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Shift — Definition, Graph & Examples

Shift
Translation
Glide

A transformation in which a graph or geometric figure is picked up and moved to another location without any change in size or orientation.

Two parabolas: original centered at origin, and same parabola shifted right and down, illustrating a translation shift.

 

Two irregular quadrilaterals showing a translation: "pre-image (original figure)" on left, "image" on right, same size and...

 

See also

Pre-image, image, horizontal shift, vertical shift

Key Formula

Point: (x,y)⟶(x+h,  y+k)\text{Point: } (x, y) \longrightarrow (x + h,\; y + k)
Where:
  • x,yx, y = The coordinates of the original point (pre-image)
  • hh = The horizontal shift — positive moves right, negative moves left
  • kk = The vertical shift — positive moves up, negative moves down
  • (x+h,y+k)(x + h, y + k) = The coordinates of the new point (image) after the shift

Worked Example

Problem: A triangle has vertices at A(1, 2), B(4, 2), and C(4, 6). Shift the triangle 3 units to the left and 5 units up. Find the new vertices.
Step 1: Identify the shift values. Moving 3 units left means h = −3. Moving 5 units up means k = 5.
h=−3,k=5h = -3, \quad k = 5
Step 2: Apply the shift rule to vertex A(1, 2).
A′=(1+(−3),  2+5)=(−2,  7)A' = (1 + (-3),\; 2 + 5) = (-2,\; 7)
Step 3: Apply the shift rule to vertex B(4, 2).
B′=(4+(−3),  2+5)=(1,  7)B' = (4 + (-3),\; 2 + 5) = (1,\; 7)
Step 4: Apply the shift rule to vertex C(4, 6).
C′=(4+(−3),  6+5)=(1,  11)C' = (4 + (-3),\; 6 + 5) = (1,\; 11)
Step 5: Verify that the shape is unchanged. The original side AB had length 3; the new side A'B' also has length 3. The original side BC had length 4; B'C' also has length 4. The triangle is identical — only its position changed.
Answer: The shifted triangle has vertices A'(−2, 7), B'(1, 7), and C'(1, 11).

Another Example

This example applies shifts to function equations rather than geometric coordinates, showing the algebraic form students encounter in algebra and precalculus.

Problem: The function f(x) = x² has its graph shifted 4 units to the right and 1 unit down. Write the equation of the new function g(x).
Step 1: Recall how shifts affect a function's equation. A horizontal shift of h units to the right replaces x with (x − h). A vertical shift of k units down subtracts k from the entire function.
g(x)=f(x−h)+kg(x) = f(x - h) + k
Step 2: Substitute h = 4 (right) and k = −1 (down).
g(x)=f(x−4)+(−1)=(x−4)2−1g(x) = f(x - 4) + (-1) = (x - 4)^2 - 1
Step 3: Check with a point. The vertex of f(x) = x² is at (0, 0). After shifting 4 right and 1 down, the vertex should be at (4, −1). Substituting x = 4 into g(x):
g(4)=(4−4)2−1=0−1=−1✓g(4) = (4 - 4)^2 - 1 = 0 - 1 = -1 \quad \checkmark
Answer: The shifted function is g(x) = (x − 4)² − 1.

Frequently Asked Questions

What is the difference between a shift and a reflection?
A shift slides every point the same distance in the same direction, keeping the figure's orientation intact. A reflection flips the figure across a line (like a mirror), which reverses its orientation. For example, a shift moves a letter 'R' to a new position and it still reads as 'R,' but a reflection would produce a backwards 'R.'
Why does shifting right use a minus sign in the equation?
When you shift a graph h units to the right, you replace x with (x − h) in the function's equation. This feels counterintuitive, but think of it this way: the new graph needs x to be h units larger to produce the same y-value the original graph produced at x. So you subtract h from x to 'compensate,' which effectively moves the output to the right.
Does a shift change the size or shape of a figure?
No. A shift is a rigid transformation (also called an isometry). It preserves all distances, angles, and the overall shape. Only the position changes — the figure is congruent to the original.

Shift (Translation) vs. Reflection

Shift (Translation)Reflection
What it doesSlides the figure to a new positionFlips the figure across a line of reflection
OrientationPreserved (same orientation)Reversed (mirror image)
Formula (2D point)(x, y) → (x + h, y + k)Depends on the line; e.g., across the y-axis: (x, y) → (−x, y)
Rigid transformation?Yes — distances and angles unchangedYes — distances and angles unchanged
Common useMoving graphs or figures without rotatingCreating mirror images or studying symmetry

Why It Matters

Shifts appear throughout algebra, geometry, and precalculus whenever you move a graph or figure without distorting it. In algebra, understanding horizontal and vertical shifts lets you quickly graph families of functions like parabolas, absolute value functions, and square roots by starting from a parent function. In geometry, translations are one of the four rigid motions used to prove congruence and analyze symmetry in figures and tessellations.

Common Mistakes

Mistake: Reversing the sign for horizontal shifts in function equations — writing f(x + 4) when trying to shift 4 units to the right.
Correction: A shift of h units to the right requires replacing x with (x − h). So shifting right by 4 gives f(x − 4), not f(x + 4). Remember: the sign inside the parentheses is opposite to the direction of the shift.
Mistake: Applying different shift amounts to different points of the same figure.
Correction: In a true shift, every point moves by exactly the same h and k values. If you add different amounts to different vertices, you are distorting the figure, not translating it.

Related Terms