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

To analyze the given quadratic function \( y = 5x^2 + 50x + 125 \), let's break down the statements based on the given information:

1. **Finding the vertex**: The standard form of a quadratic equation is \( y = ax^2 + bx + c \). Here, \( a = 5 \), \( b = 50 \), and \( c = 125 \).

The vertex \( x \)-coordinate can be found using the formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{50}{2 \cdot 5} = -\frac{50}{10} = -5.
\]
This means that the vertex is at \( x = -5 \).

2. **Opening direction**: Since \( a = 5 > 0 \), the parabola opens upward.

3. **Roots**: Since we have a repeated root at \( x = -5 \), we can conclude that the quadratic does touch the x-axis at this point, meaning the graph has exactly one x-intercept.

Now, we can evaluate the statements:

- **Statement #1**: "The graph opens downward."
- This is **false** because the parabola opens upward.

- **Statement #2**: "The graph has a vertex to the right of the x-axis."
- This is **false** because the vertex is at \( x = -5 \), which is to the left of the x-axis.

- **Statement #3**: "The graph touches the x-axis once."
- This is **true** since there is a repeated root at \( x = -5 \).

In conclusion, the only true statement is Statement #3.

So, the accurate description based on the given information is:
**Statement #3: True**.
Statements #1 and #2 are false.