Question

Which of the following accurately describes what the graph of y = 5x^2 + 50x + 125 looks like, with the repeated root of x = -5?
Statement #1: The graph opens downward.
Statement #2: The graph has a vertex to the right of the x-axis.
Statement #3: The graph touches the x-axis once.

Select the correct statement: Statement # \(\square\)

Answer 1

To analyze the graph of the quadratic function \(y = 5x^2 + 50x + 125\) with the repeated root at \(x = -5\), we can evaluate each of the statements given.

1. **Determining if the graph opens downward**:
The leading coefficient of \(x^2\) in the equation \(y = 5x^2 + 50x + 125\) is \(5\), which is positive. This means that the graph opens **upward**.
- **Statement #1**: The graph opens downward. **(False)**

2. **Finding the vertex**:
The vertex of a parabola in the form \(y = ax^2 + bx + c\) can be found using the vertex formula \(x = -\frac{b}{2a}\). Here, \(a = 5\) and \(b = 50\).
\[
x = -\frac{50}{2 \cdot 5} = -\frac{50}{10} = -5
\]
The vertex is at \(x = -5\), which is to the left of the x-axis (i.e., less than \(0\)).
- **Statement #2**: The graph has a vertex to the right of the x-axis. **(False)**

3. **Understanding the x-axis intersections**:
A quadratic function with a repeated root means it touches the x-axis at a single point. Since \(x = -5\) is the repeated root, the graph will indeed touch the x-axis at this point.
- **Statement #3**: The graph touches the x-axis once. **(True)**

Based on this analysis, the correct statement is:

**Statement #3**: The graph touches the x-axis once.

So the correct answer is Statement # \(3\).