Describing Transformations Suppose the graph of f is given....

Describing Transformations Suppose the graph of $f$ is given. Describe how the graph of each function can be obtained from the graph of $f .$
$$
\text { (a) } 2 f(x+5)-1 \quad \text { (b) } \frac{1}{4} f(x-3)+5
$$

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Key Concepts

Horizontal Translation

This transformation is applied when the input of the function is altered (e.g., f(x + c) or f(x - c)). Changing x to x + c shifts the graph horizontally to the left by c units, while changing x to x - c shifts the graph to the right by c units. Understanding how horizontal shifts are represented inside the function argument is critical for manipulating the graph's position along the x-axis.

Vertical Scaling

Vertical scaling involves multiplying the entire function by a constant. When the function is multiplied by a factor greater than 1, such as 2, the graph stretches vertically, making it taller. When multiplied by a factor between 0 and 1, such as 1/4, the graph is compressed vertically, making it appear shorter. This concept is important in modifying the amplitude or overall height of a graph without altering its horizontal aspects.

Vertical Translation

Vertical translation is performed by adding or subtracting a constant outside the function. Adding a constant, like +5, lifts the graph upward by that amount, while subtracting a constant, like -1, shifts the graph downward. This transformation adjusts the graph's vertical position without affecting its shape or orientation.

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