Which of the following graphs represents the solution set of the quadratic equation?

Question

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.

Answer 1

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 \( y \geq x^2 + 2 \).