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.

(1 point)
Statement #

Answer 1

Let’s analyze the given quadratic function \( y = 5x^2 + 50x + 125 \):

1. **Determining the Direction of the Graph:**
The leading coefficient (the coefficient of \( x^2 \)) is \( 5 \), which is positive. This means the graph opens **upward**, not downward. Therefore, **Statement #1** is **false**.

2. **Finding the Vertex:**
To find the vertex, we can use 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 **not** to the right of the x-axis (it's at \( x = -5 \)). Hence, **Statement #2** is also **false**.

3. **Identifying the Roots:**
The statement mentions that there is a repeated root at \( x = -5 \). Since it mentions that the graph touches the x-axis once, this is indeed true because a repeated root signifies that the graph just touches the x-axis at that point rather than crossing it. Therefore, **Statement #3** is **true**.

In summary:
- Statement #1: False (Graph opens upward)
- Statement #2: False (Vertex is at \( x = -5 \))
- Statement #3: True (Graph touches the x-axis at \( x = -5 \))

Thus, the accurate description is that only **Statement #3** is true.