A rental charges for every hour that has started. Staying 1.1 hours uses two charge units, while staying exactly one hour uses one. The graph must show horizontal pieces and jumps. A diagonal line would invent gradual charges that the rule does not allow. Endpoint marks are therefore part of the mathematics, not decoration.
Predict
Predict the charge at a boundary and just after it. Under a started-hour rule, exactly two hours stays on the second step; any positive extra duration moves to the third. Predict whether zero duration should have the first positive charge. That is a policy choice, and this lesson explicitly assigns zero charge at zero.
Experiment
Graph S: started-hour charge. Input x is time in hours; output y is dollars. Horizontal steps are open on the left and closed on the right, with a separate origin.
Supplied Graph S charges 3 dollars per started hour on 0≤t≤3, with C(0)=0. Open circles exclude endpoints from a segment; filled circles include them.
Duration interval, hours
Charge, dollars
Exactly 0
0
0<t≤1
3
1<t≤2
6
2<t≤3
9
Read t=1, then a point slightly to its right. Use the table as an equivalent accessible description of the graph’s endpoint conventions. The horizontal axis represents continuous elapsed time; the output can still take only four values in this window.
Understand
The ceiling of a real number x is the smallest integer greater than or equal to x. Thus ceiling(1.2)=2 and ceiling(2)=2. It differs from floor, the greatest integer less than or equal to x. A ceiling function is not ordinary rounding to the nearest integer. It answers how many whole units are sufficient to cover a requirement.
For positive duration t and a block length b>0, ceiling(t/b) counts started blocks. Multiplying by a charge c gives C(t)=c ceiling(t/b), with a separately stated zero-duration convention if needed. On the interval (kb,(k+1)b], the charge is (k+1)c for nonnegative integer k. The left endpoint is excluded because at exactly kb only k blocks are required; the right endpoint is included because (k+1)b exactly fills the current count.
To construct the graph, first mark multiples of b on the input axis and multiples of c on the output axis. Draw each horizontal segment with an open left endpoint and closed right endpoint. Then plot the origin according to the policy. Do not connect adjacent step endpoints by vertical segments: that would include multiple charges at a single input and cease to be the graph of this function.
Input and output discreteness are independent questions. Elapsed time may be any nonnegative real value while the charge is a whole multiple. In contrast, a function of a batch count n may have only integer inputs. Its graph consists of separate points even if the formula resembles a line. State the domain before deciding which marks to connect.
Apply
Compare started-block and completed-block policies at both exact boundaries and interior points. Testing only t=2,4,6 can fail to distinguish floor and ceiling rules. For a new charging scenario, state units, block size, charge per block, zero behavior and any maximum duration. Then give either a labeled step drawing or a complete interval-and-value description.
Continue
A step graph encodes a decision rule. Its jumps need exact endpoint interpretation, and its visible range depends on the chosen domain. Next, exponential and logarithmic curves introduce asymptotes: lines approached by a curve are different from included step segments or arbitrary edges of a plotting window.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
A locker costs 4 dollars per started hour, with zero charge at duration zero. What is the charge for 2.25 hours?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For the same rule, what is the charge at exactly 2 hours?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Construct C(t)=5 ceiling(t/2) for t≥0 with C(0)=0. Describe the graph through t=8, including every endpoint convention.
Structure your argument and preserve revisions
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In supplied Graph S, explain the difference between t=1 and t just larger than 1. Give the value at t=0 and the range on 0≤t≤3.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A machine processes only whole batches n=0,1,2,... and cost is 7n dollars. Should its graph include the whole line y=7x? Explain.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A learner uses 5 floor(t/2) for charging 5 dollars per started two-hour block. Diagnose the error at t=1 and t=2 and give a corrected rule.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
