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All courses Math I · F-BF.1.a 15 of 59
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
Build an explicit linear function for this situation: a plant starts at 6 cm and grows 2 cm each week. Define your 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 the variables

\[w~=~\text{weeks},\;~h(w)~=~\text{height}~\text{in}~\text{centimeters}\]

The input is the number of weeks, and the output is the plant's height in centimeters.

02

Identify the starting value and rate

\[\text{start}~=~6,\;~\text{rate}~=~2\]

At week 0 the plant is already 6 centimeters tall, and each week adds 2 centimeters.

03

Write the function

\[h(w)~=~6~+~2w\]

A linear function with a starting value adds the rate times the input to the initial amount.

04

State and check the answer

\[h(w)~=~6~+~2w;\;~h(0)~=~6,\;~h(1)~=~8\]

The explicit function is h(w) = 6 + 2w, where w is weeks and h(w) is height in centimeters. Checking gives h(0) = 6 and h(1) = 8, matching the start and weekly growth.

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Another way

  1. You can also view the rule as slope-intercept form: the initial height is the intercept, and the weekly growth is the slope.

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Common mistake

A common mistake is to write h(w) = 2 + 6w. That swaps the starting height with the weekly growth rate, but the plant starts at 6 cm and then adds 2 cm each week.