Digital SAT Math · Algebra

Linear functions

Digital SAT Math · Algebra

A linear function is one rule wearing four outfits: an equation, a table, a graph, and a sentence. The test almost never asks you to solve anything hard here — it hands you the rule in one outfit and demands it back in another, and it collects points from students who read a table’s row-to-row jump as a rate, or who answer with an output when the question asked for an input.

On the test


Domain	Algebra (score report)
What it looks like	$f(x) = mx + b$ , a table of $x$ and $f(x)$ values, a line on a grid, or a described situation
Often asked	“What is $f(6)$ ?”, “Which equation defines $f$ ?”, “For what value of $x$ is $f(x) = 12$ ?”, “What is the best interpretation of $f(6) = 42$ ?”
Format	Multiple choice and student-produced response
Calculator	Desmos earns its keystrokes on some of these and wastes them on others — see the box below

Recognition cues: the letter $f$ (or $g$ , $h$ , $P$ , $T$ ) with something in parentheses; a two-column table headed $x$ and $f(x)$ ; the phrases linear function, constant rate, rate of change, best interpretation; a straight line with two dots on it.

Where this skill ends and its neighbours begin

Three pages share this territory, so fix the boundary before you start. If the stem is a story and the job is turning words into an equation, that is linear equations in context. If the object is a line described by points, slopes and forms — no $f$ anywhere — that is linear equations in two variables. This page is what happens when the same relationship is written as a function and then translated between table, graph, equation and words. The notation is not decoration: it is what makes “input” and “output” separate, nameable things.

Pattern recognition

Sort by which outfit you were handed and which one the answer wears:

Equation → number. Substitute, or set the rule equal to a given output and solve.
Table → equation. Get the rate from two rows, then anchor the constant with one row.
Graph → number. Harvest two exact lattice points; never estimate between grid lines.
Two input-output pairs → anything. Rate first, constant second, then answer whatever was asked.
Words → equation. The per-unit quantity multiplies the input; the starting quantity stands alone.

Method

Read the notation, not the numbers. Decide in two seconds: given an input, or given an output? Inside the parentheses means going in.
Find the rate of change. $m = \dfrac{\text{change in output}}{\text{change in input}}$ . From two pairs, from two table rows, from two lattice points — the formula never changes, only the packaging.
Divide by the input step. Always. A table whose $x$ column steps by 3 and whose $f(x)$ column jumps by 12 has a rate of $4$ , not $12$ . This one division is where most of the points on this skill are won and lost.
Anchor the constant. Substitute any single pair into $f(x) = mx + b$ and solve for $b$ . Only when a row or point sits at $x = 0$ can you read $b$ straight off.
Answer the question that was asked, then check the pair. Substitute your answer back and confirm both a given input and its given output survive.

Rate of change, in each outfit

Given	Where the rate is	Watch for
$f(x) = mx + b$	the coefficient $m$	a rearranged form where $m$ is not written first
A table	$\Delta f(x) \div \Delta x$ between any two rows	$x$ steps that are not 1
A graph	rise over run between two lattice points	non-lattice crossings; a squashed vertical scale
Two pairs $(a, f(a))$ , $(c, f(c))$	$\dfrac{f(c)-f(a)}{c-a}$	subtracting in opposite orders top and bottom
A description	the quantity phrased per something	the starting amount, which is not a rate

Where Desmos pays, and where it burns time. Two genuine uses on this skill. Solving $f(x) = k$ : type the function and type y = k, and the intersection is labelled with both coordinates, which removes the input-versus-output confusion entirely because you can see which coordinate is which. Checking an equation against a table: plot your candidate function and add the table rows as points; if a point sits off the line, your rate or your constant is wrong, and you will know in one glance which rows disagree. Two places it costs you: building an equation from a table (typing four rows into a Desmos table to run a regression takes longer than $\Delta y \div \Delta x$ by hand), and interpretation items, where the answer is a sentence and there is nothing to plot. Desmos settles arithmetic; it has no opinion about which number is the rate.

Worked example 1 — a table with a step you must notice

Stem. The table gives values of the linear function $f$ . Which equation defines $f$ ?

