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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblema 19
Capitolo 6, Problema 19

Graph each inequality. y < x2 - 1

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Identify the inequality given: \(y < x^{2} - 1\). This represents all points \((x, y)\) where the \(y\)-value is less than the value of the parabola \(y = x^{2} - 1\).
Graph the boundary curve \(y = x^{2} - 1\). This is a parabola opening upwards with its vertex at the point \((0, -1)\).
Since the inequality is strictly less than ($<$) and not less than or equal to (\(\leq\)), draw the parabola as a dashed curve to indicate that points on the curve are not included in the solution.
Determine which side of the parabola to shade. Choose a test point not on the parabola, such as \((0, 0)\), and substitute into the inequality: check if \(0 < 0^{2} - 1\), which simplifies to \(0 < -1\). Since this is false, the region containing \((0, 0)\) is not part of the solution.
Shade the region on the graph that satisfies the inequality \(y < x^{2} - 1\), which will be the area below the dashed parabola curve.

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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 = x² - 1 helps in visualizing the boundary for the inequality.
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Inequalities and Their Graphs

Graphing an inequality like y < x² - 1 involves shading the region of the coordinate plane where the inequality holds true. The boundary curve y = x² - 1 is drawn as a dashed line because the inequality is strict (<), indicating points on the curve are not included.
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Linear Inequalities

Test Points for Shading Regions

To determine which side of the boundary to shade, select a test point not on the curve (often (0,0)) and substitute it into the inequality. If the inequality is true for that point, shade the region containing it; otherwise, shade the opposite side. This ensures the correct solution set is represented.
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