Write linear functions that model the relationship between two quantities from a description, table, or graph, write an equation representing a functional relationship, and evaluate functions in function notation (TEKS A.2C, A.2G, A.12B).

What this topic is asking

Modeling is the point of Reporting Category 3: turn a situation into a linear function and use it. TEKS A.2C and A.2G ask you to write a function or equation from a description, table, or graph, and A.12B asks you to evaluate functions written in function notation $f(x)$. STAAR rewards both building the correct model and interpreting its inputs and outputs in context.

Building the model: rate and initial value

A linear model has the shape $f(x) = mx + b$, and the words of the problem tell you each part.

• The rate of change is the "per" quantity (per month, per mile, per item). It is the slope $m$ and multiplies the variable.
• The initial value is the fixed, one-time, or starting amount (a sign-up fee, a starting balance). It is the $y$-intercept $b$ and stands alone.

"A plumber charges $60 to come out plus$45 an hour" becomes $C(h) = 45h + 60$: the hourly rate multiplies $h$, the call-out fee is the constant.

From a table or graph

From a table, the slope is the constant change in output per unit change in input, and the initial value is the output at input 0. From a graph, read the $y$-intercept and a slope step. Either way, assemble $f(x) = mx + b$.

Function notation

Function notation $f(x)$ is a name for the output. The letter $f$ names the function, $x$ is the input, and $f(x)$ is the corresponding output.

Function notation can also be read backward: "$f(x) = 90$, find $x$" means solve $90 = 20t + 150$, an equation you solve as usual.

How STAAR examines this topic

• Multiple choice. Choose the function matching a description; rate-and-fee swaps are the standard distractor.
• Equation editor and number entry. Build the function, evaluate $f$ at a value, or solve $f(x) = k$ for the input.
• Inline choice. Interpret the slope and intercept in context from dropdowns.

A clarifying idea is that building the model and using it are two skills the test pairs: first translate the words into $f(x) = mx + b$, then either evaluate (input given) or solve (output given). Keeping straight which quantity is known prevents the most common modeling errors.

Choosing the right variable as the input

A subtle modeling decision is which quantity is the input ($x$) and which is the output ($f(x)$). The input is the quantity you control or that drives the change, usually time, distance, or the number of items; the output is what results, usually cost, amount, or value. Reading the prompt for "after $x$ minutes" or "for each item" identifies the input, and the phrase "total cost" or "amount remaining" identifies the output. Getting this the right way around matters because it fixes which number is the slope and which is the intercept, and it determines how function notation is read.

Comparing two linear models

STAAR sometimes gives two models and asks which is greater, or when they are equal. If two phone plans are $A(m) = 0.10m + 20$ and $B(m) = 0.05m + 30$, setting them equal, $0.10m + 20 = 0.05m + 30$, gives $m = 200$ minutes as the crossover, and for fewer minutes plan A is cheaper while for more minutes plan B is. This connects modeling to solving a linear equation, and it shows why interpreting the slope (the per-unit rate) is what tells you which model pulls ahead as the input grows.

Try this

Q1. A pool starts with 500 gallons and fills at 30 gallons per minute. Write $W(t)$. [1 point]

• Cue. $W(t) = 30t + 500$.

Q2. For $f(x) = 6x - 4$, find $f(7)$. [1 point]

• Cue. $f(7) = 6(7) - 4 = 38$.