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Graph reflections are a fundamental transformation in mathematics, especially when analyzing functions and their graphs. A reflection over the y-axis involves flipping the graph horizontally. Take the original graph of the function, and imagine it is like a piece of paper folded along the y-axis. This flipping creates a mirror image on the opposite side. So if our original function is represented by \(y = f(x)\), reflecting it over the y-axis results in \(y = f(-x)\).
For example, in our exercise, transforming the graph of \(y = f(x)\) to \(y = f(-x)\) flips it horizontally. The same logic applies when continuing to transform it to \(y = -f(-x)\), that would end up flipped both over the y-axis and then the x-axis. In essence, each reflection changes the direction and orientation of the graph, allowing for a diverse exploration of graph behavior and symmetry.

Horizontal compression shrinks the graph of a function towards the y-axis. This transformation reduces the horizontal extent of the graph, making features of the graph up to twice as close if it's compressed by a factor of 2.
When you compress \( f(x) \) into \( f(kx) \) where \( k > 1 \), the compression occurs. For instance, graphing \( y = f(-2x) \) compresses the original graph horizontally by a factor of 2, as seen in the exercise.

• Imagine the graph squeezing inwards.
• This effect also involves reflecting it horizontally if the constant multiplying \(x\) is negative.

This process results in a more confined and narrower graph, sharply changing its appearance while keeping its essential shape.

Horizontal stretching extends the graph of a function away from the y-axis. Essentially, it broadens the graph horizontally. When we modify a function with \( f(kx) \) and \( 0 < k < 1 \), the function experiences a horizontal stretch.
For the exercise, in part (e), when the function \(y=f\left(-\frac{1}{2}x\right)\) is analyzed, it involves a horizontal stretch because multiplying \(x\) by \(-\frac{1}{2}\) caused the original graph to widen. The effect:

• Graph becomes more spread out.
• The domain of the function effectively enlarges, covering more x-values.

The aspect of stretching widens the graph, and if negative, it results in reflection as well, altering the graph's shape but widening its reach.

Domain restrictions determine the values for which a function is defined. In functions involving square roots like \(f(x)=\sqrt{2x-x^2}\), ensuring the expression inside the square root is non-negative is critical.
For this exercise, the domain is specifically \(0 \leq x \leq 2\), a range where the square root’s argument remains non-negative. This confinement means:

• The graph exists only within this interval on the x-axis.
• When transformations such as reflections and compressions are applied, the domain dynamically alters.
• Specific transformations might compress or stretch this interval further, as seen previously with horizontal stretches or compressions.

Understanding domain restrictions ensures precise graphing and highlights parts of the function relevant for real-world problems and calculations.