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Horizontal scaling is a type of function transformation that modifies the function's domain, specifically affecting the x-values. In simple terms, it either stretches or compresses the function horizontally depending on the value of the scaling factor. When you see an expression such as \( f(cx) \), it indicates a horizontal scaling transformation.

• If \( c > 1 \), the function compresses toward the y-axis because the x-values are getting closer.
• If \( 0 < c < 1 \), the function stretches away from the y-axis. The x-values get further apart, making the function look wider.

In our example, \( f\left(\frac{1}{5}x\right) \) causes a horizontal scaling because the factor \( \frac{1}{5} \), which is less than 1, transforms the x-values of the original function \( f(x) \). This means the x-values are stretched, making the entire graph appear wider. This factor fundamentally changes the spread of the function across the x-axis.

A horizontal stretch can be visualized as making the graph of a function appear wider. It's a specific type of horizontal scaling where the scaling factor \( c \) in \( f(cx) \) is between 0 and 1. This stretches the function by spreading out its x-values.
For \( f\left(\frac{1}{5}x\right) \), since \( c = \frac{1}{5} \), the horizontal stretch factor is actually \( \frac{1}{c} = 5 \). This means each point on the x-axis is moved five times farther away from the y-axis than it originally was.
A practical way to understand this is by imagining you are taking a piece of rubber with a curve on it and pulling it sideways. Each part of the curve moves outward, making the entire curve wider and the graph effectively 'stretched' horizontally by a factor of 5.

Function transformations encompass a broad area of changes that can be applied to a function to alter its graph in various ways. Key transformations include:

• Horizontal Scaling - Altering the width of the graph, either stretching or compressing it horizontally.
• Vertical Scaling - Similar to horizontal scaling but affects the height, either stretching or compressing it vertically.
• Translation - Moving the graph up, down, left, or right without altering its shape.
• Reflections - Flipping the graph over a line, such as the x-axis or y-axis, to produce a mirror image.

In the instance of \( f\left(\frac{1}{5}x\right) \), a horizontal scaling transformation is applied. This particular transformation changes the graph, stretching it along the x-axis, while maintaining its overall shape, just in a wider form. Understanding these transformations allows you to manipulate functions effectively and predict how graphs will change with different transformations.