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Build a function from context by determining an explicit formula, recursive rule, or calculation process.

Build an explicit linear function from a starting value and constant rate

Problem
A plant starts at \(6~\text{centimeters}\) and grows \(2~\text{centimeters}~\text{each}~\text{week}\). Write an explicit linear function for its height and define the variables.
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Hint

Let w be weeks and h(w) be the plant height. Identify the starting height and the weekly growth rate before writing the rule.

For an explicit linear function, output = starting value + rate times input. A positive growth rate adds to the starting height.

Solution walkthrough

01

Define input and output

\[w~=~\text{weeks}~\text{after}~\text{the}~\text{start};~h(w)~=~\text{plant}~\text{height}~\text{in}~\text{centimeters}\]

Time is the independent input and height is the dependent output requested by the function.

02

Identify start and rate

\[h(0)~=~6~\text{centimeters};~\text{weekly}~\text{rate}~=~+2~\text{centimeters}/\text{week}\]

The plant already has height 6 at week zero and grows, so the rate is positive 2.

03

Build the explicit rule

\[\text{height}~=~\text{start}~+~\text{rate}~\text{times}~\text{weeks}~->~h(w)~=~6~+~2w\]

A constant-rate linear function adds 2 centimeters for each elapsed week to the starting height.

04

Check and state the model

\[h(0)~=~6;~h(1)~=~8~->~h(w)~=~6~+~2w,~\text{where}~w~\text{is}~\text{weeks}~\text{and}~h~\text{is}~\text{height}~\text{in}~\text{cm}\]

The checks reproduce the stated starting height and one week of growth, and the final answer defines both variables.

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Another way

  1. Use slope-intercept form h = mw + b with slope m = 2 centimeters per week and intercept b = 6 centimeters.

!

Common mistake

Do not write h(w) = 2 + 6w. That swaps the starting height and weekly rate and fails the week-zero check.

Solution walkthrough video