The function given is \( f(x) = |x - 5| + 4 \).

This is an absolute value function, and we can analyze its properties as follows:

### 1. **Vertex**:
   - The absolute value function \( |x - 5| \) has its vertex at \( x = 5 \), where the expression inside the absolute value equals zero.
   - Therefore, the vertex of the function \( f(x) \) is at \( (5, 4) \) because when \( x = 5 \), \( f(x) = |5 - 5| + 4 = 0 + 4 = 4 \).

### 2. **Shape**:
   - The graph of \( f(x) = |x - 5| + 4 \) is a "V" shape, with the point at \( (5, 4) \) being the lowest point (vertex) of the graph.
   - The function increases linearly with a slope of 1 for \( x > 5 \) and decreases linearly with a slope of -1 for \( x < 5 \).

### 3. **Domain**:
   - The domain of the function is all real numbers, \( x \in \mathbb{R} \), because there are no restrictions on the values \( x \) can take.

### 4. **Range**:
   - Since the lowest value of \( f(x) \) is 4 (which occurs at \( x = 5 \)), the range of the function is \( [4, \infty) \).

### 5. **Intervals of Increase and Decrease**:
   - The function decreases on the interval \( (-\infty, 5] \).
   - The function increases on the interval \( [5, \infty) \).

Would you like more details on any specific part of this function?

Here are 5 related questions to explore further:

1. How would the graph change if the function were \( f(x) = |x - 5| - 4 \)?
2. What are the effects of shifting the absolute value function horizontally or vertically?
3. How would the function change if it were multiplied by a negative number, such as \( f(x) = -|x - 5| + 4 \)?
4. Can you find the points where the function intersects the x-axis?
5. How would you express this function as a piecewise function?

**Tip:** When analyzing absolute value functions, always identify the vertex first as it helps to understand the entire graph's behavior.

Explore the properties of the absolute value function f(x) = |x - 5| + 4. This problem involves understanding the vertex, shape, domain, and range of the function, making it suitable for students in Grades 8-10.

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