Question
Which of the following graphs represents the solution set of the quadratic equation?
y≥x2+2
(1 point)
Responses
The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 6 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma 6 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 6 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma 6 right parenthesis. There is shading below the curve.
The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens downward and goes through the points left parenthesis negative 2 comma, negative 1 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma negative 1 right parenthesis. There is shading above the curve.
Image with alt text: The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens downward and goes through the points left parenthesis negative 2 comma, negative 1 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma negative 1 right parenthesis. There is shading above the curve.
The coordinate grid has -axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 6 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma 6 right parenthesis. There is shading above the curve.
Image with alt text: The coordinate grid has -axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 6 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma 6 right parenthesis. There is shading above the curve.
The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a dashed curve opens upward and goes through the points left parenthesis negative 2 comma 6 right parenthesis; left parenthesis 0 comma 2 right parenthesis and left parenthesis 2comma 6 right parenthesis. There is shading above the curve.
y≥x2+2
(1 point)
Responses
The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 6 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma 6 right parenthesis. There is shading below the curve.
Image with alt text: The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 6 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma 6 right parenthesis. There is shading below the curve.
The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens downward and goes through the points left parenthesis negative 2 comma, negative 1 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma negative 1 right parenthesis. There is shading above the curve.
Image with alt text: The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens downward and goes through the points left parenthesis negative 2 comma, negative 1 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma negative 1 right parenthesis. There is shading above the curve.
The coordinate grid has -axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 6 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma 6 right parenthesis. There is shading above the curve.
Image with alt text: The coordinate grid has -axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a solid curve opens upward and goes through the points left parenthesis negative 2 comma 6 right parenthesis; left parenthesis 0 comma 2 right parenthesis; and left parenthesis 2 comma 6 right parenthesis. There is shading above the curve.
The coordinate grid has x axis from negative 5 to 5 and y axis labeled from negative 3 to 7. The graph of a dashed curve opens upward and goes through the points left parenthesis negative 2 comma 6 right parenthesis; left parenthesis 0 comma 2 right parenthesis and left parenthesis 2comma 6 right parenthesis. There is shading above the curve.
Answers
GPT-4o mini
To solve the inequality \( y \geq x^2 + 2 \), we need to analyze the equation \( y = x^2 + 2 \).
1. **Identify the graph of \( y = x^2 + 2 \)**: This is a quadratic function that opens upwards, with a vertex at (0, 2). It passes through the point (0, 2) and the points (2, 6) and (-2, 6).
2. **Determine the shading for the inequality**: Since the inequality is \( y \geq x^2 + 2 \), we shade *above* the curve to represent all points where \( y \) is greater than or equal to the function.
Now, let's match this with the given responses:
- **First option**: The curve opens upwards with points (0, 2), (2, 6), and (-2, 6), with shading below the curve. This is incorrect.
- **Second option**: The curve opens downwards, which does not match our function, and also has shading above. This is incorrect.
- **Third option**: The curve opens upwards, matches the points (0, 2), (2, 6), and (-2, 6), but it has shading above the curve. This is correct.
- **Fourth option**: This one describes a dashed curve, which typically signifies that points on the curve are not included in the solution, but since the inequality is \( \geq \), it should be a solid curve. Thus, this option is incorrect.
The correct option is the **third option**, which represents the solution set of the quadratic equation \( y \geq x^2 + 2 \).
1. **Identify the graph of \( y = x^2 + 2 \)**: This is a quadratic function that opens upwards, with a vertex at (0, 2). It passes through the point (0, 2) and the points (2, 6) and (-2, 6).
2. **Determine the shading for the inequality**: Since the inequality is \( y \geq x^2 + 2 \), we shade *above* the curve to represent all points where \( y \) is greater than or equal to the function.
Now, let's match this with the given responses:
- **First option**: The curve opens upwards with points (0, 2), (2, 6), and (-2, 6), with shading below the curve. This is incorrect.
- **Second option**: The curve opens downwards, which does not match our function, and also has shading above. This is incorrect.
- **Third option**: The curve opens upwards, matches the points (0, 2), (2, 6), and (-2, 6), but it has shading above the curve. This is correct.
- **Fourth option**: This one describes a dashed curve, which typically signifies that points on the curve are not included in the solution, but since the inequality is \( \geq \), it should be a solid curve. Thus, this option is incorrect.
The correct option is the **third option**, which represents the solution set of the quadratic equation \( y \geq x^2 + 2 \).