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 graph touches the x-axis exactly one time. The graph touches the x -axis exactly one time. The vertex is to the left of the y-axis. The vertex is to the left of the y -axis. The graph opens downward. The graph opens downward.

Answer 1

To determine the general shape of the graph of the quadratic equation \( y = -10x^2 + 20x + 80 \), we can analyze its features based on the coefficients and given solutions.

1. **Direction of Opening**: The coefficient of \( x^2 \) is -10, which is negative. This means that the graph opens downward.

2. **Number of x-intercepts**: Given that the solutions (or roots) of the equation are \( x = -2 \) and \( x = 4 \), it indicates that the graph crosses the x-axis at these two points, which means it touches the x-axis exactly two times (once at each root).

3. **Vertex Position**: The vertex of a parabola defined by the equation \( y = ax^2 + bx + c \) is located at \( x = -\frac{b}{2a} \). Here, \( a = -10 \) and \( b = 20 \):
\[
x = -\frac{20}{2 \times -10} = -\frac{20}{-20} = 1
\]
Since the vertex x-coordinate (1) is to the right of the y-axis, it doesn't help in confirming that the vertex is to the left of the y-axis.

Based on this analysis:

- **The graph opens downward.** (This is the correct identification of the general shape.)
- It touches the x-axis two times.
- The vertex is to the right of the y-axis.

Thus, the correct answer from the choices given is: **The graph opens downward.**