In Exercises 1–26, graph each inequality. y≥x²−9

Verified step by step guidance

1. Identify the inequality to graph: $y\ge x^2-9$. This represents all points where the y-coordinate is greater than or equal to $x^2\; -\; 9$.
2. Start by graphing the boundary curve $y\; =\; x^2\; -\; 9$. This is a parabola opening upwards with its vertex at the point $(0,\; -9)$.
3. Since the inequality is $y\; \ge \; x^2\; -\; 9$, the boundary line (the parabola) should be drawn as a solid curve to indicate that points on the parabola satisfy the inequality.
4. Choose a test point not on the parabola, such as $(0,\; 0)$, and substitute into the inequality: $0\; \ge \; 0^2\; -\; 9$ which simplifies to $0\; \ge \; -9$. Since this is true, shade the region above the parabola where $y$ values are greater than or equal to $x^2\; -\; 9$.
5. Label the graph clearly, showing the parabola and the shaded region representing all solutions to the inequality $y\; \ge \; x^2\; -\; 9$.

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 - 9 helps in sketching the boundary for the inequality.

Inequalities and Their Graphs

Graphing inequalities involves shading the region of the coordinate plane that satisfies the inequality. For y ≥ x² - 9, the graph includes all points on or above the parabola y = x² - 9. The boundary curve is solid because the inequality includes equality (≥).

Boundary Lines and Test Points

The boundary line or curve separates the solution region from the non-solution region. To determine which side to shade, select a test point not on the boundary and check if it satisfies the inequality. If it does, shade that side; if not, shade the opposite side.