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Graph transformations involve changing the basic graph of a function in various ways, such as translating, reflecting, stretching, or compressing it.
In our case, we are focusing on a reflection. When dealing with transformations, it's essential to understand how different algebraic modifications to the function's equation affect its graph.
Let's take a look at the absolute value function and how transformations apply to it.

The absolute value function, represented by \text{\textbar x \textbar}, forms a V-shaped graph. This graph opens upwards and has its vertex at the origin (0,0).
Mathematically, it is written as: \(y = |x|\). The function outputs the non-negative value of its input, meaning all y-values are zero or positive.
Think of the absolute value function as a mirror, reflecting all negative x-values onto the positive side while positive x-values remain unchanged. This is why the graph forms a V-shape.
Understanding this parent function is crucial before moving to more complex transformations, such as reflections.

A reflection over the x-axis is a type of graph transformation that flips the graph upside down. For absolute value functions, multiplying the entire function by -1 accomplishes this.
For instance, \(k(x) = -|x|\) reflects the graph of \(y = |x|\) over the x-axis.
This changes the y-values of the function to their opposites. So, a point \((x, y)\) on the original graph will become \((x, -y)\) on the transformed graph.
This process inverts the V-shaped graph from opening upwards to opening downwards, while the vertex remains at the origin.
Reflecting over the x-axis is one of the most common transformations and is particularly useful for manipulating the shapes and orientations of graphs.