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Linear Functions on the Digital SAT

A linear function models a quantity that changes at a constant rate: $f(x) = mx + b$, where $m$ is the rate of change (slope) and $b$ is the starting value. The Digital SAT tests whether you can evaluate the function, build it from a description or two points, and — above all — interpret what $m$ and $b$ mean in a real context. Linear questions appear in every Math module, usually early, and they are points you cannot afford to leave behind.

How the question announces itself

• “If $f(x) = 3x - 4$, what is the value of $f(5)$?” — direct evaluation.
• “The function $c(m) = 12m + 30$ gives the cost… What is the best interpretation of $30$?” — interpret slope or intercept in context.
• “The table shows values of the linear function $g$…” — find the rule from points.
• A word problem describing a starting amount plus a constant change per unit — build the function yourself.

The method

1. Name the two numbers before touching the algebra. Every linear situation has exactly two parameters: the starting value (what you have when $x = 0$) and the rate (what changes per unit of $x$). Read the problem once just to label them. Most wrong answers come from swapping these two roles, not from arithmetic.
2. Compute what is asked — not what is easy. For $f(k)$, substitute and follow order of operations: multiply before adding. For a rule from two points, slope first: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$, then recover $b$ by plugging one point back in. Reread the final sentence: the SAT loves asking for $f(5)$ when you just found $x$ such that $f(x) = 5$.
3. Verify in Desmos in ten seconds. The built-in calculator makes linear questions nearly free: define $f(x) = 3x - 4$ and evaluate $f(5)$ directly, or plot a table of points and read the slope and intercept off the line. If your algebraic answer and the graph disagree, trust the graph and find your sign error.

The named mistakes behind every wrong answer

Swapping slope and intercept

Reading $f(x) = 12x + 30$ as “starts at 12, grows by 30.” The coefficient of $x$ is always the rate; the constant is always the starting value.

Adding instead of subtracting (or vice versa)

Computing $3(5) + 4 = 19$ for $f(x) = 3x - 4$. Distractors are built from exactly this sign slip — check the operation before you move on.

Inverting the slope ratio

Writing $m = \dfrac{x_2 - x_1}{y_2 - y_1}$. Slope is rise over run — change in output over change in input.

Answering the wrong question

Finding $x$ when the test asked for $f(x)$, or reporting the intermediate value. The last sentence of the stem decides what counts.

Try one

The full Linear Functions lesson inside Satisfied adds interpretation drills, table and graph variants, and a practice set that adapts to your level — with every mistake named in the review.

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