The formulas for vertical stretching and horizontal compression in the context of transformations of functions are:

1. Vertical Stretch:

   • The formula for a vertical stretch of a function $f(x)$ by a factor of $a$ (where $a > 1$) is: $g(x) = a \cdot f(x)$ This stretches the graph of $f(x)$ vertically by the factor $a$.

2. Horizontal Compression:

   • The formula for a horizontal compression of a function $f(x)$ by a factor of $b$ (where $b > 1$) is: $g(x) = f(b \cdot x)$ This compresses the graph of $f(x)$ horizontally by the factor $b$.

Important Notes:

• For vertical stretch, the $y$-values are multiplied by $a$, causing the graph to become taller.
• For horizontal compression, the $x$-values are effectively multiplied by $1/b$, making the graph "shrink" horizontally.

Would you like further details or have any questions?

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Here are 5 related questions for expanding your understanding:

1. How does vertical compression differ from vertical stretch?
2. What happens if the factor $a$ or $b$ is less than 1?
3. How do vertical and horizontal transformations affect the function's intercepts?
4. Can vertical stretch and horizontal compression be applied together? How?
5. How do reflections relate to these transformations?

Tip: Always check the impact of transformation constants on both the domain and range of the function!