Question

Graph the function and its parent function. Then describe the transformation. \(f(x)=\frac{1}{2}|x|\)

Short answer

The function \(\frac{1}{2}|x|\) is a vertical shrink by a factor of \(\frac{1}{2}\) of the parent function \(|x|\).

Step-by-step solution

Graphing the Parent Function

Start by graphing the parent function \(|x|\). This function forms a V shape, starting from the origin (0,0) and extending to both directions in a positive slope.

Graphing the Given Function

Now graph the given function \(\frac{1}{2}|x|\). This function will also form a V shape similar to the parent function, but the slope will differ. Since the coefficient is \(\frac{1}{2}\), the slope of the lines will be half as steep as compared to the parent function \(|x|\).

Describing the Transformation

Now, compare both the graphs and describe the transformation. Here, the function \(\frac{1}{2}|x|\) is a vertical shrink by a factor of \(\frac{1}{2}\) compared to the parent function \(|x|\). This transformation has not shifted the graph right, left, up or down, but it has made the graph less steep as compared to the parent function.

Key concepts

Understanding the Parent Function

The term "parent function" refers to the simplest form of a particular type of function. In the case of the absolute value function, the parent function can be described by the equation \(|x|\). This classic absolute value function forms a V-shaped graph that has its vertex at the origin \((0,0)\). It's symmetric concerning the y-axis.
It rises at a 45-degree angle in both the positive and negative directions along the x-axis.
The parent function provides a baseline, from which we can apply transformations like shifts, stretches, and shrinks. Recognizing the shape and properties of the parent function is the first step to understanding how more complex functions are transformed versions of these basic graphs.

Exploring the Absolute Value Function

The absolute value function is characterized by the expression \(|x|\), which can be thought of as the distance of a number from zero on the number line. This function has some important properties:

• It's always non-negative.
• The graph is V-shaped.
• The vertex is located at the origin.

The absolute value of a number is always positive, as it measures distance without considering direction.
For example, both -3 and +3 have an absolute value of 3.
In graphing, this feature translates to the V-shape because the negative values of the input (x-values) are reflected into positive output values.
Understanding these properties helps in predicting how alterations to the function will modify its graphical representation.

Analyzing Vertical Shrink

A vertical shrink is a type of transformation that compresses the graph of a function towards the x-axis.
In the function \(f(x) = \frac{1}{2}|x|\), the coefficient \(\frac{1}{2}\) before the absolute value function acts as a scaling factor.
This means every y-coordinate of the parent function \(|x|\) is multiplied by \(\frac{1}{2}\), effectively halving the height of the graph at each point.

• The V-shape remains.
• The angles become less steep.
• No horizontal shift occurs.

This vertical shrink by a factor of \(\frac{1}{2}\) ensures that the graph maintains its original shape but with reduced vertical height, making it appear flatter compared to the parent function.
This transformation is crucial in shaping how graphs of absolute value functions are visually represented without distorting their core characteristics.