Function Transformations: Vertical and Horizontal Shifts

Function Transformations

Vertical and Horizontal Shifts

Transformations of functions are a fundamental concept in precalculus, allowing us to modify and analyze the graphs of functions by shifting them vertically or horizontally. Understanding these shifts helps in graphing and interpreting functions efficiently.

Vertical Shifts

• Definition: A vertical shift moves the graph of a function up or down without changing its shape.

• General Form: If $y = f(x)$, then $y = f(x) + d$ shifts the graph vertically by $d$ units.

• Direction:

  • If $d > 0$, the graph shifts upward.

  • If $d < 0$, the graph shifts downward.

• Example:

  • $y = f(x) + 1$ shifts the graph of $y = f(x)$ up by 1 unit.

  • $y = f(x) - 1$ shifts the graph of $y = f(x)$ down by 1 unit.

Horizontal Shifts

• Definition: A horizontal shift moves the graph of a function left or right.

• General Form: If $y = f(x)$, then $y = f(x + c)$ shifts the graph horizontally by $c$ units.

• Direction:

  • If $c > 0$, the graph shifts left by $c$ units.

  • If $c < 0$, the graph shifts right by $|c|$ units.

• Example:

  • $y = f(x + 2)$ shifts the graph of $y = f(x)$ left by 2 units.

  • $y = f(x - 2)$ shifts the graph of $y = f(x)$ right by 2 units.

Examples of Shifts

• Vertical Shift Example:

  • $y = \\sqrt{x}$ (original function)

  • $y = \\sqrt{x} + 2$ (shifted up by 2 units)

• Horizontal Shift Example:

  • $y = x^2$ (original function)

  • $y = (x - 1)^2$ (shifted right by 1 unit)

• Combined Shifts:

  • $y = |x - 1| + 2$ (absolute value function shifted right by 1 and up by 2)

Order of Transformations

• General Rule: When applying multiple transformations, start with the horizontal shift (inside the function) and then apply the vertical shift (outside the function).

• Example:

  • First, shift $y = |x|$ right by 1: $y = |x - 1|$

  • Then, shift up by 2: $y = |x - 1| + 2$

Summary Table: Vertical vs. Horizontal Shifts

Type of Shift	Transformation	Direction	Example
Vertical	$y = f(x) + d$	Up if $d > 0$, Down if $d < 0$	$y = x^2 + 3$
Horizontal	$y = f(x + c)$	Left if $c > 0$, Right if $c < 0$	$y = (x - 2)^2$

Applications

• Shifting functions is useful for modeling real-world phenomena, such as adjusting the starting point of a process or changing the baseline of a measurement.

• Understanding shifts helps in solving equations and inequalities involving transformed functions.

Additional info: The notes also emphasize that the order of applying transformations generally starts from the interior (horizontal) and moves to the exterior (vertical), which is a useful rule for more complex transformations.