Question

The graph of the function f(x) = (x – 4)(x + 1) is shown below.

On a coordinate plane, a parabola opens up. It goes through (negative 1, 0), has a vertex at (1.75, negative 6.2), and goes through (4, 0).

Which statement about the function is true?

The function is increasing for all real values of x where
x < 0.
The function is increasing for all real values of x where
x < –1 and where x > 4.
The function is decreasing for all real values of x where
–1 < x < 4.
The function is decreasing for all real values of x where
x < 1.5.

Answer 1

To analyze the function \( f(x) = (x - 4)(x + 1) \), we can first find its critical points and determine the intervals where it is increasing or decreasing.

### Step 1: Find the Roots
The roots (x-intercepts) of the function can be found by setting the function to zero:
\[
f(x) = (x - 4)(x + 1) = 0
\]
This gives us the roots \( x = -1 \) and \( x = 4 \).

### Step 2: Determine the Vertex
The vertex of a parabola given by a quadratic function in standard form, \( ax^2 + bx + c \), can be calculated using:
\[
x_v = -\frac{b}{2a}
\]
To convert \( f(x) \) into standard form, we expand:
\[
f(x) = x^2 - 3x - 4
\]
Here, \( a = 1 \) and \( b = -3 \). Thus:
\[
x_v = -\frac{-3}{2 \times 1} = \frac{3}{2} = 1.5
\]
Now that we know the x-coordinate of the vertex is at \( x = 1.5 \), we can compute the function value at the vertex to find its maximum or minimum point.

### Step 3: Determine the Function's Behavior
The parabola opens upwards (since \( a > 0 \)), indicating that the vertex is a minimum point.

The function will be:
- **Decreasing** on the interval \( (-\infty, x_v) \) or \( (-\infty, 1.5) \).
- **Increasing** on the interval \( (x_v, \infty) \) or \( (1.5, \infty) \).

### Step 4: Evaluate the Statements
1. **The function is increasing for all real values of x where \( x < 0 \)**: False (it's decreasing for \( x < 1.5 \)).
2. **The function is increasing for all real values of x where \( x < -1 \) and where \( x > 4 \)**: False (it’s decreasing to the left of \( x = 1.5 \)).
3. **The function is decreasing for all real values of x where \( -1 < x < 4 \)**: True (within this interval \( x = 4 \) onwards, it starts increasing).
4. **The function is decreasing for all real values of x where \( x < 1.5 \)**: True but not the most precise since it is only decreasing until reaching the vertex, after which it begins increasing.

### Conclusion
The correct statement is:
**The function is decreasing for all real values of x where \( -1 < x < 4 \).**