Consider the graph of [tex]$\frac{(x+3)^2}{1}+\frac{(y-2)^2}{9}\ \textgreater \ 1$[/tex]. Which point could confirm the graph is shaded correctly?
A. $(-4,2)$
B. $(-3,5)$
C. $(-3,1)$
D. $(-2,2)$
Answer (1)
We are given the inequality 1"> 1 ( x + 3 ) 2 + 9 ( y − 2 ) 2 > 1 and four points.
We substitute each point into the inequality to check if it satisfies the inequality.
We find that none of the points satisfy the inequality.
Since none of the points satisfy the inequality, any of them could confirm the graph is shaded correctly if the shaded region is outside the ellipse. ( − 3 , 1 )
Explanation
Analyze the problem We are given the inequality 1"> 1 ( x + 3 ) 2 + 9 ( y − 2 ) 2 > 1 and four points: ( − 4 , 2 ) , ( − 3 , 5 ) , ( − 3 , 1 ) , and ( − 2 , 2 ) . We need to determine which of these points could confirm the graph is shaded correctly. This means we need to check which point satisfies the inequality.
Test each point Let's test each point:
Point (-4, 2): 1 ( − 4 + 3 ) 2 + 9 ( 2 − 2 ) 2 = 1 ( − 1 ) 2 + 9 0 2 = 1 + 0 = 1 . Since 1"> 1 > 1 is false, the point ( − 4 , 2 ) does not satisfy the inequality.
Point (-3, 5): 1 ( − 3 + 3 ) 2 + 9 ( 5 − 2 ) 2 = 1 0 2 + 9 3 2 = 0 + 9 9 = 1 . Since 1"> 1 > 1 is false, the point ( − 3 , 5 ) does not satisfy the inequality.
Point (-3, 1): 1 ( − 3 + 3 ) 2 + 9 ( 1 − 2 ) 2 = 1 0 2 + 9 ( − 1 ) 2 = 0 + 9 1 = 9 1 . Since 1"> 9 1 > 1 is false, the point ( − 3 , 1 ) does not satisfy the inequality.
Point (-2, 2): 1 ( − 2 + 3 ) 2 + 9 ( 2 − 2 ) 2 = 1 1 2 + 9 0 2 = 1 + 0 = 1 . Since 1"> 1 > 1 is false, the point ( − 2 , 2 ) does not satisfy the inequality.
Analyze the results Since the inequality is 1"> 1 ( x + 3 ) 2 + 9 ( y − 2 ) 2 > 1 , the points that satisfy the inequality should be in the shaded region. The points that do not satisfy the inequality should not be in the shaded region. However, none of the points satisfy the inequality. This means that if the graph is shaded correctly, none of these points should be in the shaded region. Therefore, any of the points could confirm the graph is shaded correctly if they are not in the shaded region.
Final Answer Since none of the points satisfy the inequality, any of them could be used to confirm the graph is shaded correctly, assuming the shaded region is outside the ellipse. However, the question asks which point could confirm the graph is shaded correctly. Since the points do not satisfy the inequality, they should not be in the shaded region. Therefore, any of the points could confirm the graph is shaded correctly.
Examples
When designing a playground, you might want to ensure that certain equipment is placed outside of a specific elliptical area for safety reasons. By using an inequality like the one in this problem, you can determine whether a point (representing the location of the equipment) falls within or outside the ellipse, helping you to plan the layout effectively and maintain a safe environment.
