To analyze the equation \( y = -5(x)^2 + 1 \), let's break it down:
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Graphing the Equation: The equation is in the standard quadratic form \( y = ax^2 + bx + c \), where \( a = -5 \), \( b = 0 \), and \( c = 1 \).
- Since \( a < 0 \), the parabola opens downwards.
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Finding the Vertex: The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
- Here, \( b = 0 \). Therefore, the x-coordinate of the vertex is: \[ x = -\frac{0}{2(-5)} = 0 \]
- To find the y-coordinate of the vertex, we substitute \( x = 0 \) back into the equation: \[ y = -5(0)^2 + 1 = 1 \]
- Thus, the vertex is at \( (0, 1) \).
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Statements Assessment:
- The first statement, "The parabola will open downwards," is correct.
- The second statement, "The parabola will have a vertex at (1, 0)," is incorrect.
- The third statement, "The parabola will have a vertex at (−1, 0)," is incorrect.
- The fourth statement, "The parabola will have a vertex at (0, 0)," is also incorrect.
Based on the analysis, the correct answer is:
The parabola will open downwards.