Graph the following inequality: [tex]y \geq x^2+2[/tex]

The inequality y g e x 2 + 2 represents the region above the parabola y = x 2 + 2 .
The parabola opens upwards with a vertex at ( 0 , 2 ) .
The region above the parabola is shaded to represent y g e x 2 + 2 .
The parabola is drawn as a solid line because the inequality includes 'equal to'.

Explanation

Understanding the Inequality We are asked to graph the inequality y g e x 2 + 2 . This means we need to graph the parabola y = x 2 + 2 and shade the region above it, since y is greater than or equal to x 2 + 2 .

Identifying the Parabola The equation y = x 2 + 2 represents a parabola. The vertex of the parabola is at ( 0 , 2 ) . Since the coefficient of x 2 is positive, the parabola opens upwards.

Determining the Shaded Region Since the inequality is y g e x 2 + 2 , we need to shade the region above the parabola. The parabola itself is included in the solution because of the 'equal to' part of the inequality, so we draw a solid curve for the parabola.

Final Graph Therefore, the graph of the inequality y g e x 2 + 2 is the parabola y = x 2 + 2 and the region above it.


Examples
Understanding inequalities like y g e x 2 + 2 is crucial in various real-world applications. For instance, in physics, you might use such inequalities to describe the region where a projectile can land, given its initial velocity and launch angle. In economics, these inequalities can define feasible production regions, showing the possible combinations of goods a company can produce with its resources. Moreover, in computer graphics, inequalities help determine which parts of an object are visible on the screen. These examples show how visualizing and understanding inequalities can help solve practical problems in diverse fields.

Answered by GinnyAnswer | 2025-07-05