Graph the following inequality.
$y \geq x^2+5$
Answer (1)
Identify the parabola: Recognize that y = x 2 + 5 is a parabola with vertex at ( 0 , 5 ) opening upwards.
Determine the shading region: Since the inequality is y g e x 2 + 5 , shade the region above the parabola.
Draw the parabola: Use a solid line for the parabola because the inequality includes 'equal to'.
Final graph: The graph of the inequality y g e x 2 + 5 is the parabola y = x 2 + 5 and the region above it, with the parabola drawn as a solid line.
Explanation
Understanding the Inequality We are asked to graph the inequality y g e x 2 + 5 . This means we need to sketch the parabola y = x 2 + 5 and shade the region that satisfies the inequality.
Finding the Vertex The equation y = x 2 + 5 represents a parabola. The basic parabola y = x 2 has its vertex at the origin ( 0 , 0 ) . The given equation is a vertical translation of the basic parabola by 5 units upwards. Therefore, the vertex of the parabola y = x 2 + 5 is at ( 0 , 5 ) .
Determining the Direction of Opening Since the coefficient of the x 2 term is positive (1), the parabola opens upwards. This means the vertex is the lowest point on the graph.
Determining the Shaded Region Now we need to determine the region to shade. The inequality is y g e x 2 + 5 , which means we want to shade the region where the y -values are greater than or equal to the corresponding values on the parabola. This corresponds to the region above the parabola.
Drawing the Parabola Since the inequality includes 'equal to', we draw the parabola as a solid line. If the inequality was strictly 'greater than' ( x^2 + 5"> y > x 2 + 5 ), we would draw the parabola as a dashed line to indicate that the points on the parabola are not included in the solution.
Final Graph In summary, we graph the parabola y = x 2 + 5 with a solid line and shade the region above the parabola to represent the solution to the inequality y g e x 2 + 5 .
Examples
Understanding inequalities like y g e x 2 + 5 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 limited resources. Moreover, in engineering, they help determine the safe operating areas for machines, ensuring that temperature or pressure stays within acceptable limits. Graphing these inequalities provides a visual representation of the solution set, making it easier to understand and apply in practical scenarios.
