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Horizontal translation is a geometric transformation that shifts the graph of a function left or right along the -axis without changing the shape of the graph. Imagine sliding an object horizontally on a table, its shape remains the same, only its position changes.
If we have a function , then horizontal translation produces a new function where is the translation constant.
For any function , horizontal translation is defined as:
Where:
Let's see how horizontal translation 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:
Horizontal translation preserves the original shape of the graph. The vertical distance between points on the graph remains the same, only the horizontal position changes.
If point is on the graph of , then after horizontal translation by , that point becomes on the graph of .
If the domain of the original function is , then the domain after horizontal translation becomes .
Let's look at horizontal translation on the exponential function .
For exponential functions:
It's important to understand the difference between horizontal and vertical translation:
Given the function . Determine the equation of the function resulting from horizontal translation to the right by .
If the graph of function is translated horizontally to the left by , determine:
Function undergoes horizontal translation such that point becomes . Determine the translation constant value and the equation of the resulting translated function.
Horizontal translation to the right by :
Equation of the resulting translated function:
Point on becomes , meaning horizontal translation by to the right. Equation of the translation result:
Visualization:
