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Vertical reflection is a geometric transformation that reflects the graph of a function across the -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 , then vertical reflection produces a new function which is the reflection of the original function across the -axis.
For any function , vertical reflection is defined as:
This transformation changes every point on the original graph to on the reflected graph.
Let's see how vertical reflection works on the quadratic function .
From the visualization above, we can observe:
Now let's apply the same concept to the linear function .
Notice that:
Vertical reflection uses the -axis as the mirror line. Every point on the original graph has the same distance to the -axis as the corresponding point on the reflected graph.
If point is on the graph of , then the corresponding point on the graph of is .
If the range of the original function is , then the range after vertical reflection becomes .
Let's look at vertical reflection on the exponential function .
For exponential functions:
Let's see how vertical reflection affects the function.
Notice that:
Given the function . Determine the equation of the function resulting from vertical reflection.
If the graph of function is reflected across the -axis, determine:
Function undergoes vertical reflection. Determine the -intercept of the resulting reflected function.
Vertical reflection:
Equation of the resulting reflected function:
The original function has a -intercept at because . After vertical reflection: , the -intercept becomes .
Visualization:
