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

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 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.

GPT-4o mini · Answer

The given equation is a quadratic function of the form $y = a x^{2} + b x + c$ , where $a = - 10$ , $b = 20$ , and $c = 80$ .

Direction of Opening: Since the coefficient of $x^{2}$ (which is $a = - 10$ ) is negative, the graph of the quadratic opens downward.
Solutions: The equation has solutions (or roots) at $x = - 2$ and $x = 4$ . This means the graph intersects the x-axis at these points.
Vertex: The vertex lies midway between the roots of the quadratic. The midpoint between $x = - 2$ and $x = 4$ is: $x_{v e r t e x} = \frac{- 2 + 4}{2} = 1$ This means the vertex is located at $x = 1$ , which is to the right of the y-axis (not to the left).
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.