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Un tractorista limpia un terreno siguiendo una 
trayectoria definida por la siguiente función como 
se muestra en la figura:
F(x) = –2|x – 3| + 16
Calcule «a + b + c + d».


We are given the function \( F(x) = -2|x - 3| + 16 \), which describes the trajectory of a tractor clearing a field, as shown in the figure. The question asks for the sum \( a + b + c + d \), where these variables correspond to specific points and segments on the graph.

### Step-by-step solution:

1. **Understand the function**: 
   - \( F(x) = -2|x - 3| + 16 \) is a piecewise linear function (an absolute value function), which forms a "V" shape on the graph.
   - The vertex of the function occurs at \( x = 3 \), making the vertex point \( (3, 16) \).

2. **Find the intercepts**:
   - The function will intersect the x-axis (where \( F(x) = 0 \)) at two points. To find these, solve for \( x \) when \( F(x) = 0 \):

   \[
   0 = -2|x - 3| + 16
   \]
   \[
   2|x - 3| = 16
   \]
   \[
   |x - 3| = 8
   \]

   This gives two solutions:
   \[
   x - 3 = 8 \quad \Rightarrow \quad x = 11
   \]
   \[
   x - 3 = -8 \quad \Rightarrow \quad x = -5
   \]

   Therefore, the x-intercepts are \( (-5, 0) \) and \( (11, 0) \).

   These points correspond to \( c = -5 \) and \( d = 11 \).

3. **Determine the value of \( a \) and \( b \)**:
   - \( a \) corresponds to the y-coordinate of the vertex, which is \( a = 16 \).
   - \( b \) corresponds to the x-coordinate of the vertex, which is \( b = 3 \).

4. **Calculate the sum**:
   \[
   a + b + c + d = 16 + 3 + (-5) + 11 = 25
   \]

Thus, the correct answer is \( \boxed{25} \).

### Relative Questions:
1. How do you determine the vertex of an absolute value function algebraically?
2. What is the significance of the slope in the given piecewise function?
3. How do you find x-intercepts for functions involving absolute values?
4. Why does the graph of an absolute value function create a "V" shape?
5. How does the transformation of functions affect their graphs?

### Tip:
When dealing with absolute value functions, always identify the vertex first as it helps in understanding the shape and solving for intercepts.

Un tractorista limpia un terreno siguiendo una trayectoria definida por la siguiente función como se muestra en la figura: F(x) = –2|x – 3| + 16. Calcule «a + b + c + d».

Learn how to calculate the sum a + b + c + d for the absolute value function F(x) = –2|x – 3| + 16. This step-by-step solution explains how to find the vertex, x-intercepts, and sum. Suitable for high school math students.

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