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)

Question

Responses

The graph opens downward.
The graph opens downward.

The vertex is to the left of the y-axis.
The vertex is to the left of the y -axis.

The graph opens upward.
The graph opens upward.

The graph touches the x-axis exactly one time.

Answer 1

The given equation is a quadratic function of the form \(y = ax^2 + bx + c\), where \(a = -10\), \(b = 20\), and \(c = 80\).

1. Direction of Opening: Since the coefficient of \(x^2\) (which is \(a = -10\)) is negative, the graph of the quadratic opens downward.

2. Solutions: The equation has solutions (or roots) at \(x = -2\) and \(x = 4\). This means the graph intersects the x-axis at these points.

3. Vertex: The vertex lies midway between the roots of the quadratic. The midpoint between \(x = -2\) and \(x = 4\) is: \[ x_{vertex} = \frac{-2 + 4}{2} = 1 \] This means the vertex is located at \(x = 1\), which is to the right of the y-axis (not to the left).

4. Touching the x-axis: Since there are two distinct real roots, the graph intersects the x-axis at two points, meaning it touches the x-axis exactly two times, not once.

Based on this analysis, the appropriate response regarding the general shape of the associated graph is:

The graph opens downward.