Horizontal Reflection

Basic Concepts of Horizontal Reflection

Horizontal reflection is a geometric transformation that reflects the graph of a function across the y-axis, like seeing the reflection of an object in a vertical mirror. Imagine standing in front of a mirror, your right hand will appear as your left hand in the mirror, similarly with function graphs that are reflected horizontally.

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:

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

Visualization of Horizontal Reflection

Let's see how horizontal reflection works on the quadratic function $f(x) = (x - 2)^2$.

From the visualization above, we can observe:

• The original function $f(x) = (x - 2)^2$ (purple) has its vertex at $(2, 0)$
• The reflected function $g(x) = (-x - 2)^2$ (orange) has its vertex at $(-2, 0)$
• Both graphs are symmetric across the y-axis

Horizontal Reflection on Linear Functions

Now let's apply the same concept to the linear function $f(x) = 2x + 3$.

Notice that:

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

Important Properties of Horizontal Reflection

Y-axis as Axis of Symmetry

Horizontal reflection uses the y-axis as the mirror line. Every point on the original graph has the same distance to the y-axis as the corresponding point on the reflected graph.

Effect on Coordinate Points

If point $(a, b)$ is on the graph of $f(x)$, then the corresponding point on the graph of $f(-x)$ is $(-a, b)$.

Domain and Range

• Domain: Changes to the opposite of the original domain
• Range: Does not change after horizontal reflection

If the domain of the original function is $[c, d]$, then the domain after horizontal reflection becomes $[-d, -c]$.

Application Examples

Exponential Function Example

Let's look at horizontal reflection on the exponential function $f(x) = 2^x$.

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

Let's see how horizontal reflection affects the square root function.

Notice that:

• 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

1. Given the function $f(x) = x^2 + 3x + 2$. Determine the equation of the function resulting from horizontal reflection.

2. If the graph of function $g(x) = 3^x + 1$ is reflected across the y-axis, determine:

   • The equation of the resulting reflected function
   • The domain of the function after reflection

3. Function $h(x) = |x + 2|$ undergoes horizontal reflection. Determine the vertex of the resulting reflected function.

Answer Key

1. 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

   The original parabola is reflected across the y-axis producing a parabola with different orientation.

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2. 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.

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3. 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.

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