$x$	3	7	11
$f(x)$	5	17	29

Step 1 — the input step, first. The $x$ -values climb by 4 each time. Write “ $+4$ ” down before looking at the outputs.

Step 2 — the rate. The outputs climb by 12 each time, and 12 is the change across 4 units of input:

m = \frac{17 - 5}{7 - 3} = \frac{12}{4} = 3

Step 3 — anchor the constant. Use any row; $(3, 5)$ is closest:

5 = 3(3) + b \quad\Rightarrow\quad b = 5 - 9 = -4

So $f(x) = 3x - 4$ .

Check on the row that was not the anchor: $3(11) - 4 = 33 - 4 = 29$ . ✓ Answer: $f(x) = 3x - 4$ .

Trap watch. Reading the jump 12 as the rate and anchoring at $(3,5)$ gives $f(x) = 12x - 31$ , which reproduces the anchor row perfectly and then predicts 53 where the table says 17 — exactly why the check must use a different row. Reading the first output 5 as the constant gives $f(x) = 3x + 5$ , which fails at $x = 3$ by 9, the value of $m$ times the missing distance back to $x = 0$ .

Worked example 2 — a graph, a rate with units, and a zero

Stem. A bricklayer works from a full pallet. The graph of $y = B(t)$ is a line through $(0, 240)$ and $(8, 80)$ , where $B(t)$ is the number of bricks left on the pallet after $t$ hours. (a) What is the best interpretation of the rate of change of $B$ ? (b) After how many hours is the pallet empty?

(a) Rate, then units.

m = \frac{80 - 240}{8 - 0} = \frac{-160}{8} = -20

The output is bricks and the input is hours, so the rate is $-20$ bricks per hour: for each additional hour of work, the number of bricks left decreases by 20. The sign is not a detail — it is the difference between laying bricks and receiving them.

(b) A zero of the function. The point $(0, 240)$ sits on the vertical axis, so the constant can be read directly: $B(t) = 240 - 20t$ . “Empty” means the output is 0, so this is an input question:

240 - 20t = 0 \quad\Rightarrow\quad 20t = 240 \quad\Rightarrow\quad t = 12

Check. $B(12) = 240 - 20(12) = 240 - 240 = 0$ ✓, and $B(8) = 240 - 160 = 80$ , which is the second plotted point. Answers: the pallet loses 20 bricks per hour; it is empty after 12 hours.

Trap watch. Part (b) has two attractive wrong numbers. $240$ is the output when the input is 0 — the mirror image of the question, and it is offered as a choice on this archetype every time. $20$ is the rate, an answer in bricks per hour to a question asked in hours. When a question says “0”, find out which axis the zero is on before you solve.

Worked example 3 — two pairs, then a question in the other direction

Stem. For the linear function $f$ , $f(4) = 19$ and $f(10) = 37$ . For what value of $x$ does $f(x) = 52$ ?

Step 1 — pairs are points. $f(4) = 19$ is the point $(4, 19)$ ; $f(10) = 37$ is $(10, 37)$ . Rewriting them this way is what makes the next step obviously a slope.

Step 2 — rate.

m = \frac{37 - 19}{10 - 4} = \frac{18}{6} = 3

Step 3 — constant. Anchor at $(4, 19)$ : $19 = 3(4) + b$ , so $b = 7$ and $f(x) = 3x + 7$ .

Step 4 — now read the direction of the question. The 52 is an output, so $x$ is the unknown:

3x + 7 = 52 \quad\Rightarrow\quad 3x = 45 \quad\Rightarrow\quad x = 15

Check. $f(15) = 3(15) + 7 = 45 + 7 = 52$ ✓, and the equation still returns the givens: $f(10) = 37$ ✓. Answer: $x = 15$ .

Trap watch. Computing $f(52) = 163$ answers the reversed question. Answering $52$ repeats the number the stem handed you. And skipping straight from “rate is 3” to $52 \div 3$ forgets the constant entirely — the shortcut only works when $b = 0$ , which the test rarely allows.

Practice

Answer before you open the explanation. Three items hand you a figure — two tables and one graph — and the numbers you need are only there, not in the sentence. Two items are student-produced response: no choices, so an input-for-output slip has nowhere to hide. Every wrong choice below is one specific, named error; when you miss one, log the error’s name, not the item number.

