Which are points on the graph of [tex]$y=1.5+\lceil x\rceil$[/tex]? Select three options.
A. (-4.5, -2.5)
B. (-0.8, 0.5)
C. (7.9, 9.5)
D. (4.5, 6)
E. (1.3, 3.5)
Answer (1)
Calculate ⌈ x ⌉ for each point.
Calculate 1.5 + ⌈ x ⌉ for each point.
Check if the calculated y value matches the given y value.
The points that satisfy the equation are ( − 4.5 , − 2.5 ) , ( 7.9 , 9.5 ) , and ( 1.3 , 3.5 ) . Therefore, the answer is ( − 4.5 , − 2.5 ) , ( 7.9 , 9.5 ) , ( 1.3 , 3.5 ) .
Explanation
Understanding the Problem We are given the equation y = 1.5 + ⌈ x ⌉ and five points: ( − 4.5 , − 2.5 ) , ( − 0.8 , 0.5 ) , ( 7.9 , 9.5 ) , ( 4.5 , 6 ) , and ( 1.3 , 3.5 ) . We need to determine which three points satisfy the equation. The ceiling function, denoted by ⌈ x ⌉ , returns the smallest integer greater than or equal to x .
Checking Point (-4.5, -2.5) For the point ( − 4.5 , − 2.5 ) , we have x = − 4.5 . The ceiling of x is ⌈ − 4.5 ⌉ = − 4 . Therefore, y = 1.5 + ( − 4 ) = − 2.5 . This matches the given y value, so ( − 4.5 , − 2.5 ) is on the graph.
Checking Point (-0.8, 0.5) For the point ( − 0.8 , 0.5 ) , we have x = − 0.8 . The ceiling of x is ⌈ − 0.8 ⌉ = 0 . Therefore, y = 1.5 + 0 = 1.5 . This does not match the given y value of 0.5 , so ( − 0.8 , 0.5 ) is not on the graph.
Checking Point (7.9, 9.5) For the point ( 7.9 , 9.5 ) , we have x = 7.9 . The ceiling of x is ⌈ 7.9 ⌉ = 8 . Therefore, y = 1.5 + 8 = 9.5 . This matches the given y value, so ( 7.9 , 9.5 ) is on the graph.
Checking Point (4.5, 6) For the point ( 4.5 , 6 ) , we have x = 4.5 . The ceiling of x is ⌈ 4.5 ⌉ = 5 . Therefore, y = 1.5 + 5 = 6.5 . This does not match the given y value of 6 , so ( 4.5 , 6 ) is not on the graph.
Checking Point (1.3, 3.5) For the point ( 1.3 , 3.5 ) , we have x = 1.3 . The ceiling of x is ⌈ 1.3 ⌉ = 2 . Therefore, y = 1.5 + 2 = 3.5 . This matches the given y value, so ( 1.3 , 3.5 ) is on the graph.
Final Answer The points on the graph are ( − 4.5 , − 2.5 ) , ( 7.9 , 9.5 ) , and ( 1.3 , 3.5 ) .
Examples
The ceiling function is used in many real-world applications, such as determining the number of containers needed to ship a certain number of items. For example, if you have 10.5 items to ship and each container can hold only one item, you would need ⌈ 10.5 ⌉ = 11 containers. This concept is also used in pricing, where a fractional amount is rounded up to the nearest whole number. Understanding the ceiling function helps in resource allocation, logistics, and various business calculations where overestimation is necessary to avoid shortages or inefficiencies.
