Mathematics from zero
Functions: model a real-world rate
Reading about functions is not the same as catching one in the wild. Pick a real situation that grows at a steady rate, turn it into a function, and use that function the way the whole unit promised — feed inputs in, read outputs out, and read answers straight off a graph.
Turn the unit’s idea into a working tool: choose a real linear situation, write its rule with a starting value and a rate, build a table and a graph, evaluate the function with notation, and make and check a prediction — proving each step instead of just asserting it.
Take one real situation that changes at a steady rate — a phone plan with a monthly fee plus a per-gigabyte charge, a savings jar with a starting balance plus a fixed weekly deposit, a taxi with a base fare plus a per-kilometre rate, or your own — and model it end to end as a linear function: rule, table, graph, evaluation, and a verified prediction.
- The input and output are named correctly, and the starting value and rate are each tied to a concrete real-world meaning, not just numbers.
- The written rule matches the table: every input substituted into the rule gives exactly the output listed, with no arithmetic slips.
- The plotted points fall on a single straight line, with the axes labelled input and output and the pairs read across-then-up.
- The out-of-table prediction computed with notation agrees with the value read off the graph at the same input.
- Model a second situation with a different rate and the same starting value, plot both lines on one grid, and explain in a sentence why the steeper line has the larger rate.
- Add a situation where the output goes down over time (a negative rate, like a balance being spent), and describe how its graph differs from a rising one.
- Pick a real input that does not make sense for your situation (for example a negative number of weeks) and explain why it lies outside the domain you would allow.
- Find a real bill, receipt, or app screen that hides a linear function, recover its rate and starting value from the numbers shown, and write the rule that produced them.
This is the loop you will run whenever something grows at a steady rate: name the input and output, pull out the starting value and the rate, write the rule, build a table, plot the straight line, and then use the function both ways — evaluate it with notation to predict, and read the graph to confirm. Doing it once on a situation you chose turns the unit’s machine from a definition into a tool you reach for.