Choice A: Inverts the ratio, dividing the change in input by the change in output (8 / 24). Rate of change always reports output per input, so the output difference belongs on top.
Choice C: Reports 10 - 2, the change in the input alone. That is the width of the interval, not how fast the function moves across it.
Choice D: Reports 54 - 30, the total change in output across the whole interval. That change took 8 units of input to accumulate, so it must be divided by 8 before it describes a per-unit rate.

Question 6 Harder

The table gives four values of x and their corresponding values of the linear function f. Which equation defines f?

x	f(x)
2	9
5	21
8	33
11	45

Selected values of the linear function f.

Show the answer Choice A

Why it is right

The outputs rise by 12 from row to row, but the inputs rise by 3, so the rate is 12 / 3 = 4 per unit of x, not 12. Anchor with any row to find the constant: 9 = 4(2) + b gives b = 9 - 8 = 1. So f(x) = 4x + 1. Every row confirms it: 4(2) + 1 = 9, 4(5) + 1 = 21, 4(8) + 1 = 33, and 4(11) + 1 = 45. Note that the table never shows x = 0, so the constant had to be computed rather than read.

Why each other choice fails

Choice B: Takes the first output, 9, as the constant term. The constant is the output at x = 0, and the table starts at x = 2; this equation gives 17 at x = 2, not 9.
Choice C: Uses the right rate but adds when isolating the constant: 9 = 4(2) + b becomes b = 9 + 8. Subtracting 8 from both sides is what isolates b, so the sign is reversed.
Choice D: Reads the output difference 12 as the rate, ignoring that the inputs step by 3. Anchored at (2, 9) it reproduces the first row exactly, which is why it survives a one-row check, but at x = 5 it gives 45 instead of 21.

Question 7 Harder

A conveyor scanner logs packages at a constant rate. The table gives the total number of packages scanned, f(x), at selected times x minutes after 8:00 a.m. How many packages have been scanned 20 minutes after 8:00 a.m.?

x (minutes)	f(x) (packages)
2	17
6	29
10	41
14	53

Total packages scanned x minutes after 8:00 a.m.

Show the answer Choice D

Why it is right

The totals rise by 12 while the times rise by 4, so the scanner logs 12 / 4 = 3 packages per minute. From the last row, x = 14 with 53 packages, the jump to x = 20 is 6 minutes, which adds 6(3) = 18 packages: 53 + 18 = 71. The equation route agrees: the constant is 17 - 3(2) = 11, so f(x) = 3x + 11 and f(20) = 60 + 11 = 71, which also reproduces all four table rows.

Why each other choice fails

Choice A: Answers the reversed question. Solving 3x + 11 = 20 gives x = 3, the time at which 20 packages had been scanned; the stem gives a time and asks for a count.
Choice B: Adds the per-minute rate once, 53 + 3, as though 20 were one step past 14. The gap is 6 minutes, so the rate applies six times.
Choice C: Continues the table by one more row instead of going to the stated time: adding 12 to 53 lands on x = 18, not x = 20, because each printed row advances 4 minutes.

Question 8 Harder

The graph of y = f(x) is shown in the xy-plane, where f is a linear function. Two points on the graph are marked. What is the value of x for which f(x) = 12?

The graph of y = f(x), with two lattice points marked.

Show the answer Choice B

Why it is right

Build the function from the two marked lattice points. From (0, 6) to (5, 16) the output rises 10 while the input rises 5, so the rate is 10 / 5 = 2, and the marked point on the vertical axis gives the constant directly: f(x) = 2x + 6. The question fixes the output at 12 and asks for the input, so solve 2x + 6 = 12, giving 2x = 6 and x = 3. On the graph this is the point (3, 12): start at 12 on the vertical axis, move across to the line, then read down to the horizontal axis.

Why each other choice fails

Choice A: Reads the wrong axis. The value 6 is where the line meets the vertical axis, which is f(0), the output when the input is 0 -- not the input that produces an output of 12.
Choice C: Repeats the number given in the question. The 12 is the output already stated; the answer must be the matching input, read on the horizontal axis.
Choice D: Evaluates f(12) = 2(12) + 6 = 30 instead of solving f(x) = 12. That treats 12 as an input, which is the reverse of what the notation f(x) = 12 says.

