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

Question

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.

Answer 1

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.