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 vertex is to the left of the y -axis.
The graph touches the x-axis exactly one time.
The graph touches the x -axis exactly one time.
The graph opens downward.
The graph opens downward.
The graph opens upward.
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 vertex is to the left of the y -axis.
The graph touches the x-axis exactly one time.
The graph touches the x -axis exactly one time.
The graph opens downward.
The graph opens downward.
The graph opens upward.
Answers
GPT-4o mini
To analyze the given equation \( y = -10x^2 + 20x + 80 \), we can identify key characteristics:
1. **Shape of the Graph**: The equation is a quadratic function of the form \( y = ax^2 + bx + c \) where \( a = -10 \). Since \( a \) is negative, the graph opens downward.
2. **Roots/Solutions**: The solutions \( x = -2 \) and \( x = 4 \) indicate that the graph intersects the x-axis at these points. Since there are two distinct real roots, the graph touches the x-axis exactly two times.
3. **Vertex Location**: The vertex of a parabola described by \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{20}{2 \times -10} = 1
\]
This indicates that the vertex is at \( x = 1 \), which is to the right of the y-axis.
Given these analyses, the correct response to the question identifying the general shape of the associated graph would be:
**The graph opens downward.**
1. **Shape of the Graph**: The equation is a quadratic function of the form \( y = ax^2 + bx + c \) where \( a = -10 \). Since \( a \) is negative, the graph opens downward.
2. **Roots/Solutions**: The solutions \( x = -2 \) and \( x = 4 \) indicate that the graph intersects the x-axis at these points. Since there are two distinct real roots, the graph touches the x-axis exactly two times.
3. **Vertex Location**: The vertex of a parabola described by \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{20}{2 \times -10} = 1
\]
This indicates that the vertex is at \( x = 1 \), which is to the right of the y-axis.
Given these analyses, the correct response to the question identifying the general shape of the associated graph would be:
**The graph opens downward.**