In Exercises 1–26, graph each inequality. y≥x^2−9

Verified step by step guidance

1. Step 1: Identify the inequality to graph: \( y \geq x^2 - 9 \).
2. Step 2: Recognize that the boundary of the inequality is the parabola \( y = x^2 - 9 \).
3. Step 3: Graph the parabola \( y = x^2 - 9 \) as a solid line because the inequality is \( \geq \), indicating that points on the line are included.
4. Step 4: Determine which side of the parabola to shade. Since the inequality is \( y \geq x^2 - 9 \), shade the region above the parabola.
5. Step 5: Verify by selecting a test point not on the boundary, such as the origin (0,0). Substitute into the inequality: \( 0 \geq 0^2 - 9 \), which simplifies to \( 0 \geq -9 \), a true statement, confirming the correct region is shaded.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as '≥' (greater than or equal to) and '≤' (less than or equal to) to indicate the direction of the relationship. Understanding how to interpret and graph inequalities is essential for visualizing solutions in a coordinate plane.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'. In the given inequality, y ≥ x^2 - 9, the quadratic function is y = x^2 - 9, which opens upwards and has its vertex at the point (0, -9).

Graphing Techniques

Graphing techniques involve plotting points and understanding the shape of functions to represent them visually on a coordinate plane. For inequalities, it is important to determine the boundary line (or curve) and then shade the appropriate region that satisfies the inequality. In this case, after graphing the parabola y = x^2 - 9, the area above or on the curve will be shaded to represent the solution set for y ≥ x^2 - 9.