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An absolute value function is a function that involves the absolute value of a variable. The general form of an absolute value function is given by:
\[ f(x) = |x| \]
The absolute value of a number is its distance from zero on a number line, regardless of direction. This means that:

• If the input is positive or zero, the output is the same as the input: \( |x| = x \) if \( x \geq 0 \).
• If the input is negative, the output is the positive counterpart: \( |x| = -x \) if \( x < 0 \).

The graph of an absolute value function has a characteristic 'V' shape that changes direction at the origin (0,0). This point is where the function is split into two linear pieces.

A horizontal shrink is a transformation that compresses a graph towards the y-axis. If you have a function \( f(x) \), then the function \( g(x) = f(cx) \) where \( c > 1 \), represents a horizontal shrink by a factor of \( c \).

For the function \( g(x) = |2x| \), we can interpret it as a horizontal shrink of the absolute value function \( f(x) = |x| \). Specifically, the factor is 2 because before applying the absolute value, the input is multiplied by 2. This shrinks the graph horizontally by a factor of 2, meaning every point on the graph of \( f(x) \) is moved half the distance to the y-axis. So:
\[ g(x) = |2x| = f(2x) \]
Every point \( (x, |x|) \) on the graph of \( f(x) = |x| \) becomes \( (x/2, |x|) \) on the graph of \( g(x) = f(2x) \).

A vertical stretch is a transformation that elongates a graph away from the x-axis. If you have a function \( f(x) \), then the function \( g(x) = c f(x) \) where \( c > 1 \), represents a vertical stretch by a factor of \( c \).

For the function \( g(x) = |2x| \), we can also see it as a vertical stretch of the absolute value function \( f(x) = |x| \). To show this, let's rewrite \( g(x) \) as:
\[ g(x) = |2x| = 2|x| \]
This form makes it clear that the output of the function is scaled by a factor of 2 after taking the absolute value of \( x \). Therefore, the graph is stretched vertically by a factor of 2. The result is that every point \( (x, |x|) \) on the graph of \( f(x) \) becomes \( (x, 2|x|) \) on the graph of \( g(x) \).

Graph transformations involve changes to the position, size, or shape of the graph of a function. Key transformations include translations, reflections, stretches, and shrinks:

• Translations: Shifting the graph horizontally or vertically without changing its shape.
• Reflections: Flipping the graph over a specific axis.
• Stretches and Shrinks: Changing the size of the graph, either making it taller/narrower (vertical stretch/shrink) or wider/thinner (horizontal stretch/shrink).

In our example, the function \( g(x) = |2x| \) is transformed from the original function \( f(x) = |x| \) through both a horizontal shrink and a vertical stretch. This allows us to view the same new graph from two different perspectives: compressing it towards the y-axis or elongating it away from the x-axis. Understanding these transformations helps interpret how functions change and predict their graphs' behaviors.