Graph each inequality. y < 3x² + 2

Verified step by step guidance

1. Identify the boundary equation by replacing the inequality symbol with an equal sign: \(y = 3x^{2} + 2\).
2. Graph the parabola \(y = 3x^{2} + 2\). Since the coefficient of \(x^{2}\) is positive, the parabola opens upward, and the vertex is at the point \((0, 2)\).
3. Determine the type of boundary line. Because the inequality is strictly less than (\(<\)), draw the parabola as a dashed curve to indicate that points on the curve are not included in the solution.
4. Choose a test point not on the boundary to determine which side of the parabola to shade. A common test point is \((0, 0)\).
5. Substitute the test point into the inequality \(y < 3x^{2} + 2\) to check if it satisfies the inequality. If it does, shade the region containing the test point; if not, shade the opposite side.

Key Concepts

Graphing Quadratic Functions

A quadratic function is a polynomial of degree two, typically written as y = ax² + bx + c. Its graph is a parabola that opens upward if a > 0 and downward if a < 0. Understanding the shape and position of the parabola y = 3x² + 2 helps in graphing the related inequality.

Inequalities in Two Variables

An inequality like y < 3x² + 2 represents all points (x, y) where y is less than the quadratic expression. Graphing this involves shading the region below the parabola y = 3x² + 2, indicating all solutions that satisfy the inequality.

Boundary Curves and Shading

The boundary curve y = 3x² + 2 separates the plane into two regions. Since the inequality is strict (y < 3x² + 2), the boundary is dashed to show points on the curve are not included. Proper shading below the curve visually represents the solution set.