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 > 3The 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 where the function \( f(x) = (x - 3)(x + 1) \) is positive and decreasing, let's break it down:

1. **Identify the roots**: The function has roots at \( x = -1 \) and \( x = 3 \) because that is where \( f(x) = 0 \).

2. **Vertex of the parabola**: The vertex of the parabola \( f(x) = (x - 3)(x + 1) \) occurs at \( x = 1 \), where \( f(1) = (1 - 3)(1 + 1) = (-2)(2) = -4 \). The vertex is at the point \( (1, -4) \).

3. **Nature of the parabola**: Since the parabola opens upwards (the coefficient of \( x^2 \) is positive), it will be:
- Negative between the roots \( (-1, 3) \)
- Positive outside the roots (i.e., when \( x < -1 \) and \( x > 3 \))

4. **Determine where the graph is decreasing**:
- The function decreases from \( x = -1 \) to \( x = 1 \) (it reaches its minimum at the vertex).
- The function starts increasing again after the vertex at \( x = 1 \).

5. **Finding the interval**:
- From the analysis:
- The graph is **positive** in the intervals \( x < -1 \) and \( x > 3 \).
- The graph is **decreasing** in the interval \( x < 1 \).
- The only overlap where the graph is both positive and decreasing is for \( x < -1 \).

Thus, the correct description for all of the values for which the graph is positive and decreasing is:

**all real values of x where \( x < -1 \)**.