The domain of the reflected function $g(x) = \sqrt{-x}$ is $(-\infty, 0]$
Both curves meet at the origin $(0, 0)$

Exercises

Given the function $f(x) = x^2 + 3x + 2$ . Determine the equation of the function resulting from horizontal reflection.
The graph of the function $g(x) = 3^x + 1$ is reflected across the $y$ -axis. Determine the equation of the resulting function, then determine the domain of that function.
Function $h(x) = |x + 2|$ undergoes horizontal reflection. Determine the vertex of the resulting reflected function.

Answer Key

Horizontal reflection: $f'(x) = f(-x) = (-x)^2 + 3(-x) + 2 = x^2 - 3x + 2$

Function $f(x) = x^2 + 3x + 2$ and Its Reflection Result
Reflection across the $y$ -axis moves the vertex to the opposite side while the parabola still opens upward.
Equation of the resulting reflected function:
- Horizontal reflection: $g'(x) = g(-x) = 3^{-x} + 1$
- Domain after reflection: Remains $(-\infty, \infty)$ because exponential functions are defined for all real numbers
Visualization:

Function $g(x) = 3^x + 1$ and Its Reflection Result
The increasing exponential curve is reflected to become a decreasing curve with the same asymptote.
The original function $h(x) = |x + 2|$ has its vertex at $(-2, 0)$ . After horizontal reflection: $h'(x) = |-x + 2| = |-(x - 2)| = |x - 2|$ , the vertex becomes $(2, 0)$ .

Function $h(x) = |x + 2|$ and Its Reflection Result
The absolute value function is reflected across the $y$ -axis producing a function with opposite vertex position.