Let's analyze the given functions $f(x)$ and $g(x)$.

1. Function $f(x) = |x|$:

   • This is the absolute value function. It is defined as: x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$
   • The graph of $f(x) = |x|$ is a V-shaped curve with its vertex at the origin $(0, 0)$, opening upwards.

2. Function $g(x) = |x - 4| - 4$:

   • This function involves a transformation of the basic absolute value function $|x|$.
   • First, consider $|x - 4|$: x - 4 & \text{if } x \geq 4 \\ 4 - x & \text{if } x < 4 \end{cases}$$
   • Then subtract 4 from $|x - 4|$: $g(x) = |x - 4| - 4$
   • This translates the graph of $|x|$ to the right by 4 units and then shifts it downward by 4 units.

Analyzing $g(x)$ in Detail

• To understand $g(x)$ better, let's break it into cases based on the definition of $|x - 4|$:

1. For $x \geq 4$: $g(x) = |x - 4| - 4 = (x - 4) - 4 = x - 8$

2. For $x < 4$: $g(x) = |x - 4| - 4 = (4 - x) - 4 = -x$

So, we can write $g(x)$ as a piecewise function:

x - 8 & \text{if } x \geq 4 \\ - x & \text{if } x < 4 \end{cases}$$ ### Graphing $$g(x)$$ - **For $$x < 4$$:** The function $$g(x) = -x$$ is a straight line with a slope of -1 passing through the points $$(4, 0)$$ and extending leftward. - **For $$x \geq 4$$:** The function $$g(x) = x - 8$$ is a straight line with a slope of 1 passing through the point $$(4, -4)$$ and extending rightward. ### Comparison and Transformation - The graph of $$g(x) = |x - 4| - 4$$ is obtained by: 1. Shifting $$|x|$$ horizontally to the right by 4 units to get $$|x - 4|$$. 2. Then shifting $$|x - 4|$$ vertically downward by 4 units to get $$|x - 4| - 4$$. ### Summary - $$f(x) = |x|$$ is a V-shaped curve with vertex at $$(0,0)$$. - $$g(x) = |x - 4| - 4$$ is a V-shaped curve with vertex at $$(4, -4)$$. Would you like to see a detailed graph of these functions, or do you have any specific questions about these transformations? ### Related Questions 1. How do you graph transformations of absolute value functions? 2. What are the properties of the absolute value function? 3. How does shifting a function horizontally affect its graph? 4. How does shifting a function vertically affect its graph? 5. What are piecewise functions and how are they defined? 6. How do you find the vertex of an absolute value function? 7. How do you determine the slope of linear segments in a piecewise function? 8. Can you provide examples of other transformed absolute value functions? ### Tip When analyzing transformations of functions, remember that horizontal shifts involve changes within the function's argument, while vertical shifts involve changes outside of the function.