To determine the shape of the graph associated with the equation \( y = -10x^2 + 20x + 80 \), we can analyze the equation, which is a quadratic equation of the form \( y = ax^2 + bx + c \).
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Coefficient of \( x^2 \): The coefficient \( a \) is \(-10\), which is negative. This means that the parabola opens downward.
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Roots/Solutions: The given solutions \( x = -2 \) and \( x = 4 \) indicate that the graph intersects the x-axis at these points. Since the equation has two real solutions (roots), the graph does not touch the x-axis exactly one time; it crosses it at two points.
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Vertex Location: The vertex of the parabola opens downward, and it can be found by using the formula \( x = -\frac{b}{2a} \), where \( b = 20 \) and \( a = -10 \): \[ x = -\frac{20}{2 \times -10} = 1 \] Since the vertex \( x \) value (1) is to the right of the y-axis, we cannot conclude that the vertex is to the left of the y-axis.
Based on this analysis, the correct response regarding the general shape of the associated graph is:
The graph opens downward.