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Vertical Reflection

Vertical Reflection

A reflection in which a plane figure flips over vertically. Note: A vertical reflection has a horizontal axis of reflection.

 

 

See also

Horizontal reflection

Key Formula

If the axis is the x-axis: (x,y)→(x,−y)\text{If the axis is the } x\text{-axis: } (x, y) \to (x, -y)
Where:
  • xx = The horizontal coordinate of the original point (unchanged)
  • yy = The vertical coordinate of the original point (negated to produce the reflected point)

Worked Example

Problem: Triangle ABC has vertices A(1, 2), B(4, 2), and B(2, 5). Reflect the triangle vertically over the x-axis.
Step 1: Apply the vertical reflection rule. When the axis of reflection is the x-axis, every point (x, y) maps to (x, −y). The x-coordinate stays the same; the y-coordinate changes sign.
(x,y)→(x,−y)(x, y) \to (x, -y)
Step 2: Reflect vertex A(1, 2).
A(1,2)→A′(1,−2)A(1, 2) \to A'(1, -2)
Step 3: Reflect vertex B(4, 2).
B(4,2)→B′(4,−2)B(4, 2) \to B'(4, -2)
Step 4: Reflect vertex C(2, 5).
C(2,5)→C′(2,−5)C(2, 5) \to C'(2, -5)
Answer: The reflected triangle A'B'C' has vertices A'(1, −2), B'(4, −2), and C'(2, −5). The triangle has flipped upside-down across the x-axis.

Another Example

Problem: The graph of f(x) = x² is reflected vertically over the x-axis. What is the equation of the resulting graph?
Step 1: A vertical reflection over the x-axis replaces every y-value with its opposite. This means y = f(x) becomes y = −f(x).
y=f(x)→y=−f(x)y = f(x) \to y = -f(x)
Step 2: Substitute f(x) = x².
y=−x2y = -x^2
Step 3: The original parabola opens upward. After the vertical reflection, it opens downward, with the vertex still at the origin.
Answer: The reflected graph has equation y = −x². The parabola is flipped upside-down.

Frequently Asked Questions

Why does a vertical reflection use a horizontal axis?
The word 'vertical' describes the direction each point moves — straight up or straight down. For points to move vertically, the mirror line (axis) must run horizontally. Think of it like a lake: the water surface is a horizontal line, and your reflection appears directly below you — a vertical flip.
How do you reflect a function vertically?
To reflect the graph of y = f(x) vertically over the x-axis, replace y with −y, giving the new equation y = −f(x). Every output value is negated, which flips the entire graph upside-down while keeping the x-intercepts unchanged.

Vertical Reflection vs. Horizontal Reflection

A vertical reflection flips a figure up/down across a horizontal axis; the standard coordinate rule is (x, y) → (x, −y). A horizontal reflection flips a figure left/right across a vertical axis; the standard rule is (x, y) → (−x, y). The naming refers to the direction the points move, not the orientation of the axis. For functions, a vertical reflection gives y = −f(x), while a horizontal reflection gives y = f(−x).

Why It Matters

Vertical reflections appear throughout algebra and calculus whenever you negate a function's output. Recognizing that y = −f(x) is just a vertical reflection of y = f(x) lets you sketch transformed graphs quickly without plotting many points. In physics and engineering, vertical reflections model symmetry about a ground plane or baseline, such as an object and its image in still water.

Common Mistakes

Mistake: Confusing which coordinate changes: negating x instead of y when reflecting over the x-axis.
Correction: In a vertical reflection over the x-axis, the y-coordinate is negated because points move vertically. The rule is (x, y) → (x, −y), not (−x, y).
Mistake: Thinking the axis of reflection in a vertical reflection is vertical.
Correction: The axis is horizontal. 'Vertical' refers to the direction of the flip, not the orientation of the mirror line. A horizontal mirror line causes points to flip vertically.

Related Terms