Transformations of Functions: A Precalculus Study Guide

Transformations of Functions

Introduction

Transformations are operations that alter the position, size, or orientation of the graph of a function. Understanding these transformations is essential for graphing and analyzing functions in precalculus. The main types of transformations include shifts, reflections, and stretches or shrinks, both vertically and horizontally.

Vertical Shifts

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

• Upward Shift: The graph of $y = f(x) + c$ is the graph of $y = f(x)$ shifted up by $c$ units, where $c > 0$.

• Downward Shift: The graph of $y = f(x) - c$ is the graph of $y = f(x)$ shifted down by $c$ units, where $c > 0$.

• Example: If $f(x) = x^2$, then $f(x) + 3 = x^2 + 3$ shifts the parabola up by 3 units.

Horizontal Shifts

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

• Left Shift: The graph of $y = f(x + c)$ is the graph of $y = f(x)$ shifted left by $c$ units, where $c > 0$.

• Right Shift: The graph of $y = f(x - c)$ is the graph of $y = f(x)$ shifted right by $c$ units, where $c > 0$.

• Example: If $f(x) = \sqrt{x}$, then $f(x - 2) = \sqrt{x - 2}$ shifts the graph right by 2 units.

Reflections

Reflections flip the graph of a function over a specific axis.

• Reflection about the x-axis: The graph of $y = -f(x)$ is the graph of $y = f(x)$ reflected over the x-axis.

• Reflection about the y-axis: The graph of $y = f(-x)$ is the graph of $y = f(x)$ reflected over the y-axis.

• Example: If $f(x) = x^3$, then $-f(x) = -x^3$ reflects the graph over the x-axis.

Vertical Stretching and Shrinking

Vertical transformations change the steepness or flatness of a graph by multiplying the function by a constant.

• Vertical Stretch: If $c > 1$, the graph of $y = c f(x)$ is a vertical stretch by a factor of $c$ (each y-coordinate is multiplied by $c$).

• Vertical Shrink: If $0 < c < 1$, the graph of $y = c f(x)$ is a vertical shrink by a factor of $c$ (each y-coordinate is multiplied by $c$).

• Example: If $f(x) = x^2$, then $2f(x) = 2x^2$ is a vertical stretch by 2; $0.5f(x) = 0.5x^2$ is a vertical shrink by 0.5.

Horizontal Stretching and Shrinking

Horizontal transformations compress or expand the graph along the x-axis by changing the input variable.

• Horizontal Shrink: If $c > 1$, the graph of $y = f(c x)$ is a horizontal shrink by a factor of $1/c$ (each x-coordinate is divided by $c$).

• Horizontal Stretch: If $0 < c < 1$, the graph of $y = f(c x)$ is a horizontal stretch by a factor of $1/c$ (each x-coordinate is divided by $c$).

• Example: If $f(x) = x^2$, then $f(2x) = (2x)^2 = 4x^2$ is a horizontal shrink by 1/2; $f(0.5x) = (0.5x)^2 = 0.25x^2$ is a horizontal stretch by 2.

Order of Transformations

When multiple transformations are applied to a function, the order affects the final graph. The recommended order is:

1. Horizontal shifting

2. Stretching or shrinking (horizontal and vertical)

3. Reflections

4. Vertical shifting

Example: For $y = -2f(x + 3) - 1$:

• Shift left by 3 units ($x + 3$)

• Vertical stretch by 2 ($2f(x)$)

• Reflect over x-axis ($-f(x)$)

• Shift down by 1 unit ($-1$)

Summary Table: Types of Function Transformations

Transformation	Equation	Effect on Graph
Vertical Shift Up	$y = f(x) + c$	Up by $c$ units
Vertical Shift Down	$y = f(x) - c$	Down by $c$ units
Horizontal Shift Right	$y = f(x - c)$	Right by $c$ units
Horizontal Shift Left	$y = f(x + c)$	Left by $c$ units
Vertical Stretch	$y = c f(x)$, $c > 1$	Stretched vertically by $c$
Vertical Shrink	$y = c f(x)$, $0 < c < 1$	Shrunk vertically by $c$
Horizontal Stretch	$y = f(c x)$, $0 < c < 1$	Stretched horizontally by $1/c$
Horizontal Shrink	$y = f(c x)$, $c > 1$	Shrunk horizontally by $1/c$
Reflection over x-axis	$y = -f(x)$	Flipped over x-axis
Reflection over y-axis	$y = f(-x)$	Flipped over y-axis

Domain and Range under Transformations

• Vertical and horizontal shifts do not change the shape of the graph, but may change the domain and range depending on the function.

• Stretches and shrinks can expand or contract the range and/or domain.

• Reflections may change the sign of the range or domain values.

• Example: For $f(x) = \sqrt{x}$, the domain is $[0, \infty)$ and the range is $[0, \infty)$. For $f(x - 2)$, the domain is $[2, \infty)$.

Additional info: The order of transformations is crucial when combining multiple operations. Always apply horizontal shifts and stretches before vertical ones for consistent results.