The function f is defined by f (x) = (x + 3)(x + 1). The graph of f in the xy-plane is a parabola. Which of the following intervals contains the x-coordinate of the vertex of the graph of f ?

Options

A. −4 lt x lt −3
B. −3 lt x lt 1
C. 1 lt x lt 3
D. 3 lt x lt 4

Explanation

To find the x-coordinate of the vertex of the function \( f(x) = (x + 3)(x + 1) \), we can follow these steps:

### Step 1: Expand the function
First, we need to expand the function \( f(x) \).

\[ f(x) = (x + 3)(x + 1) = x^2 + 1x + 3x + 3 = x^2 + 4x + 3 \]

### Step 2: Identify coefficients
Now, we identify the coefficients \( a \) and \( b \) from the standard form of a quadratic equation \( ax^2 + bx + c \). Here, we have:

- \( a = 1 \)
- \( b = 4 \)
- \( c = 3 \)

### Step 3: Use the vertex formula
The x-coordinate of the vertex of a parabola given by the equation \( y = ax^2 + bx + c \) can be found using the formula:

\[ x = -\frac{b}{2a} \]

Substituting the values of \( a \) and \( b \):

\[ x = -\frac{4}{2 \cdot 1} = -\frac{4}{2} = -2 \]

### Step 4: Determine the interval
Now, we need to find which of the given intervals contains the x-coordinate of the vertex, which we found to be \( -2 \).

Assuming the intervals provided are:
1. \( (-4, -3) \)
2. \( (-3, 1) \)
3. \( (-1, 0) \)

We see that \( -2 \) falls within the interval \( (-3, 1) \).

### Conclusion
Thus, the interval that contains the x-coordinate of the vertex of the graph of \( f \) is \( (-3, 1) \).