The sum of all the integral values of a {where `a in (-10, 10)}` such that the graph of the function `f(x)=||x-2|-a|-3` has exastly three x-intercepts is

Options

A. `10`
B. `5`
C. `3`
D. `0`

Answer

To solve the problem, we need to find the integral values of \( a \) such that the function \( f(x) = ||x - 2| - a| - 3 \) has exactly three x-intercepts. Let's break down the solution step by step.

### Step 1: Understand the function
The function can be rewritten as:
\[ f(x) = ||x - 2| - a| - 3 \]
This function involves absolute values, which means the graph will have certain characteristics based on the values of \( a \).

### Step 2: Analyze the inner absolute value
The expression \( |x - 2| \) has a vertex at \( x = 2 \) and is symmetric about this point. It equals zero when \( x = 2 \) and increases as \( x \) moves away from 2.

### Step 3: Determine the effect of \( a \)
The term \( |x - 2| - a \) will shift the graph of \( |x - 2| \) vertically. If \( a < 0 \), the graph shifts upwards, and if \( a > 0 \), it shifts downwards.

### Step 4: Identify conditions for x-intercepts
For the function \( f(x) \) to have exactly three x-intercepts, the graph must touch the x-axis at three distinct points. This occurs when the vertex of the graph of \( ||x - 2| - a| \) is positioned such that it intersects the line \( y = 3 \) at exactly three points.

### Step 5: Analyze the critical points
The critical points occur when:
1. \( |x - 2| - a = 3 \)
2. \( |x - 2| - a = -3 \)

From these equations, we can derive the values of \( x \):
1. \( |x - 2| = a + 3 \)
2. \( |x - 2| = a - 3 \)

### Step 6: Find the conditions for three x-intercepts
To have exactly three x-intercepts, the value of \( a \) must be such that:
1. \( a + 3 \) must be equal to the distance from \( 2 \) to the x-axis, which gives two solutions.
2. \( a - 3 \) must be equal to the distance from \( 2 \) to the x-axis, which gives one solution.

### Step 7: Solve for \( a \)
From the conditions:
- For \( a + 3 = 0 \) (the lowest point of the graph), we have \( a = -3 \) (not valid since \( a \) must be in the range (-10, 10)).
- For \( a - 3 = 0 \), we have \( a = 3 \).

### Step 8: Check the range of \( a \)
Since \( a \) must be in the range \( (-10, 10) \), we check:
- \( a = 3 \) is valid.

### Step 9: Sum of integral values of \( a \)
The only integral value of \( a \) that satisfies the condition is \( 3 \). Therefore, the sum of all integral values of \( a \) is:
\[ \text{Sum} = 3 \]

### Final Answer
The sum of all integral values of \( a \) such that the graph of the function \( f(x) \) has exactly three x-intercepts is \( 3 \).