Question

The graph of the function f(x) = (x − 3)(x + 1) is shown.

On a coordinate plane, a parabola opens up. It goes through (negative 1, 0), has a vertex at (1, negative 4), and goes through (3, 0).
Which describes all of the values for which the graph is positive and decreasing?

all real values of x where x < −1
all real values of x where x < 1
all real values of x where 1 < x < 3
all real values of x where x > 3

Answer 1

To analyze the function \( f(x) = (x - 3)(x + 1) \), we need to consider the properties of the parabola: its roots, vertex, and intervals where it is positive or negative, as well as where it is increasing or decreasing.

### Step 1: Identify the roots
The roots of the function are at \( x = -1 \) and \( x = 3 \), where the function intersects the x-axis:
- \( f(-1) = 0 \)
- \( f(3) = 0 \)

### Step 2: Find the vertex
The vertex is given at \( (1, -4) \). A parabola that opens upwards (since the leading coefficient is positive) will reach its minimum at the vertex.

### Step 3: Determine intervals
- **For \( x < -1 \)**:
- Choose a test point like \( x = -2 \):
\[
f(-2) = (-2 - 3)(-2 + 1) = (-5)(-1) = 5 \quad (\text{positive})
\]
- The function is positive.

- **For \( -1 < x < 1 \)**:
- Choose a test point like \( x = 0 \):
\[
f(0) = (0 - 3)(0 + 1) = (-3)(1) = -3 \quad (\text{negative})
\]
- The function is negative.

- **For \( 1 < x < 3 \)**:
- Choose a test point like \( x = 2 \):
\[
f(2) = (2 - 3)(2 + 1) = (-1)(3) = -3 \quad (\text{negative})
\]
- The function is negative.

- **For \( x > 3 \)**:
- Choose a test point like \( x = 4 \):
\[
f(4) = (4 - 3)(4 + 1) = (1)(5) = 5 \quad (\text{positive})
\]
- The function is positive.

### Step 4: Analyze increasing and decreasing intervals
The vertex \( (1, -4) \) is where the function transitions from decreasing to increasing.

- **Decreasing**: On the interval \( (-\infty, 1) \)
- The function is decreasing from \( (-\infty, -1) \) where it is positive, through \( (-1, 3) \) where it is negative up until the vertex.

- **Increasing**: From \( (1, \infty) \) where it is positive again.

### Conclusion
The function is positive in the intervals \( (-\infty, -1) \) and \( (3, \infty) \), but it is decreasing only in the interval \( (-\infty, 1) \).

### Final Answer
The correct description of all values for which the graph is positive and decreasing is:
**all real values of \( x \) where \( x < -1 \)**.