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 vertex is to the left of the y -axis.

The graph opens upward.

The graph opens downward.

The graph touches the x-axis exactly one time.

Answer 1

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

1. **Direction of opening**: Since the coefficient \( a \) is negative (\( a = -10 \)), the graph opens downward.

2. **Vertex location**: To find the vertex, we can use the formula for the x-coordinate of the vertex \( x = -\frac{b}{2a} = -\frac{20}{2(-10)} = 1 \). Since \( 1 \) is to the right of the y-axis, we can conclude that the vertex is to the right of the y-axis.

3. **Solutions (roots)**: The equation has two solutions \( x = -2 \) and \( x = 4 \), which means the graph intersects the x-axis at two distinct points (as it is a quadratic with real solutions).

Based on the above reasoning, the correct response from the options provided is:

**The graph opens downward.**