Graph each inequality. y \u003c 3x2 + 2

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 21

6장, 문제 21

Graph each inequality. y < 3x² + 2

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Identify the boundary equation by replacing the inequality symbol with an equal sign: \(y = 3x^{2} + 2\).

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 \((0, 2)\).

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.

Choose a test point not on the boundary to determine which side of the parabola to shade. A common test point is \((0, 0)\).

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Graph each inequality. y < 3x2 + 2