Graph the solution set.
[tex]y \geq-(x-2)^2-1[/tex]
Graph the related equation [tex]y=-(x-2)^2-1[/tex]. The equation represents a parabola opening (Choose one) with vertex at ( , ). Because the inequality symbol [tex]\geq[/tex] allows equality, draw the parabola as a (Choose one) curve.
Answer (2)
The parabola opens downward.
The vertex of the parabola is at ( 2 , − 1 ) .
The parabola is drawn as a solid curve because the inequality includes equality.
Shade the region above the parabola to represent the solution set of y g e q − ( x − 2 ) 2 − 1 .
Explanation
Analyzing the Inequality and Related Equation The given inequality is y g e q − ( x − 2 ) 2 − 1 . We need to graph the solution set for this inequality. First, we analyze the related equation y = − ( x − 2 ) 2 − 1 . This equation represents a parabola.
Finding the Vertex and Direction of Opening The equation y = − ( x − 2 ) 2 − 1 is in vertex form, y = a ( x − h ) 2 + k , where the vertex of the parabola is ( h , k ) . In our case, h = 2 and k = − 1 , so the vertex is ( 2 , − 1 ) . Since the coefficient a = − 1 is negative, the parabola opens downward.
Determining the Type of Curve The inequality symbol is ≥ , which means y is greater than or equal to − ( x − 2 ) 2 − 1 . Because the inequality includes equality, we draw the parabola as a solid curve. If the inequality was "> > or < , we would draw a dashed curve to indicate that the points on the parabola are not included in the solution set.
Determining the Region to Shade To graph the solution set, we need to determine which region to shade. Since y g e q − ( x − 2 ) 2 − 1 , we shade the region above the parabola. This represents all the points ( x , y ) where the y -value is greater than or equal to the corresponding value on the parabola.
Conclusion Therefore, the parabola opens downward, the vertex is at ( 2 , − 1 ) , and the parabola is drawn as a solid curve.
Examples
Consider designing a protective structure that must provide a certain level of coverage. The parabolic shape, defined by an inequality like the one in this problem, can model the boundary of the protected area. By understanding how to graph and interpret such inequalities, engineers can determine the safe zone within the structure, ensuring that objects or people inside are adequately shielded. This principle applies in various fields, from architecture to creating safety barriers.
The parabola described by the equation y = − ( x − 2 ) 2 − 1 opens downward and has its vertex at ( 2 , − 1 ) . The curve is drawn as solid because the inequality allows equality ( y ≥ ). The area above the parabola should be shaded to represent the solution set of the given inequality.
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