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Horizontal Shift

Horizontal Shift
Horizontal Translation

A shift in which a plane figure moves horizontally.

 

 

See also

Vertical shift

Key Formula

y=f(x−h)y = f(x - h)
Where:
  • f(x)f(x) = The original function before the shift
  • hh = The number of units the graph shifts horizontally. When h > 0, the graph shifts right; when h < 0, the graph shifts left.
  • xx = The independent variable (input)

Worked Example

Problem: The function f(x) = x² has its vertex at the origin. Find the equation after shifting the graph 3 units to the right, and identify the new vertex.
Step 1: To shift a graph h units to the right, replace x with (x − h) in the function.
y=f(x−h)y = f(x - h)
Step 2: Here h = 3 (shift right by 3), so replace x with (x − 3).
y=(x−3)2y = (x - 3)^2
Step 3: The original vertex was at (0, 0). Every point moves 3 units to the right, so the new vertex is at (3, 0).
(0,0)→(3,0)(0, 0) \rightarrow (3, 0)
Answer: The shifted function is y = (x − 3)², and its vertex is at (3, 0).

Another Example

Problem: Shift the graph of y = √x four units to the left and write the new equation.
Step 1: A shift to the left by 4 means h = −4. Replace x with (x − (−4)) = (x + 4).
y=x+4y = \sqrt{x + 4}
Step 2: The original starting point was (0, 0). Moving 4 units left gives a new starting point of (−4, 0).
(0,0)→(−4,0)(0, 0) \rightarrow (-4, 0)
Answer: The shifted function is y = √(x + 4), starting at (−4, 0).

Frequently Asked Questions

Why does (x − h) shift the graph to the right instead of the left?
It seems backward, but think about what input produces the same output. For y = (x − 3)², you need x = 3 to get the same value that x = 0 gave in the original y = x². Every key output now happens 3 units further to the right, so the entire graph moves right. The subtraction inside the function works opposite to what you might expect.
How do you tell a horizontal shift from a vertical shift?
A horizontal shift changes the input: y = f(x − h). A vertical shift changes the output: y = f(x) + k. The horizontal shift moves the graph left or right, while the vertical shift moves it up or down. Both preserve the shape of the graph.

Horizontal Shift vs. Vertical Shift

A horizontal shift replaces x with (x − h) inside the function, moving the graph left or right. A vertical shift adds a constant k outside the function, written y = f(x) + k, moving the graph up or down. Horizontal shifts affect the input; vertical shifts affect the output. Both are rigid transformations that preserve the graph's shape and size.

Why It Matters

Horizontal shifts let you reposition graphs without re-deriving entire equations. They appear constantly when modeling real-world timing — for instance, if a process starts 5 seconds late, you shift the model by replacing t with (t − 5). Understanding this transformation is also essential for graphing by hand and for combining multiple transformations (shifts, stretches, and reflections) in algebra and precalculus.

Common Mistakes

Mistake: Thinking y = f(x − 3) shifts the graph 3 units to the left.
Correction: Subtracting inside the function shifts the graph in the positive (right) direction. To shift left by 3, you write y = f(x + 3). The sign inside the parentheses is opposite to the direction of the shift.
Mistake: Confusing a horizontal shift with a horizontal stretch.
Correction: A horizontal shift adds or subtracts a constant from x, like f(x − 2). A horizontal stretch multiplies x by a constant, like f(2x). The shift slides the graph; the stretch compresses or widens it.

Related Terms