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 \( c = 1 \).
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Opening Direction: Since the coefficient of \( x^2 \) (which is \( a = -5 \)) is negative, the parabola opens downwards.
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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:
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The parabola will open downwards.
- Correct: This statement is true because \( a < 0 \).
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The parabola will have a vertex at (1,0).
- Incorrect: The vertex is at \( (0, 1) \).
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The parabola will have a vertex at (−1,0).
- Incorrect: The vertex is at \( (0, 1) \).
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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.
1 answer