Build a function that models a relationship, write a linear function from a context, table, or two points, and interpret its parameters in context (Ohio F-BF.1, F-LE.2, F-IF.7).

What this topic is asking

Ohio standards F-BF.1 and F-LE.2 ask you to build a function that models a relationship, often a linear function written from a description, a table, or two points, and to interpret its parts. This is modeling in function notation: identify the rate (slope) and the starting value (intercept), write $f(x) = mx + b$, and use it to predict. It is a core Functions skill across both parts.

Building from a description

Most linear models hand you the two parts in words: a fixed starting amount and a constant rate.

For "a tank holds $50$ liters and drains $4$ liters per hour," the rate is $-4$ (draining) and the start is $50$, so $L(t) = 50 - 4t$.

Building from two points

When you are given two input-output pairs, find the slope, then the intercept.

Building from a table

A table is linear when the output changes by a constant amount for each equal input step. That constant change is the slope; the output at $x = 0$ (or extended back to it) is the intercept. If a table shows $(0, 7), (1, 10), (2, 13)$, the output rises $3$ each step, so $f(x) = 3x + 7$.

Using the function to predict

Once the rule is built, evaluate it to predict an output, or solve $f(x) = k$ to find when the output reaches a target. For $L(t) = 50 - 4t$, the tank is empty when $50 - 4t = 0$, so $t = 12.5$ hours.

How Ohio examines this topic

• Equation response. Write the function rule from a context or two points, then evaluate it.
• Multiple choice and multiple-select. Match a description, table, or pair of points to the correct rule.
• Tables. Complete a function table consistent with a linear rule.

Why the slope is the rate and the intercept is the start

The structure $f(x) = mx + b$ maps cleanly onto a real situation, which is what makes it such a useful model. The term $mx$ grows in proportion to the input, so $m$ is exactly the amount of change per one unit of input, the rate. The term $b$ stands alone, unaffected by $x$, so it is the output when $x = 0$, the starting value before any change has happened. Reading a word problem this way, "starts at" supplies $b$ and "per" supplies $m$, lets you write the rule almost directly. Interpreting them back, with units, is the modeling habit the test rewards: $m$ dollars per month, $b$ dollars at the start.

Why two points are enough for a line

A straight line is completely determined by two points, which is why "passes through these two points" is enough to recover the whole rule. The two points fix the slope (rise over run between them), and either point then fixes the intercept, leaving no freedom. This is the same idea as writing a line's equation, now phrased in function notation. It also means a third given point is a check: if it does not satisfy the rule you built, either the data is not linear or you made an arithmetic slip, so substituting the extra point is a quick way to catch an error before it costs the item.

Try this

Q1. A gym charges \25$to join plus$\$10$ per month. Write $C(m)$ and find $C(6)$. [2 points]

• Cue. $C(m) = 25 + 10m$; $C(6) = 25 + 60 = 85$.

Q2. Write the linear function through $(0, -1)$ and $(4, 7)$. [2 points]

• Cue. $m = \frac{7 - (-1)}{4} = 2$, $b = -1$, so $f(x) = 2x - 1$.