Question
Which of the following graphs represents the solution set of the quadratic equation?
y≤2x2+2x+1
(1 point)
Responses
The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 3 right parenthesis; left parenthesis negative 0.5 comma negative 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; left parenthesis 1 comma 1 right parenthesis; and left parenthesis 2 comma negative 3 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 3 right parenthesis; left parenthesis negative 0.5 comma negative 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; left parenthesis 1 comma 1 right parenthesis; and left parenthesis 2 comma negative 3 right parenthesis. There is shading below the curve.
The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading below the curve.
The coordinate grid has x and y axes from negative 5 to 5. The graph of a dashed curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has x and y axes from negative 5 to 5. The graph of a dashed curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading below the curve.
The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading above the curve.
Image with alt text: The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading above the curve.
y≤2x2+2x+1
(1 point)
Responses
The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 3 right parenthesis; left parenthesis negative 0.5 comma negative 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; left parenthesis 1 comma 1 right parenthesis; and left parenthesis 2 comma negative 3 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens downward and goes through the points left parenthesis negative 1 comma negative 3 right parenthesis; left parenthesis negative 0.5 comma negative 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; left parenthesis 1 comma 1 right parenthesis; and left parenthesis 2 comma negative 3 right parenthesis. There is shading below the curve.
The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading below the curve.
The coordinate grid has x and y axes from negative 5 to 5. The graph of a dashed curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has x and y axes from negative 5 to 5. The graph of a dashed curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading below the curve.
The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading above the curve.
Image with alt text: The coordinate grid has x and y axes from negative 5 to 5. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 5 right parenthesis; left parenthesis negative 1 comma 1 right parenthesis; left parenthesis negative 0.5 comma 0.5 right parenthesis; left parenthesis 0 comma 1 right parenthesis; and left parenthesis 1 comma 5 right parenthesis. There is shading above the curve.
Answers
GPT-4o mini
To determine which graph represents the solution set of the given quadratic inequality \( y \leq 2x^2 + 2x + 1 \), we first need to analyze the quadratic equation \( y = 2x^2 + 2x + 1 \).
1. **Nature of the parabola**: The coefficient of \( x^2 \) is positive (2), which means the parabola opens upward.
2. **Shading**: Since the inequality is \( y \leq \), we will shade below the parabola.
3. **Graph characteristics**:
- The parabola represented by the equation should open upward.
- We need to find a graph with shading below the curve.
Based on the provided descriptions of the graphs:
- The **first graph** has a solid curve that opens downward with shading below. This does not fit as the parabola should open upward.
- The **second graph** has a solid curve opening upward and shading below. This is a possible match, but we need to check the points it passes through.
- The **third graph** has a dashed curve opening upward and shading below. Since the inequality is \( \leq \), this graph cannot be the solution.
- The **fourth graph** has a solid curve opening upward but shading above the curve. This also does not satisfy our requirements.
Now, we need to think through whether the **second graph**, which opens upward and has shading below, accurately represents the specific quadratic inequality.
Given that the equation \( y = 2x^2 + 2x + 1 \) has its vertex and roots, you can double-check how it plots with the points listed. Even if it passes through certain points, what we seek is an upward-opening solid parabola where shading is below.
Thus, the correct answer based on the analysis is:
**The second graph**, which has a solid curve that opens upward and shading below the curve.
1. **Nature of the parabola**: The coefficient of \( x^2 \) is positive (2), which means the parabola opens upward.
2. **Shading**: Since the inequality is \( y \leq \), we will shade below the parabola.
3. **Graph characteristics**:
- The parabola represented by the equation should open upward.
- We need to find a graph with shading below the curve.
Based on the provided descriptions of the graphs:
- The **first graph** has a solid curve that opens downward with shading below. This does not fit as the parabola should open upward.
- The **second graph** has a solid curve opening upward and shading below. This is a possible match, but we need to check the points it passes through.
- The **third graph** has a dashed curve opening upward and shading below. Since the inequality is \( \leq \), this graph cannot be the solution.
- The **fourth graph** has a solid curve opening upward but shading above the curve. This also does not satisfy our requirements.
Now, we need to think through whether the **second graph**, which opens upward and has shading below, accurately represents the specific quadratic inequality.
Given that the equation \( y = 2x^2 + 2x + 1 \) has its vertex and roots, you can double-check how it plots with the points listed. Even if it passes through certain points, what we seek is an upward-opening solid parabola where shading is below.
Thus, the correct answer based on the analysis is:
**The second graph**, which has a solid curve that opens upward and shading below the curve.