# Linear Functions on the SAT: Examples & Practice | 1600.now Source: https://1600.now/sat-skill/linear-functions --- Math · Algebra ## Linear Functions A linear function changes by the same amount for each equal increase in its input. Its slope is that rate of change; its intercept is the output when the input is zero. Written by [Luke Finigan](https://1600.now/about) 2 min read Updated Oct 2, 2026 ### Read slope as a rate with units A delivery service charges $C(d)=4+1.5d$, where $d$ is distance in miles and $C$ is cost in dollars. The slope 1.5 means each additional mile adds 1.50 dollars. The intercept 4 is the fixed charge when the distance is zero. A question about the coefficient of $d$ asks about change, not total cost. At 6 miles, the total is $C(6)=13$ dollars, but the rate is still 1.50 dollars per mile. ### Build the rule from a table Suppose a table contains $(2,11)$ and $(6,23)$. The slope is $m=\frac{23-11}{6-2}=3$. Use either point in $y=3x+b$: $11=6+b$, so $b=5$. The function is $f(x)=3x+5$. The output difference 12 is not the slope because the input changed by 4. Divide changes in the same order: second output minus first output over second input minus first input. ### Use the intercept that matches the question The vertical intercept comes from setting the input to zero. The horizontal intercept comes from setting the output to zero. For $f(x)=3x+5$, the vertical intercept is $(0,5)$ and the horizontal intercept is $(-\frac{5}{3},0)$. In a context, a horizontal intercept may be the time when a quantity reaches zero. Check whether that time is within the model's stated domain before treating it as a real event. ### Distinguish constant change from constant percent change Outputs 10, 15, 20 at inputs 0, 1, 2 have constant differences, so they fit a linear rule. Outputs 10, 15, 22.5 have constant ratios of 1.5, so they fit an exponential rule instead. When the table's input gaps vary, compare rates rather than raw output differences. A longer input interval should produce a proportionally larger change in a linear model. ### Check your understanding Try it yourself A tank has $V(t)=80-4t$ liters after $t$ minutes. What does $-4$ mean? A The tank starts with 4 liters. B The volume decreases by 4 liters per minute. C The tank empties after 4 minutes. D The volume decreases by 80 liters per minute. ### Sample linear functions questions Linear Functions · Easy: $f(x) = 90x + 6$. What is the value of $f(x)$ when $x = 8$? · A 104 B 138 C 720 D 726 - [Open the question and explanation](https://1600.now/bank/math/5a244d60) Linear Functions · Easy: $f(x) = 7(2x + 3)$. For what value of $x$ does $f(x) = 77$? · A 4 B 7 C 11 D 37 - [Open the question and explanation](https://1600.now/bank/math/6b946355) ### Practice this skill 1. Write the units of each slope and intercept before selecting an interpretation. 2. Find a rule from two points, then use a third point to check it. 3. Practice a mix of equations, tables, and verbal descriptions. [Practice linear functions](https://1600.now/bank/math/skill/Linear%20functions) [Print the linear functions worksheet and worked answers](https://1600.now/sat-linear-functions-worksheet). Then use a [mixed practice module](https://1600.now/modules) to check whether you can choose this method among other question types. ### Source and question labels The bank label for this guide is "Linear functions" in Algebra. The worked examples above are written for this guide. - [College Board: Types of Math tested](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types) ### Related skills - **[Linear Equations in Two Variables](https://1600.now/sat-skill/linear-equations-two-variables)** A two-variable linear equation describes a line. - **[Nonlinear Functions](https://1600.now/sat-skill/nonlinear-functions)** The form of a nonlinear function tells you what you can read directly. - **[Two-Variable Data: Models and Scatterplots](https://1600.now/sat-skill/two-variable-data)** A scatterplot compares paired variables.