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 graph touches the x-axis exactly one time.
The graph touches the x -axis exactly one time.

The graph opens upward.
The graph opens upward.

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

Answer 1

To determine the shape of the graph associated with the equation \(y = -10x^2 + 20x + 80\), let's analyze the equation:

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

2. Find the number of solutions (roots): The given solutions are \(x = -2\) and \(x = 4\). This means the graph intersects the x-axis at these two points, so it touches the x-axis exactly two times (not one time).

3. Determine the vertex position: The vertex of the parabola can be found using the formula \(x = -\frac{b}{2a}\), where \(a = -10\) and \(b = 20\). Here: \[ x = -\frac{20}{2 \times -10} = \frac{20}{20} = 1 \] The x-coordinate of the vertex is \(1\), which is to the right of the y-axis.

From this analysis, we can conclude:

• The graph opens downward.
• The graph touches the x-axis exactly two times.
• The graph opens upward is incorrect since we've established it opens downward.
• The vertex is at \(x = 1\), which is to the right of the y-axis.

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