To describe transformations on a function $f(x)$, we can use specific operations to manipulate its graph. Transformations can include shifts, stretches, compressions, and reflections. Here’s a general guide on how different modifications affect the graph of $f(x)$:

1. Vertical Shifts:

   • $f(x) + k$: Shifts the graph up by $k$ units if $k > 0$, and down by $|k|$ units if $k < 0$.

2. Horizontal Shifts:

   • $f(x - h)$: Shifts the graph to the right by $h$ units if $h > 0$, and to the left by $|h|$ units if $h < 0$.

3. Vertical Stretch and Compression:

   • $a \cdot f(x)$: If $|a| > 1$, the graph is vertically stretched (made taller) by a factor of $a$. If $0 < |a| < 1$, the graph is vertically compressed (made shorter).

4. Horizontal Stretch and Compression:

   • $f(bx)$: If $|b| > 1$, the graph is horizontally compressed (narrowed) by a factor of $\frac{1}{b}$. If $0 < |b| < 1$, the graph is horizontally stretched (widened).

5. Reflections:

   • $-f(x)$: Reflects the graph across the x-axis.
   • $f(-x)$: Reflects the graph across the y-axis.

These transformations can be combined. For example, the function $g(x) = -2f(x + 3) - 5$ involves several transformations:

• $f(x + 3)$: Shift 3 units to the left.
• $-2f(x + 3)$: Reflect across the x-axis and vertically stretch by a factor of 2.
• $-2f(x + 3) - 5$: Shift 5 units downward.

Would you like a specific transformation described, or do you have an example function in mind?

Related Questions:

1. How does the transformation $f(2x + 1)$ affect the graph of $f(x)$?
2. What is the difference between a vertical stretch and a horizontal compression?
3. How do combined transformations, like $3f(-x + 2) - 4$, change the graph?
4. What transformations would reflect the graph of $f(x) = x^2$ across the y-axis and shift it up by 3 units?
5. Can a function have both horizontal and vertical compressions? Provide an example.

Tip: Always apply transformations in the correct order to understand their full effect on a graph! Start with horizontal shifts, followed by stretches/compressions, reflections, and finally vertical shifts.