Graph each inequality. y \u003c x2 - 1

Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 19

Chapter 6, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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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Graph each inequality. y < x2 - 1

Verified video answer for a similar problem:

Key Concepts

Graphing Quadratic Functions

Inequalities and Their Graphs

Test Points for Shading Regions

Graph each inequality. y < x²- 1