Graph each inequality. y≥x²−9

Verified step by step guidance

1. Identify the inequality to be graphed: \(y \geq x^{2} - 9\). This means we are dealing with a parabola and the region above or on the parabola.
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 \(y \geq x^{2} - 9\), the boundary line (the parabola) is included in the solution set, so draw the parabola as a solid curve.
4. Determine which side of the parabola to shade by choosing a test point not on the parabola, commonly \((0,0)\), and check if it satisfies the inequality: substitute \(x=0\) and \(y=0\) into \(y \geq x^{2} - 9\).
5. If the test point satisfies the inequality, shade the region containing that point; if not, shade the opposite side. This shaded region represents all solutions to 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 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 where the inequality holds true. 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 Shading Regions

The boundary line or curve separates the solution region from the non-solution region. For inequalities with '≥' or '≤', the boundary is included and drawn solid. After graphing the boundary, test points determine which side to shade, representing all solutions to the inequality.