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Transformations Use the graph of f in the figure to plot the...

Key Concepts

Horizontal Scaling (Compression/Stretching)

Horizontal scaling involves altering the graph’s width by compressing or stretching it. When the function is transformed by replacing x with a constant multiple—such as y = f(ax)—the graph is horizontally compressed by a factor of 1/a if a > 1, and stretched if 0 < a < 1. This modification affects the rate at which the graph moves along the x-axis.

Reflection

Reflection in function transformations involves 'flipping' the graph across a particular axis. For instance, the operation y = -f(x) reflects the graph of the original function across the x-axis, turning every point (x, y) into (x, -y). This concept is widely used to alter the orientation of the function's graph without changing its shape.

Horizontal Shifts

Horizontal shifts translate the graph of a function left or right. Changing the input variable through an expression like f(x + c) or f(x - c) results in the function's graph moving horizontally. Specifically, f(x + c) shifts the graph c units to the left, while f(x - c) shifts it c units to the right, without affecting the shape or vertical position of the original graph.

Vertical Shifts

Vertical shifts raise or lower the graph of a function by adding or subtracting a constant to the function's output. For example, the transformation y = f(x) + k moves the graph upward by k units when k is positive and downward when k is negative. This concept helps in repositioning graphs for better interpretation and comparison.

Vertical Scaling (Compression/Stretching)

Vertical scaling modifies the graph’s height, stretching or compressing it along the y-axis. In a transformation like y = kf(x), the graph is vertically stretched if k > 1, meaning that all y-values are multiplied by k, and compressed if 0 < k < 1. This is used to adjust the amplitude of a function without altering its horizontal properties.

Composite Transformations

Composite transformations occur when two or more transformation types, such as reflections, shifts, and scalings, are applied in succession to a function. Understanding the order and effect of each transformation is critical, as the final graph is the result of these combined changes. This concept is central in graphing more complex functions and analyzing their behavior based on simpler, well-understood transformations.

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