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In function transformations, a horizontal stretch affects the width of the graph. When we have a function such as \( g(x) = f\left(\frac{1}{5} x\right) \), it indicates a horizontal change. Particularly, the expression \( \frac{1}{5} x \) signifies a horizontal stretch because the coefficient is a fraction less than 1.

Consider the original graph of \( f(x) \). In a horizontal stretch, each point on the graph moves farther away from the y-axis, making the graph wider. In our example, every point on \( f(x) \) is moved 5 times further away in the horizontal direction. This is achieved by multiplying each x-value by 5, resulting in a more stretched, wider graph.

Remember, when the coefficient of \( x \) in the function is a fraction (e.g., \( \frac{1}{5} \)), it leads to a stretch. Conversely, if the coefficient were greater than 1, the graph would undergo a compression. So, always check the coefficient to determine if it's a stretch or compression.

Graph transformations involve altering the basic position and form of a graph. Transformations can be vertical, horizontal, or both, changing where a graph is positioned and its shape.

For our function \( g(x) = f\left(\frac{1}{5}x\right) \), we're only seeing a horizontal transformation. In this case, transformation refers exclusively to the horizontal stretch applied to \( f(x) \). This process involves effectively pulling the graph apart along the x-axis, resulting in a wider graph.

Different transformations include:

• Translations: Shifts the entire graph horizontally or vertically.
• Reflections: Flips the graph across a specific axis.
• Stretches and Compressions: Expand or contract the graph along the x or y-axis.

Graph transformations are foundational because they allow us to predict and understand changes in any graph, enabling a deeper insight into the function's behavior.

The coefficient of \( x \) plays a crucial role in determining the nature of a graph transformation. In mathematical terms, the coefficient of \( x \) directly influences whether a graph undergoes a horizontal stretch or compression.

For the function \( g(x) = f\left(\frac{1}{5}x\right) \), the coefficient is \( \frac{1}{5} \). This coefficient is less than 1, indicating a horizontal stretch. The interpretation here is important: the smaller the coefficient relative to 1, the more pronounced the stretch will be.

Key effects of different coefficients include:

• If the coefficient \( c < 1 \), the graph of the function \( f(x) \) will stretch horizontally.
• If the coefficient \( c > 1 \), the graph will compress horizontally.
• If the coefficient \( c = 1 \), the graph remains unchanged in terms of width.
• Negative coefficients also indicate reflection across the y-axis along with any stretch/compression effects.

Understanding the coefficient of \( x \) allows one to quickly determine the resulting changes in the graph's width, making it an essential aspect of graph transformations.