Linear functions: interpret slope as a rate of change and the y-intercept as an initial value, use function notation, and build linear models from a rate and a starting amount.

What this skill is asking

A linear function has the form $f(x) = mx + b$: a constant rate of change $m$ (the slope) and a starting value $b$ (the output when $x = 0$). The Digital SAT (Algebra domain) tests whether you can evaluate a linear function, read its slope as a rate of change, interpret its intercept, and build a linear model from a description. The algebra is light; the emphasis is on meaning.

Reading a linear function

Each part of $f(x) = mx + b$ has a fixed meaning.

A worked model-building question

The SAT often asks you to assemble a linear function from words.

Slope as a rate of change

The single most tested idea here is that slope is a rate. If a function's output is distance in miles and its input is time in hours, the slope is a speed in miles per hour. If the output is cost in dollars and the input is items, the slope is a price per item. When a question asks what a coefficient "represents", name the rate with its units and its direction (increasing for positive, decreasing for negative). This is also how you compute a slope from a table or two points: the change in output divided by the change in input.

Intercepts in context

The $y$-intercept is the value when the input is zero, often a fixed or starting amount: a sign-up fee, an initial population, a starting balance. The $x$-intercept (where $f(x) = 0$) is where the modelled quantity reaches zero, such as the time a draining tank is empty or a depreciating asset is worth nothing. Reading both intercepts in the language of the problem (a starting fee, a break-even time) is exactly the interpretation the SAT rewards, and it is usually faster than any computation.

Tables, graphs and equations of the same function

The SAT presents linear functions in three forms and asks you to move between them: an equation ($f(x) = mx + b$), a table of input-output pairs, and a graph. All three carry the same slope and intercept, so a question may give a table and ask for the equation, or give a graph and ask for $f$ at some input. To get the slope from a table, divide any change in output by the matching change in input, $m = \frac{\Delta f}{\Delta x}$, then read the intercept as the output when the input is $0$ (or work back to it). From a graph, read the $y$-intercept where the line crosses the vertical axis and the slope as rise over run between two clear lattice points. Because a linear function has a constant rate, the differences in a table are equal for equal input steps; if they are not, the relationship is not linear. Being able to translate freely among equation, table and graph is one of the most frequently tested ideas in the whole Algebra domain.

Comparing two linear functions

Some questions give two linear models and ask where one overtakes the other, or which has the greater rate. The function with the larger slope grows faster and, far enough along, produces the larger output regardless of starting value; the function with the larger $y$-intercept starts higher. The input where they are equal is found by setting the two expressions equal and solving (the same idea as a system of equations). Reading "which plan is cheaper after 12 months" or "when do the two populations match" in these terms turns a wordy comparison into a quick slope-and-intercept reading or a one-line equation.