Question

graph the equation y = -5(x)^2 + 1. which of the following statements about the graph is correct
a. the parabola will have a vertex at (1, 0).
b. the parabola will have a vertex at (-1, 0).
c. the parabola will have a vertex at (0, 0).
d. the parabola will open downwards

Answer 1

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

1. **Vertex**: The vertex of a quadratic function \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]

Plugging in \( a \) and \( b \):
\[
x = -\frac{0}{2 \cdot -5} = 0
\]

To find the corresponding \( y \)-coordinate of the vertex, substitute \( x = 0 \) back into the equation:
\[
y = -5(0)^2 + 1 = 1
\]

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

2. **Direction of Opening**: Since \( a = -5 \) (a negative value), the parabola opens downwards.

Now let's evaluate the statements:

a. The parabola will have a vertex at (1, 0). **False** (vertex is at (0, 1))
b. The parabola will have a vertex at (-1, 0). **False** (vertex is at (0, 1))
c. The parabola will have a vertex at (0, 0). **False** (vertex is at (0, 1))
d. The parabola will open downwards. **True**

Therefore, the correct statement is:
**d. the parabola will open downwards.**