Graph the equation y=−5(x)2+1 . Which of the following statements about the graph is correct?(1 point)

Question

Responses

The parabola will open downwards.
The parabola will open downwards.

The parabola will have a vertex at (−1,0).
The parabola will have a vertex at left parenthesis negative 1 comma 0 right parenthesis .

The parabola will have a vertex at (1,0).
The parabola will have a vertex at left parenthesis 1 comma 0 right parenthesis .

The parabola will have a vertex at (0,0).

Answer 1

To analyze the quadratic equation \(y = -5x^2 + 1\), we can identify its key features.

1. Direction of the Parabola: The coefficient of \(x^2\) is \(-5\), which is negative. This means the parabola opens downward.

2. Vertex of the Parabola: The general form of a quadratic equation is \(y = ax^2 + bx + c\). Here, \(a = -5\), \(b = 0\), and \(c = 1\). The vertex can be found using the formula \(x = -\frac{b}{2a}\). Substituting in the values:

   \[ x = -\frac{0}{2 \cdot -5} = 0 \]

   To find the y-coordinate of the vertex, substitute \(x = 0\) back into the equation:

   \[ y = -5(0)^2 + 1 = 1 \]

   So, the vertex is at \((0, 1)\).

Based on this analysis, we conclude that:

• The parabola indeed opens downwards.
• The vertex is at \((0, 1)\), not any of the points given in the options (including \( (−1,0) \), \( (1,0) \), or \( (0,0) \) ).

Thus, the correct response is:

The parabola will open downwards.