Answer: See graph below.
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Explanation:
- x = number of students in a school
- y = number of classrooms
Each classroom holds 36 students on average. If we had 36 students in the school, then we'd need only one classroom. We can express this as 36/36 = 1 so we can build to the next step.
If we had 72 students, then we'd need 2 classrooms because 72/36 = 2.
If we had 108 students, then we'd need 3 classrooms since 108/36 = 3.
And so on. The values 36,72,108 are multiples of 36. The generalized rule is that if we had x students, then we'd need x/36 classrooms. If you get a decimal value, round up to the nearest whole number. If you round down, then you won't have enough room to fit the leftover students.
For example, let's say we had x = 50 students. Computing y = x/36 leads to y = 50/36 = 1.389 approximately. The temptation is to round down to y = 1 since 1.389 is closer to 1 than it is to 2; however, we'll have 50-36 = 14 students left over without a classroom. So that's why we round 1.389 up to 2 instead.
This action of rounding up to the nearest integer tells us that we'll apply the ceiling function to the result of x/36. Most books would show this notation as [tex]y = \lceil x/36\rceil[/tex]. The "bracket" bars on either side aren't exactly the standard square brackets you may be used to. Instead, they're only half of such. A piece of the lower portion isn't drawn. The upper portion is only drawn to indicate we round up rather than round down. Most graphing calculators and graphing software has a "ceil" or "ceiling" function built in.
What results is known as a step function or staircase function. We have a staircase without the vertical pieces because otherwise we wouldn't have a function (refer to the vertical line test). Note the use of open vs closed holes to indicate which points are excluded or included respectively.
