Horizontal Translations of Linear Functions
A horizontal translation slides a graph left or right. The signal is a change inside the parentheses: shifts the graph of horizontally by units. The shape and the slope stay exactly the same — the line just picks up and moves sideways.
The direction is the part everyone trips on, because it runs opposite to the sign you see. moves the graph units right, and moves it units left. This article explains why, and how to write the shifted equation.
Why the direction is reversed
Think about what input needs to reproduce an old output. To get the value used to give at , you now need , which means . Every output happens units later than it used to — so the whole graph slides right.
The same logic flips for : the output that used to happen at now happens at , so the graph slides left. Inside the parentheses, subtracting moves right and adding moves left.
Simplifying the shifted equation
To write for a specific linear function, replace every in the formula with and simplify. For , the shift becomes .
Notice the result: the slope is untouched, so the lines are parallel, and only the constant term changed. For lines, a horizontal shift always simplifies to something that looks like a vertical shift — the graph above shows in blue and in red, which you could equally describe as units right or units down.
Worked examples
Example 1: shift right and simplify
Let . Find and simplify.
Answer:
Example 2: shift left and simplify
Let . Find and simplify.
Answer:
Example 3: name the direction
The function is transformed to . Which way does the graph move?
Answer: units left
Try one yourself
Common questions
Why does move right instead of left?
Because the input has to be bigger to make equal what used to be. Every output arrives units later on the -axis, which slides the whole graph right.
Does a horizontal translation change the slope?
No. Sliding a line sideways doesn't tilt it, so the new line is parallel to the original. Only the constant term of the equation changes after you simplify.
Why does my shifted line look like it moved down instead of right?
For lines, both descriptions are true at once. Shifting right gives , which is also shifted down . The two moves land the line in the same place — that only happens because a line's shape never changes.
How do I handle when ?
Substitute everywhere appears: . Replace, then simplify — that's the entire procedure.
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