Graph each inequality. y \u003e (x - 1)2 + 2

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 23

6장, 문제 23

Graph each inequality. y > (x - 1)² + 2

검증된 단계별 안내

Identify the boundary curve by rewriting the inequality as an equation: \(y = (x - 1)^{2} + 2\). This is a parabola opening upwards with vertex at \((1, 2)\).

Graph the parabola \(y = (x - 1)^{2} + 2\) on the coordinate plane. Since the inequality is strict (\(>\)), draw the parabola as a dashed curve to indicate points on the curve are not included.

Determine which side of the parabola to shade by testing a point not on the curve, such as the point \((1, 3)\). Substitute into the inequality: \(3 > (1 - 1)^{2} + 2\) simplifies to \(3 > 2\), which is true.

Since the test point satisfies the inequality, shade the region above the parabola where \(y\) values are greater than \((x - 1)^{2} + 2\).

Label the graph clearly, showing the dashed parabola and the shaded region above it to represent all solutions to the inequality \(y > (x - 1)^{2} + 2\).

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주요 개념

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

Graphing Quadratic Functions

A quadratic function is represented by an equation of the form y = ax² + bx + c, producing a parabola when graphed. Understanding the shape and vertex of the parabola y = (x - 1)² + 2 helps in visualizing the boundary curve for the inequality.

Inequalities and Their Graphs

Graphing inequalities involves shading the region of the coordinate plane where the inequality holds true. For y > (x - 1)² + 2, the area above the parabola is shaded, indicating all points where y-values are greater than the quadratic expression.

Boundary Lines and Open/Closed Curves

The boundary of the inequality y > (x - 1)² + 2 is the parabola y = (x - 1)² + 2. Since the inequality is strict (greater than, not greater than or equal to), the boundary curve is drawn as a dashed line to show points on the curve are not included in the solution.

Graph each inequality. y > (x - 1)2 + 2