Graph each inequality. y < x² - 1

Verified step by step guidance

1. 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 \(y = x^{2} - 1\).
2. Graph the boundary curve \(y = x^{2} - 1\). This is a parabola opening upwards with its vertex at the point \((0, -1)\).
3. 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.
4. 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.
5. Shade the region on the graph that satisfies the inequality \(y < x^{2} - 1\), which will be the area below the dashed parabola curve.

Key Concepts

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.