Question

Graph the equation y=−5(x)^2+1 . Which of the following statements about the graph is correct?(1 point)
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 1 comma 0 right parenthesis .

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 (0,0).

Answer 1

To analyze the equation \( y = -5x^2 + 1 \), we recognize that this is a quadratic equation in the standard form \( y = ax^2 + bx + c \), where \( a = -5 \), \( b = 0 \), and \( c = 1 \).

1. **Opening Direction**: Since the coefficient of \( x^2 \) (which is \( a = -5 \)) is negative, the parabola opens **downwards**.

2. **Vertex**: The vertex of a parabola given by the formula \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
Plugging in our values:
\[
x = -\frac{0}{2(-5)} = 0
\]
To find the y-coordinate of the vertex, substitute \( x = 0 \) back into the equation:
\[
y = -5(0)^2 + 1 = 1
\]
So the vertex of the parabola is at the point \( (0, 1) \).

Now, let's evaluate the given statements:

- **The parabola will open downwards.**
- **Correct**: This statement is true because \( a < 0 \).

- **The parabola will have a vertex at (1,0).**
- **Incorrect**: The vertex is at \( (0, 1) \).

- **The parabola will have a vertex at (−1,0).**
- **Incorrect**: The vertex is at \( (0, 1) \).

- **The parabola will have a vertex at (0,0).**
- **Incorrect**: The vertex is at \( (0, 1) \).

The **correct statement** is: **The parabola will open downwards.**