Vertical Reflection

Basic Concepts of Vertical Reflection

Vertical reflection is a geometric transformation that reflects the graph of a function across the x-axis, like seeing the reflection of an object on the surface of calm water. Imagine an object reflected in a horizontal mirror, its shape remains the same but its position is flipped vertically.

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

Rules of Vertical Reflection

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

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

Visualization of Vertical Reflection

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

From the visualization above, we can observe:

• The original function $f(x) = x^2$ opens upward with vertex at $(0, 0)$
• The reflected function $g(x) = -x^2$ opens downward with vertex still at $(0, 0)$
• Both graphs are symmetric across the x-axis

Vertical 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 positive slope and intersects the y-axis at $(0, 3)$
• The reflected function $g(x) = -2x - 3$ has negative slope and intersects the y-axis at $(0, -3)$
• Both lines intersect at the x-axis

Important Properties of Vertical Reflection

X-axis as Axis of Symmetry

Vertical reflection uses the x-axis as the mirror line. Every point on the original graph has the same distance to the x-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: Does not change after vertical reflection
• Range: Changes to the opposite of the original range

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

Application Examples

Exponential Function Example

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

For exponential functions:

• The horizontal asymptote remains at $y = 0$ for both functions (since the x-axis reflects onto itself)
• The y-intercept changes from $(0, 1)$ to $(0, -1)$
• The function that was originally increasing becomes decreasing

Vertical Reflection on Trigonometric Functions

Let's see how vertical reflection affects the sine function.

Notice that:

• The amplitude remains the same but the wave direction is inverted
• The period and frequency do not change
• Maximum points become minimum points and vice versa

Exercises

1. Given the function $f(x) = x^2 + 4x + 3$. Determine the equation of the function resulting from vertical reflection.

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

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

3. Function $h(x) = \sqrt{x + 1}$ undergoes vertical reflection. Determine the y-intercept of the resulting reflected function.

Answer Key

1. Vertical reflection: $f'(x) = -f(x) = -(x^2 + 4x + 3) = -x^2 - 4x - 3$

   Function $f(x) = x^2 + 4x + 3$ and Its Reflection Result

   The original parabola opening upward is reflected to become a parabola opening downward.

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2. Equation of the resulting reflected function:

   • Vertical reflection: $g'(x) = -(3^x + 2) = -3^x - 2$
   • Range after reflection: Since the original range of $g(x) = 3^x + 2$ is $(2, \infty)$, the range after reflection is $(-\infty, -2)$

   Visualization:

   Function $g(x) = 3^x + 2$ and Its Reflection Result

   The increasing exponential curve is reflected to become a decreasing curve with a new horizontal asymptote.

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3. The original function $h(x) = \sqrt{x + 1}$ has a y-intercept at $(0, 1)$ because $h(0) = \sqrt{0 + 1} = 1$. After vertical reflection: $h'(x) = -\sqrt{x + 1}$, the y-intercept becomes $(0, -1)$.

   Function $h(x) = \sqrt{x + 1}$ and Its Reflection Result

   The square root curve is reflected across the x-axis resulting in a curve opening downward.

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