Question 9 Harder Student-produced response

For the linear function f, f(3) = 11 and f(7) = 27. What is the value of f(12)?

Your answer

Show the answer 47

Why it is right

The two given pairs are the points (3, 11) and (7, 27). The output changes by 27 - 11 = 16 while the input changes by 7 - 3 = 4, so the rate is 16 / 4 = 4 per unit. From x = 7 to x = 12 is 5 units, which adds 5(4) = 20: f(12) = 27 + 20 = 47. The equation route agrees: the constant is 11 - 4(3) = -1, so f(x) = 4x - 1 and f(12) = 48 - 1 = 47.

Answers students type instead

16: Reports the total change in output between the two given points rather than a value of the function.
31: Adds the per-unit rate once, 27 + 4, instead of five times. The rate applies to every unit of input between 7 and 12.
43: Repeats the jump of 16 one more time, as if x = 12 were the same distance past 7 that 7 was past 3. The first gap is 4 units and the second is 5, so the jumps are not equal.

Question 10 Hardest

For the linear function f, the value of f(x) increases by 21 when the value of x increases by 6. What is the value of f(10) - f(4)?

Show the answer Choice C

Why it is right

For a linear function the change in output depends only on how far the input moved, never on where it started. The inputs 4 and 10 differ by 10 - 4 = 6, which is exactly the increase described in the stem, so the output difference is the same 21. Working through the rate gives the same number: the rate is 21 / 6 = 3.5 per unit, and 3.5(6) = 21. The constant term never enters, which is why the question is answerable without knowing a single value of f.

Why each other choice fails

Choice A: Reports the rate of change, 21 / 6 = 3.5. That is the change per single unit of x; the question spans 6 units, so the rate still has to be multiplied by 6.
Choice B: Multiplies the rate by the input 4 instead of by the interval width, 3.5(4) = 14. Differences of outputs depend on the gap between the inputs, not on either input by itself.
Choice D: Multiplies the rate by the input 10, 3.5(10) = 35. This is what f(10) - f(0) would be, and the question starts at 4, not at 0.

Question 11 Hardest Student-produced response

The linear function d models the depth of snow, in centimeters, on a mountain trail t days after a storm. The model gives a depth of 24 centimeters when t = 0 and a depth of 6 centimeters when t = 9. According to the model, how many days after the storm is the snow depth 0 centimeters?

Your answer

Show the answer 12

Why it is right

The depth falls from 24 to 6 centimeters over 9 days, so the rate is (6 - 24) / 9 = -18 / 9 = -2 centimeters per day, and the depth at t = 0 is 24. The model is d(t) = 24 - 2t. Setting the output to 0 gives 24 - 2t = 0, so 2t = 24 and t = 12 days. Check: d(12) = 24 - 24 = 0, and d(9) = 24 - 18 = 6, which matches the second given value.

Answers students type instead

2: Reports the rate of melting, 2 centimeters per day, rather than the number of days.
3: Finds that the remaining 6 centimeters take 3 more days but reports only those extra days. The question counts from the storm, so the 9 days already elapsed must be included: 9 + 3 = 12.
24: Reports the depth at t = 0 instead of the time at which the depth is 0. The two zeros sit on different axes -- one is an output at input 0, the other is the input that produces output 0.

Question 12 Hardest

The function f is linear, and f(x + 1) - f(x) = 7 for every value of x. If f(2) = 20, what is the value of f(0)?

Show the answer Choice C

Why it is right

The expression f(x + 1) - f(x) is the change in output produced by a one-unit increase in input, so the statement says the rate of change is 7. Going from x = 2 down to x = 0 is two unit steps in the negative direction, and each step lowers the output by 7: f(0) = 20 - 2(7) = 20 - 14 = 6. The equation confirms it: f(x) = 7x + 6, which gives f(2) = 14 + 6 = 20 and f(3) - f(2) = 27 - 20 = 7.

Why each other choice fails

Choice A: Assumes f(0) equals f(2). A rate of change of 7 is not zero, so the function cannot hold the same value at two different inputs.
Choice B: Subtracts one step of 7 instead of two. The distance from x = 2 back to x = 0 is 2 units, so the rate applies twice; 13 would be f(1).
Choice D: Adds 14 instead of subtracting it. Moving from x = 2 to x = 0 decreases the input, and with a positive rate a decreasing input means a decreasing output; 34 is f(4).

