Function transformations let you change a graph's shape and position. You can shift it up, down, left, or right, stretch or compress it, and even flip it around. These tricks help you understand how functions behave.

Mastering transformations is key to analyzing complex functions. By combining different transformations, you can create new functions from familiar ones. This skill is crucial for solving real-world problems and understanding advanced math concepts.

Function Transformations

Vertical and horizontal shifts

• Vertical shifts move the graph up or down by adding a constant $k$ to the function: $f(x) + k$
  • Positive $k$ shifts the graph up (e.g., $f(x) + 3$ shifts the graph up by 3 units)
  • Negative $k$ shifts the graph down (e.g., $f(x) - 2$ shifts the graph down by 2 units)
• Horizontal shifts move the graph left or right by adding a constant $h$ inside the function: $f(x + h)$
  • Positive $h$ shifts the graph left (e.g., $f(x + 4)$ shifts the graph left by 4 units)
  • Negative $h$ shifts the graph right (e.g., $f(x - 1)$ shifts the graph right by 1 unit)
• These transformations may affect the domain and range of the function

Reflections across axes

• Reflecting a function across the x-axis flips the graph vertically by multiplying the function by -1: $-f(x)$
  • Example: if $f(x) = x^2$, then $-f(x) = -x^2$ reflects the parabola across the x-axis
• Reflecting a function across the y-axis flips the graph horizontally by replacing $x$ with $-x$ in the function: $f(-x)$
  • Example: if $f(x) = x^3$, then $f(-x) = -x^3$ reflects the cubic function across the y-axis

Even and odd functions

• Even functions are symmetric about the y-axis and satisfy $f(-x) = f(x)$ for all $x$ in the domain
  • Example: $f(x) = x^2$ is an even function because $f(-x) = (-x)^2 = x^2 = f(x)$
• Odd functions are symmetric about the origin and satisfy $f(-x) = -f(x)$ for all $x$ in the domain
  • Example: $f(x) = x^3$ is an odd function because $f(-x) = (-x)^3 = -x^3 = -f(x)$
• Functions that are neither even nor odd do not have symmetry about the y-axis or origin and do not satisfy the conditions for even or odd functions
  • Example: $f(x) = x^2 + x$ is neither even nor odd because $f(-x) \neq f(x)$ and $f(-x) \neq -f(x)$

Compressions and stretches

• Vertical scaling multiplies the function by a constant $a$: $a \cdot f(x)$
  • $|a| > 1$ stretches the graph vertically (e.g., $2f(x)$ stretches the graph vertically by a factor of 2)
  • $0 < |a| < 1$ compresses the graph vertically (e.g., $\frac{1}{3}f(x)$ compresses the graph vertically by a factor of 3)
  • Negative $a$ reflects the graph across the x-axis and scales vertically (e.g., $-f(x)$ reflects and stretches the graph vertically)
• Horizontal scaling divides $x$ by a constant $b$ inside the function: $f(\frac{x}{b})$
  • $|b| > 1$ compresses the graph horizontally (e.g., $f(2x)$ compresses the graph horizontally by a factor of 2)
  • $0 < |b| < 1$ stretches the graph horizontally (e.g., $f(\frac{x}{4})$ stretches the graph horizontally by a factor of 4)
  • Negative $b$ reflects the graph across the y-axis and scales horizontally (e.g., $f(-2x)$ reflects and compresses the graph horizontally)

Combining function transformations

• When applying multiple transformations to a function, follow this order:
  1. Horizontal scaling and reflection
  2. Horizontal shift
  3. Vertical scaling and reflection
  4. Vertical shift
• The resulting transformed function will have the general form: $a \cdot f(\frac{x - h}{b}) + k$
  • $a$: vertical scaling and reflection factor
  • $b$: horizontal scaling and reflection factor
  • $h$: horizontal shift value
  • $k$: vertical shift value
• Example: given $f(x) = x^2$, the transformed function $-2f(3x + 1) - 4$ can be broken down as follows:
  • $a = -2$ (vertical scaling and reflection)
  • $b = \frac{1}{3}$ (horizontal scaling)
  • $h = -\frac{1}{3}$ (horizontal shift)
  • $k = -4$ (vertical shift)

Advanced Function Concepts

• Function composition combines two or more functions to create a new function
• Inverse functions "undo" the effect of a function, reversing its input and output
• Piecewise functions are defined by different expressions for different parts of the domain
• One-to-one functions have a unique output for each input, allowing for the creation of an inverse function