Question

Which of the following accurately describes what the graph of y=5x2+50x+125 looks like, with the repeated root of x=−5 ?(1 point) Responses The graph opens downward. The graph opens downward. The graph touches the x-axis once. The graph touches the x-axis once. The graph has a vertex to the right of the x-axis.

Answer 1

To analyze the graph of the quadratic function \( y = 5x^2 + 50x + 125 \), we can look at its key features.

1. Opening Direction: The coefficient of \( x^2 \) is positive (5), which means the graph opens upward.

2. Roots: The question states that there is a repeated root at \( x = -5 \). This indicates that the quadratic has a double root, meaning it touches the x-axis but does not cross it. This is consistent with the fact that the graph opens upward.

3. Vertex: The vertex of the quadratic function can be calculated using the vertex formula \( x = -\frac{b}{2a} \). In this case, \( a = 5 \) and \( b = 50 \):

   \[ x = -\frac{50}{2 \cdot 5} = -\frac{50}{10} = -5 \]

   The vertex is at \( x = -5 \), which is exactly on the x-axis.

So, based on these characteristics:

• The graph opens upward, not downward.
• The graph touches the x-axis once (since it has a double root at \( x = -5 \)).
• The vertex is at \( x = -5 \), which is not to the right of the x-axis.

With that analysis, the correct response based on the description of the graph would be that "The graph touches the x-axis once."