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 \( c = 1 \).
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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 \( (0, 1) \).
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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.
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