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

Answer 1

To analyze the equation \( y = -5(x)^2 + 1 \), let's identify important features of the graph.

1. **Form of the quadratic equation**: The equation is in the standard form \( y = ax^2 + bx + c \), where in this case, \( a = -5 \), \( b = 0 \), and \( c = 1 \).

2. **Vertex**: The vertex of a quadratic equation \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Substituting \( b = 0 \) and \( a = -5 \):
\[
x = -\frac{0}{2 \times -5} = 0
\]
To find the \( y \)-coordinate of the vertex, substitute \( x = 0 \) back into the equation:
\[
y = -5(0)^2 + 1 = 1
\]
Thus, the vertex is at \( (0, 1) \).

3. **Direction of the parabola**: Since \( a = -5 \), which is negative, the parabola opens downwards.

Now, let's summarize:

- The vertex is at \( (0, 1) \).
- The parabola opens downwards.

Based on the response options given, the correct statements are:

- **None of the vertex options are correct**, since they all suggest a vertex at \( (0, 0) \), \( (1, 0) \), or \( (-1, 0) \).
- The correct statement about the direction of the parabola is: **The parabola will open downwards.**

So, the final answer is that the parabola will open downwards.