Question

Given the equation y=−10x2+20x+80 with solutions of x=−2 and x=4 , which of the following identifies the general shape of its associated graph?(1 point) Responses The graph opens upward. The graph opens upward. The vertex is to the left of the y-axis. The vertex is to the left of the y -axis. The graph touches the x-axis exactly one time. The graph touches the x -axis exactly one time. The graph opens downward. The graph opens downward.

Answer 1

To analyze the equation \( y = -10x^2 + 20x + 80 \) and determine the shape of its graph, we need to consider several factors:

1. Leading Coefficient: The coefficient of the \( x^2 \) term is \(-10\). Since this coefficient is negative, it indicates that the parabola opens downward.

2. Solutions (Roots): The given solutions \( x = -2 \) and \( x = 4 \) are the x-intercepts of the graph. Since there are two distinct solutions, the graph will cross the x-axis at these two points.

3. Vertex: To find the vertex, we can use the formula for the x-coordinate of the vertex, which is given by \( x = -\frac{b}{2a} \), where \( a = -10 \) and \( b = 20 \). Plugging in the values: \[ x = -\frac{20}{2(-10)} = -\frac{20}{-20} = 1 \] The x-coordinate of the vertex is \( x = 1\), which is to the right of the y-axis, not to the left.

Based on this analysis, the correct identification of the general shape of the graph is:

• The graph opens downward.

So, the correct response is: The graph opens downward.