Function transformations are powerful tools for manipulating graphs. They let you shift, reflect, stretch, and compress functions without changing their core shape. These techniques are crucial for understanding how functions behave and relate to each other.

By mastering transformations, you can quickly sketch complex functions and solve equations. This skill is essential for analyzing real-world data and modeling various phenomena in fields like physics, economics, and engineering. It's a fundamental building block for advanced math.

Function Transformations

Vertical and horizontal shifts

• Vertical shifts move the graph of a function up or down along the y-axis without changing its shape
  • Add a positive constant $k$ to the function $f(x)$ to shift the graph up by $k$ units ($f(x) + k$)
  • Subtract a positive constant $k$ from the function $f(x)$ to shift the graph down by $k$ units ($f(x) - k$)
• Horizontal shifts move the graph of a function left or right along the x-axis without changing its shape
  • Replace $x$ with $x - h$ in the function $f(x)$ to shift the graph right by $h$ units ($f(x - h)$)
  • Replace $x$ with $x + h$ in the function $f(x)$ to shift the graph left by $h$ units ($f(x + h)$)
• These shifts can affect the domain and range of the function

Reflections across axes

• Reflecting a function across the x-axis produces a mirror image of the graph above or below the x-axis
  • Multiply the function $f(x)$ by -1 to reflect the graph across the x-axis ($-f(x)$)
• Reflecting a function across the y-axis produces a mirror image of the graph on the opposite side of the y-axis
  • Replace $x$ with $-x$ in the function $f(x)$ to reflect the graph across the y-axis ($f(-x)$)

Even and odd functions

• Even functions have graphs that are symmetric about the y-axis
  • A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain
  • Examples of even functions: $f(x) = x^2$, $f(x) = \cos(x)$
• Odd functions have graphs that are symmetric about the origin
  • A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain
  • Examples of odd functions: $f(x) = x^3$, $f(x) = \sin(x)$
• Functions that are neither even nor odd do not have symmetry about the y-axis or origin
  • Example of a neither even nor odd function: $f(x) = 2^x$

Compressions and stretches

• Vertical compressions and stretches change the height of the graph without changing its width
  • Multiply the function $f(x)$ by a constant $0 < |a| < 1$ to vertically compress the graph by a factor of $\frac{1}{|a|}$ ($af(x)$)
  • Multiply the function $f(x)$ by a constant $|a| > 1$ to vertically stretch the graph by a factor of $|a|$ ($af(x)$)
• Horizontal compressions and stretches change the width of the graph without changing its height
  • Replace $x$ with $\frac{x}{b}$ in the function $f(x)$ where $|b| > 1$ to horizontally compress the graph by a factor of $\frac{1}{|b|}$ ($f(\frac{x}{b})$)
  • Replace $x$ with $\frac{x}{b}$ in the function $f(x)$ where $0 < |b| < 1$ to horizontally stretch the graph by a factor of $\frac{1}{|b|}$ ($f(\frac{x}{b})$)

Combining multiple transformations

• When applying multiple transformations to a function, follow this order:

1. Compressions and stretches
2. Reflections
3. Horizontal shifts
4. Vertical shifts

• Combine transformations by applying them to the function in the correct order
  • Example: $-2f(3x - 1) + 4$ represents a function with these transformations:
    1. Horizontal compression by a factor of $\frac{1}{3}$ ($3x$)
    2. Horizontal shift right by $\frac{1}{3}$ units ($x - 1$)
    3. Vertical stretch by a factor of 2 ($-2f$)
    4. Reflection across the x-axis ($-2f$)
    5. Vertical shift up by 4 units ($+ 4$)

Function Composition and Inverse Functions

• Function composition involves applying one function to the output of another function
• The inverse function reverses the effect of a function, effectively "undoing" its operation
• Composition and inverse functions are closely related to transformations, as they can change the domain and range of the original function