Writing a Linear Function from Four Kinds of Clue
F.LE.2
Read the standard text
F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Section 1 · LearnLearn the idea
In this lesson you will: Build f(x)=mx+b from a description, a graph, a table, or two points; Find the rate of change when the inputs do not step by one; Say what the input and the output of your function stand for.
Every clue is really two numbers
A linear function is settled once you know two things: how fast the output changes, and where it starts. Everything in this lesson is a different disguise for those same two numbers.
A sentence hides the rate behind the word per or each. A graph shows it as rise over run. A table shows it as a change down the output column. Two points make you compute it. The starting value is whatever the output is when the input is zero — and sometimes you have to work back to it.
Two points is the general case
If you can do it from two points, you can do it from anything, because a graph and a table are just two points wearing a costume.
Take (2,11) and (6,3). The output fell 8 while the input rose 4, so m=-8/4=-2. Now substitute either point into y=mx+b: 11=-2(2)+b gives b=15. The function is f(x)=-2x+15, and f(6)=-12+15=3 confirms it.
Tables that step by more than one
A table is only a list of points, so the method does not change. What does change is that the input column may step by 4 or 5 instead of 1, and then the drop down the output column is not the rate.
For inputs 3,7,11,15 and outputs 5,17,29,41, the output climbs 12 each time — but the input climbs 4 each time. The rate is 12 / 4=3. Substituting (3,5) gives 5=3(3)+b, so b=-4 and f(x)=3x-4.
Common mistake — Using the change in output as the rate
A table lists inputs 0,5,10,15 with outputs 12,27,42,57. The output rises by 15 every row, so a student writes m=15 and f(x)=15x+12.
It feels right because on most tables the input does step by one, and on those tables the change in output is the rate. The habit forms quickly and then transfers to a table where it does not hold.
Here the input steps by 5, so 15 is the change over five units, not one. The rate is 15 / 5=3, and the function is f(x)=3x+12. Test the wrong version at x=15: it predicts 237 where the table says 57.
How to catch it: Write the input step under each gap before you compute anything. If those steps are not all 1, you owe yourself a division.
Section 2 · DoPractice this standard
These published practice sets are scored on the student site and cover F.LE.2 directly.