Graph Transformations: Shifting

Definition

A graph can be translated in the coordinate plane by adding constants to either the input (horizontal) or the output (vertical). For $k, h > 0$:

• Vertical Shift: $y = f(x) \pm k$ (Up/Down).

• How to read: “The value y is equal to f of x plus or minus k.”
• Meaning: Adding $k$ to the output shifts the graph up; subtracting shifts it down.

• Horizontal Shift: $y = f(x \pm h)$ (Left/Right; note the “opposite” sign intuition).

• How to read: “The function f of the quantity x plus h, or f of the quantity x minus h.”
• Meaning: $f(x - h)$ delays the input by $h$, sliding the graph right; signs feel reversed because the input must compensate.

Why It Matters

Shifting is the math of re-centering our perspective; whether adjusting for a delay in a signal or moving an object in a digital space, shifting allows us to re-index our models to match the starting conditions of the real world.

Core Concepts

• Rigid Motion: Shifting is a rigid transformation; it changes the position of the graph but does not affect its shape, size, or orientation.
• Vertical Mechanism: Adding $k$ to the output $f(x)$ directly increases every $y$-coordinate, moving the graph up.
• Horizontal Mechanism: Replacing $x$ with $(x-h)$ requires the input to be $h$ units larger to produce the same output as the original $f(x)$, thus shifting the graph to the right.

Connected Concepts

• Graph Transformations
• Graph Scaling and Graph Reflecting