Graph the solution set of each system of inequalities. y ≥ x2 + 4x + 4 y \u003c -x2

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 51

6장, 문제 51

Graph the solution set of each system of inequalities.
y ≥ x² + 4x + 4
y < -x²

검증된 단계별 안내

Step 1: Identify the two inequalities in the system: \(y \geq x^{2} + 8x + 19\) and \(y < -(x - 1)^{2}\).

Step 2: Recognize that the first inequality represents the region on or above the parabola \(y = x^{2} + 8x + 19\), which opens upward.

Step 3: Recognize that the second inequality represents the region below the parabola \(y = -(x - 1)^{2}\), which opens downward and is shifted right by 1 unit.

Step 4: To graph the solution set, first sketch both parabolas: the upward-opening parabola \(y = x^{2} + 8x + 19\) and the downward-opening parabola \(y = -(x - 1)^{2}\).

Step 5: Shade the region on or above the first parabola and below the second parabola. The solution set is the intersection of these two shaded regions.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

16m

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주요 개념

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

Graphing Quadratic Inequalities

Graphing quadratic inequalities involves plotting the parabola defined by the quadratic equation and then shading the region that satisfies the inequality. For 'y ≥ f(x)', shade above or on the parabola, and for 'y < f(x)', shade below the parabola. The boundary line is solid if the inequality includes equality (≥ or ≤) and dashed if it does not (> or <).

Parabola Vertex and Transformations

The vertex form of a quadratic function, y = a(x - h)² + k, helps identify the vertex (h, k) and the direction the parabola opens. This is crucial for graphing and understanding the shape. For example, y < -(x - 1)² has a vertex at (1, 0) and opens downward, while y ≥ x² + 8x + 19 can be rewritten in vertex form to find its vertex and orientation.

Solution Set of a System of Inequalities

The solution set of a system of inequalities is the region where the shaded areas of all inequalities overlap. To find this, graph each inequality separately, then identify the common region that satisfies all conditions simultaneously. This intersection represents all points that make every inequality true.

Graph the solution set of each system of inequalities. y ≥ x2 + 4x + 4 y < -x2