In Exercises 1–26, graph each inequality. y < x² - 1

Verified step by step guidance

1. Identify the inequality given: . This means we are looking at all points (x, y) where y is less than the parabola defined by $y\; =\; x^2\; -\; 1$.
2. First, 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 strict (<), draw the parabola as a dashed curve to indicate that points on the curve are not included in the solution set.
4. Determine which side of the parabola to shade. Pick a test point not on the curve, such as (0,0), and substitute into the inequality: becomes , which is false. So, the region that satisfies the inequality is on the opposite side of the parabola from the test point (0,0).
5. Shade the region below the parabola (since y is less than $x^2\; -\; 1$) to represent all points (x, y) that satisfy the inequality.

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, which 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 inequalities involves shading the region of the coordinate plane where the inequality holds true. For y < x² - 1, you graph the parabola y = x² - 1 as a boundary and shade the area below it, indicating all points where y is less than the parabola's value.

Boundary Lines and Open/Closed Regions

When graphing inequalities, the boundary line (or curve) is drawn solid if the inequality includes equality (≤ or ≥) and dashed if it does not (< or >). Since the inequality is y < x² - 1, the parabola is dashed, showing points on the curve are not included in the solution.