Question

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 function \( y = 5x^2 + 50x + 125 \) and assess the provided statements.

1. **Identify the nature of the roots**:
The function can be rewritten in the vertex form to find its vertex and determine the nature of its roots. However, we can also use the fact that it is a quadratic equation.

The formula for the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 5 \), \( b = 50 \), and \( c = 125 \).
We can calculate the discriminant:
\[
b^2 - 4ac = 50^2 - 4(5)(125) = 2500 - 2500 = 0
\]
Since the discriminant is 0, there is exactly one real root (a repeated root) at \( x = -5 \).

2. **Graph characteristics**:
- **Opening direction**: Since the leading coefficient \( a = 5 \) is positive, the graph opens upwards.
- **Vertex**: To find the vertex, use the formula for the x-coordinate of the vertex, given by:
\[
x = \frac{-b}{2a} = \frac{-50}{2 \times 5} = \frac{-50}{10} = -5
\]
Therefore, the vertex is at \( (-5, y(-5)) \). We can calculate \( y(-5) \):
\[
y(-5) = 5(-5)^2 + 50(-5) + 125 = 5(25) - 250 + 125 = 125 - 250 + 125 = 0
\]
This confirms that the vertex is at \( (-5, 0) \).

3. **Examine each statement**:
- **Statement #1**: The graph opens downward.
- **False**: The graph opens upward since the coefficient of \( x^2 \) is positive.

- **Statement #2**: The graph has a vertex to the right of the x-axis.
- **False**: The vertex \( (-5, 0) \) is exactly on the x-axis, not to the right.

- **Statement #3**: The graph touches the x-axis once.
- **True**: The graph touches the x-axis exactly at the repeated root \( x = -5 \).

Thus, the only accurate statement about the graph of \( y = 5x^2 + 50x + 125 \) is **Statement #3**. The other statements are false.