Common mistakes

Reading a table’s output jump as the rate — the difference between consecutive $f(x)$ values is the rate only when the $x$ column steps by exactly 1. Divide by the step, every time.
Answering an output when an input was asked — computing $f(12)$ when the question said $f(x) = 12$ . The number inside the parentheses is the only one going in.
Reading the constant off the wrong axis — reporting the $y$ -intercept when the question asked where the graph meets the horizontal axis, or vice versa. “Zero” is a value on one axis and a location on the other.
Taking the first table output as $b$ — legitimate only when that row sits at $x = 0$ . Most tables start somewhere else, and the constant has to be computed.
Inverting the rate — dividing change in input by change in output. Rate of change always reports output per input, so the output difference goes on top.
Advancing by rows instead of by units — a table stepping by 4 does not put $x = 20$ one row past $x = 14$ . Count units of $x$ , not printed lines.
Applying the rate once instead of $n$ times — adding a single step of $m$ to jump from $x = 7$ to $x = 12$ , which is five steps.
Losing the sign of a decreasing model — writing $+20t$ for a pallet that is emptying. The direction lives in the sign of $m$ , and a check against any given pair exposes it instantly.
Interpreting a single pair as a rate — “ $f(6) = 42$ ” says where the function is at one input, not how fast it is changing. One point can never carry a rate.
Combining a start and a rate into one number — adding the starting amount to the per-unit amount because both are printed in the same sentence. They have different units and can never be added.

FAQ

Is $f(x)$ just another name for $y$ ? For graphing purposes, yes: the graph of $y = f(x)$ is the same picture you would draw for $y = mx + b$ . The notation buys you something the $y$ form cannot say, though — $f(6)$ names one specific output without drawing anything, which is why the test asks its interpretation questions in function language.

How do I know a table really is linear? Compare first differences with the input step held constant. If $x$ climbs by the same amount each row and $f(x)$ also climbs by the same amount each row, the function is linear. If the $x$ -steps are uneven, divide each output jump by its own input jump and check that those quotients agree.

What does the constant term mean in a real context? It is the output when the input is 0: the starting amount, the fixed fee, the depth before the melt began. It is not the smallest value in the table, and it is not the first row unless that row happens to sit at $x = 0$ .

The question says “for what value of $x$ ” — can I still just plug in the choices? Yes, and on a nasty-looking item that is often fastest: substitute each choice into the rule and keep the one that returns the stated output. It also immunizes you against the input/output mix-up, because you are always feeding the choices in as inputs.

Do I need to memorize $f(x) = mx + b$ ? You need to recognize it in any arrangement. $f(x) = 12 - 5x$ is the same shape as $f(x) = -5x + 12$ ; the rate is $-5$ in both, and reading the rate as $12$ because it is printed first is a real and common loss.

Can a grid-in answer here be negative or fractional? Both. Rates of change on this skill are frequently negative, and a fraction like -3/2 should be entered exactly rather than rounded — an exact entry always scores.

Editorial team · last reviewed 2026-08-06

Linear functions

On the test

Where this skill ends and its neighbours begin

Pattern recognition

Method

Rate of change, in each outfit

Worked example 1 — a table with a step you must notice

Worked example 2 — a graph, a rate with units, and a zero

Worked example 3 — two pairs, then a question in the other direction

Practice

Question 1 Warm-up

Why it is right

Why each other choice fails

Question 2 Standard

Why it is right

Why each other choice fails

Question 3 Standard

Why it is right

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Question 4 Standard

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Why each other choice fails

Question 5 Standard

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Question 6 Harder

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Why each other choice fails

Question 7 Harder

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Why each other choice fails

Question 8 Harder

Why it is right

Why each other choice fails

Question 9 Harder Student-produced response

Why it is right

Answers students type instead

Question 10 Hardest

Why it is right

Why each other choice fails

Question 11 Hardest Student-produced response

Why it is right

Answers students type instead

Question 12 Hardest

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Why each other choice fails

Common mistakes

FAQ

Related skills