Section 1.6: Transformations of Functions

Graphs of Common Functions

Understanding the basic graphs of common functions is essential for analyzing transformations. These functions serve as the foundation for shifts, reflections, stretches, and shrinks.

• Constant Function: $f(x) = c$ - Domain: $(-\infty, \infty)$ - Range: The single number $c$ - Constant on $(-\infty, \infty)$ - Even function

• Identity Function: $f(x) = x$ - Domain: $(-\infty, \infty)$ - Range: $(-\infty, \infty)$ - Increasing on $(-\infty, \infty)$ - Odd function

• Absolute Value Function: $f(x) = |x|$ - Domain: $(-\infty, \infty)$ - Range: $[0, \infty)$ - Decreasing on $(-\infty, 0)$, increasing on $(0, \infty)$ - Even function

• Standard Quadratic Function: $f(x) = x^2$ - Domain: $(-\infty, \infty)$ - Range: $[0, \infty)$ - Decreasing on $(-\infty, 0)$, increasing on $(0, \infty)$ - Even function

• Square Root Function: $f(x) = \sqrt{x}$ - Domain: $[0, \infty)$ - Range: $[0, \infty)$ - Increasing on $(0, \infty)$ - Neither even nor odd

• Standard Cubic Function: $f(x) = x^3$ - Domain: $(-\infty, \infty)$ - Range: $(-\infty, \infty)$ - Increasing on $(-\infty, \infty)$ - Odd function

• Cube Root Function: $f(x) = \sqrt[3]{x}$ - Domain: $(-\infty, \infty)$ - Range: $(-\infty, \infty)$ - Increasing on $(-\infty, \infty)$ - Odd function

Shifts

Vertical Shifts

Vertical shifts move the graph up or down. The transformation $y = f(x) + c$ shifts the graph of $y = f(x)$ up $c$ units, while $y = f(x) - c$ shifts it down $c$ units.

• Example: $f(x) = x^2$, $g(x) = x^2 + 2$, $h(x) = x^2 - 3$ - $g(x)$ is shifted up 2 units. - $h(x)$ is shifted down 3 units.

Horizontal Shifts

Horizontal shifts move the graph left or right. The transformation $y = f(x + c)$ shifts the graph of $y = f(x)$ left $c$ units, while $y = f(x - c)$ shifts it right $c$ units. The direction is opposite to the sign inside the function.

• Example: $f(x) = x^2$, $g(x) = (x + 2)^2$, $h(x) = (x - 3)^2$ - $g(x)$ is shifted left 2 units. - $h(x)$ is shifted right 3 units.

Reflections of Graphs

Reflection about the x-axis

Reflecting a graph about the x-axis changes the sign of all y-values. The transformation $y = -f(x)$ reflects $y = f(x)$ about the x-axis.

• Example: $f(x) = x^2$, $g(x) = -x^2$ - Each point $(x, y)$ becomes $(x, -y)$.

Reflection about the y-axis

Reflecting a graph about the y-axis changes the sign of all x-values. The transformation $y = f(-x)$ reflects $y = f(x)$ about the y-axis.

• Example: $f(x) = \sqrt{x}$, $h(x) = \sqrt{-x}$ - Each point $(x, y)$ becomes $(-x, y)$.

Stretching and Shrinking

Vertical Stretching and Shrinking

Vertical stretching multiplies all y-values by $c$ ($c > 1$), making the graph taller. Vertical shrinking multiplies all y-values by $c$ ($0 < c < 1$), making the graph shorter.

• Example: $f(x) = x^2$, $g(x) = 2x^2$, $h(x) = \frac{1}{2}x^2$ - $g(x)$ is vertically stretched. - $h(x)$ is vertically shrunk.

• General illustration:

Horizontal Stretching and Shrinking

Horizontal stretching divides all x-values by $c$ ($0 < c < 1$), making the graph wider. Horizontal shrinking divides all x-values by $c$ ($c > 1$), making the graph narrower.

• General illustration:

• Example: $g(x) = f(2x)$ is a horizontal shrink by a factor of 2. $h(x) = f(\frac{1}{2}x)$ is a horizontal stretch by a factor of 2.

Sequences of Transformations

Order of Transformations

When multiple transformations are applied to a function, the order matters. Typically, transformations are applied in the following sequence:

1. Horizontal shifting

2. Horizontal stretching/shrinking and reflection

3. Vertical stretching/shrinking and reflection

4. Vertical shifting

• Example: $g(x) = 2(x-1)^2 + 3$ - Shift right by 1 - Vertical stretch by 2 - Shift up by 3

Summary Table: Types of Transformations

Key Formulas

• Vertical Shift: $y = f(x) + c$

• Horizontal Shift: $y = f(x + c)$

• Vertical Stretch/Shrink: $y = c f(x)$

• Horizontal Stretch/Shrink: $y = f(cx)$

• Reflection about x-axis: $y = -f(x)$

• Reflection about y-axis: $y = f(-x)$

Additional info: The study of transformations is foundational for understanding more advanced topics in precalculus, such as composition of functions, inverse functions, and graphing complex equations.