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Vertical translation is a geometric transformation that shifts the graph of a function up or down along the -axis without changing the shape of the graph. Imagine lifting or lowering an object vertically: the shape stays the same, only its position changes.
If we have a function , then vertical translation produces a new function where is the translation constant.
For any function , vertical translation is defined as:
Where:
Let's see how vertical translation works on the linear function .
From the visualization above, we can observe:
Now let's apply the same concept to the quadratic function .
Notice that:
Vertical translation preserves the original shape of the graph. The distance between points on the graph remains the same, only the vertical position changes.
If point is on the graph of , then after vertical translation by , that point becomes on the graph of .
If the range of the original function is , then the range after vertical translation becomes .
Let's look at vertical translation on the exponential function .
For exponential functions:
Given the function . Determine the equation of the function resulting from vertical translation upward by .
If the graph of function is translated vertically downward by , determine:
Function undergoes vertical translation such that point becomes . Determine the translation constant value and the equation of the resulting translated function.
Vertical translation upward by :
Equation of the resulting translated function:
Point on becomes , meaning vertical translation by upward. Equation of the translation result:
Visualization:
