The given function is:

$f(x) = -\frac{1}{4} |x - 3|$

1. Graphing the Function:

This is a transformation of the absolute value function, $f(x) = |x|$, with the following modifications:

• The $(x - 3)$ term translates the graph horizontally to the right by 3 units.
• The $-\frac{1}{4}$ term reflects the graph across the x-axis and compresses it vertically by a factor of 4.

Steps to Graph:

• The vertex of the function is at $(3, 0)$.
• The slope to the left of $x = 3$ is $-\frac{1}{4}$, and to the right of $x = 3$, the graph also has a slope of $-\frac{1}{4}$.

2. Domain and Range:

Domain:

Since the function involves an absolute value and linear operations, the domain is all real numbers: $\text{Domain} = (-\infty, \infty)$

Range:

The graph opens downward due to the negative coefficient $-\frac{1}{4}$, and the highest point is at the vertex $(3, 0)$. The range extends from negative infinity up to 0: $\text{Range} = (-\infty, 0]$

Would you like me to graph this function, or is there anything you would like to clarify?

Related Questions:

1. How do vertical and horizontal transformations affect the graph of a function?
2. What is the significance of the negative coefficient in absolute value functions?
3. Can you describe how to find the vertex of a transformed absolute value function?
4. How does the slope affect the shape of absolute value functions on both sides of the vertex?
5. What are some examples of functions with restricted domains, and how are they determined?

Tip:

When graphing piecewise or absolute value functions, start by identifying the transformations (translations, reflections, stretches/compressions) to easily predict the shape.