Question

Which of the following accurately describes what the graph of y=5x2+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.

Answer 1

Let's analyze the equation \( y = 5x^2 + 50x + 125 \) and its properties, particularly focusing on the repeated root \( x = -5 \).

1. **Determining the Direction of the Graph:**
The coefficient of \( x^2 \) is \( 5 \), which is positive. Therefore, the graph opens **upward**. This means Statement #1 is **false**.

2. **Finding the Vertex of the Parabola:**
To find the vertex, we can use the formula for the x-coordinate of the vertex in a quadratic equation \( ax^2 + bx + c \), given by \( x = -\frac{b}{2a} \).

Here, \( a = 5 \) and \( b = 50 \):
\[
x = -\frac{50}{2 \times 5} = -\frac{50}{10} = -5
\]
The vertex is at \( x = -5 \). This is exactly at the repeated root, and thus is not to the right of the x-axis. Therefore, Statement #2 is **false**.

3. **Examining the Root:**
Since \( x = -5 \) is a repeated root, the graph touches the x-axis at this point and does not cross it. Therefore, the graph touches the x-axis only once, which makes Statement #3 **true**.

In summary:
- Statement #1: **False** (the graph opens upward)
- Statement #2: **False** (the vertex is at \( x = -5 \))
- Statement #3: **True** (the graph touches the x-axis at one point, \( x = -5 \))