To analyze the quadratic equation \( y = -2x^2 + 4x + 6 \), we can evaluate each statement one by one.
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Finding the zeros: To find the zeros, we set \( y = 0 \): \[ -2x^2 + 4x + 6 = 0 \] Dividing the entire equation by -2 to make calculations easier: \[ x^2 - 2x - 3 = 0 \] Factoring: \[ (x - 3)(x + 1) = 0 \] Therefore, the zeros are \( x = 3 \) and \( x = -1 \), which corrects statement A to be true.
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Axis of symmetry: The axis of symmetry for a quadratic \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = -2 \) and \( b = 4 \): \[ x = -\frac{4}{2(-2)} = -\frac{4}{-4} = 1 \] Therefore, statement B is false.
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Finding the vertex (maximum/minimum): The vertex is at the axis of symmetry \( x = 1 \). We substitute \( x = 1 \) into the equation to find the y value: \[ y = -2(1^2) + 4(1) + 6 = -2 + 4 + 6 = 8 \] Since the coefficient of \( x^2 \) is negative, this indicates a maximum. Thus, the maximum value is indeed 8, making statement C true.
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Minimum value: Since the parabola opens downwards, there is no minimum value. Thus, statement D is false.
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Y-intercept: The y-intercept occurs when \( x = 0 \): \[ y = -2(0^2) + 4(0) + 6 = 6 \] This means the y-intercept is \( (0, 6) \), making statement E also true.
In conclusion, the true statements are:
- A: True (zeros at -1 and 3)
- C: True (maximum at 8)
- E: True (y-intercept at (0, 6))
So, the answer is that statements A, C, and E are true.
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