Function Transformations

Function transformations provide a unified framework for taking a "parent" function and altering its graph to create a new function. By understanding a few basic rules, you can sketch the graph of complex functions without having to plot points from scratch.

The General Transformation Formula

Any function $f(x)$ can be transformed into a new function $y$ using the general formula:

$y = a f(b(x - h)) + k$

Each parameter—$a$, $b$, $h$, and $k$—controls a specific change to the graph.

Vertical Changes (Outside the Function)

Changes that occur outside the function notation affect the graph vertically (the y-values).

• $a$ (Vertical Stretch/Compression & Reflection):
  • If $|a| > 1$, the graph is vertically stretched by a factor of $a$.
  • If $0 < |a| < 1$, the graph is vertically compressed by a factor of $a$.
  • If $a$ is negative, the graph is reflected across the x-axis.
• $k$ (Vertical Translation):
  • If $k > 0$, the graph shifts up by $k$ units.
  • If $k < 0$, the graph shifts down by $k$ units.

Horizontal Changes (Inside the Function)

Changes that occur inside the function notation affect the graph horizontally (the x-values). Note: Horizontal changes generally act opposite to what you might intuitively expect.

• $b$ (Horizontal Stretch/Compression & Reflection):
  • If $|b| > 1$, the graph is horizontally compressed by a factor of $\frac{1}{|b|}$.
  • If $0 < |b| < 1$, the graph is horizontally stretched by a factor of $\frac{1}{|b|}$.
  • If $b$ is negative, the graph is reflected across the y-axis.
• $h$ (Horizontal Translation):
  • Notice the formula uses $(x - h)$.
  • If you have $(x - 3)$, then $h = 3$, meaning the graph shifts right 3 units.
  • If you have $(x + 3)$, then $h = -3$, meaning the graph shifts left 3 units.

The Order of Transformations

When applying multiple transformations, the order matters. Always follow the order of operations (PEMDAS/BEDMAS). A standard sequence is:

1. Horizontal/Vertical Stretches and Compressions
2. Reflections (over the x-axis or y-axis)
3. Translations (shifts left/right and up/down)

Tip: Always factor out $b$ inside the function before determining your horizontal shift. For example, $f(2x - 4)$ should be rewritten as $f(2(x - 2))$ to see that the shift is right 2 units, not 4.

Example Problems

Example 1: Describing Transformations

Problem: Describe the transformations from the parent function $y = \sqrt{x}$ to $y = -2\sqrt{x + 3} - 1$.

Solution: Let's identify our parameters based on $y = a f(x - h) + k$:

• $a = -2$: The negative sign means a reflection over the x-axis. The 2 means a vertical stretch by a factor of 2.
• $h = -3$: Because it is $(x + 3)$, the graph is shifted left 3 units.
• $k = -1$: The graph is shifted down 1 unit.

Example 2: Sketching a Transformed Graph

Problem: Given the graph of $f(x)$, sketch $y = f(2x) - 3$.

Solution:

1. Identify the parameters: $b = 2$ and $k = -3$.
2. Horizontal Compression: The $2x$ inside the function means every x-coordinate of the original graph is divided by 2 (a horizontal compression by a factor of $\frac{1}{2}$).
3. Vertical Shift: The $-3$ outside the function means every y-coordinate is decreased by 3 (shifted down 3 units).
4. To sketch: Take key points $(x, y)$ from the original graph and apply the mapping $(x, y) \rightarrow (\frac{x}{2}, y - 3)$.