Graph the given functions, f and g, in the same rectangular coordinate system. Describe how the graph of g is related to the graph of f.
f(x) =| x|
g(x) = |x|-5

We are given two functions to graph and compare:

1. **\( f(x) = |x| \)**, which is the absolute value function.
2. **\( g(x) = |x| - 5 \)**, which is a transformation of the absolute value function.

### Step 1: Graph of \( f(x) = |x| \)
The graph of \( f(x) = |x| \) is a V-shaped graph with its vertex at the origin (0,0). It has two linear pieces:
- For \( x \geq 0 \), \( f(x) = x \), which is a line with a slope of 1.
- For \( x < 0 \), \( f(x) = -x \), which is a line with a slope of -1.

### Step 2: Graph of \( g(x) = |x| - 5 \)
The function \( g(x) = |x| - 5 \) is obtained by shifting the graph of \( f(x) \) downward by 5 units. This is because the "-5" outside the absolute value shifts the entire graph vertically.

### Step 3: Relationship between the Graphs
- The graph of \( g(x) \) is the same V-shape as \( f(x) \), but it is shifted down by 5 units. 
- The vertex of \( g(x) \) is at \( (0, -5) \), whereas the vertex of \( f(x) \) is at \( (0, 0) \).
- The slopes of the two arms of the V-shape for both functions remain the same as in \( f(x) \).

I can also generate the graph if needed. Would you like a visual representation of these functions? 

### Relative Questions:
1. What other transformations can we apply to absolute value functions?
2. How does shifting a graph horizontally differ from shifting it vertically?
3. How would the graph of \( g(x) = |x| + 5 \) compare to \( f(x) = |x| \)?
4. How does the slope of the arms of the V-shaped graph affect its appearance?
5. What happens to the graph of \( f(x) = |x| \) if we reflect it across the x-axis?

**Tip**: Vertical shifts in a function do not affect the slope or the horizontal position of the graph, only the vertical positioning.

Graph the given functions, f and g, in the same rectangular coordinate system. Describe how the graph of g is related to the graph of f.
f(x) = |x|
g(x) = |x| - 5

Learn how to graph the absolute value functions f(x) = |x| and g(x) = |x| - 5 on the same coordinate plane. Explore how the graph of g(x) is related to f(x) by applying vertical shifts. Suitable for students in Grades 8-10.

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