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 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.
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.
Answers
GPT-4o mini
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.**
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.**