Linear Functions on the Digital SAT: the Desmos Method
A linear function is a straight line, so every question about one is really a question about a point on that line. Graph the line, pin down the coordinate you were given, and read off the coordinate you were asked for.
12 min read·Algebra·65 practice problems in DesPrep
How often it comes up
Roughly 3 to 5 of the 44 math questions across the two modules, and usually among the quicker ones. You should be finishing them in under a minute each.
Why this is on the test
Algebra is the biggest of the four domains on the digital SAT math section, and linear functions are the part of it the College Board tests most plainly. They ask it because it checks one thing: can you move between a rule, a graph and a pair of numbers without getting lost. These questions are not hard. They are fast, and they are worth exactly the same as the hardest question on the test. Every one you do by hand costs you two things. It costs a minute you needed later, and it hands you a sign error, because f(-4) for f(x) = -8x - 9 has two negatives in it and your pencil only tracks one. On the graph there are no signs to track. You type the rule, you click a dot, you read a number.
The Desmos playbook for linear functions
When you see the question on the left, type the middle line into Desmos and read the number on the right. This is the whole skill, in 6 moves.
| When you see | Type in Desmos | Read off |
|---|---|---|
| The question asks for f(a), a single function value | y = the rule, then x = a in a second line | Click the crossing and read the y-coordinate. That is the function value. |
| The question gives you the output: for what value of x is f(x) = k | y = the rule, then y = k in a second line | Click the crossing and read the x-coordinate. The k is already in the question, so it is not the answer. |
| You are given two points and asked for a value somewhere else | Work out rise over run, then y = m(x - x1) + y1, then x = the value you need. If you type the slope as a fraction, press the right arrow to leave the denominator before you type anything more. | Click the crossing and read the y-coordinate. |
| The question wants a y-intercept, a starting amount, a joining fee or a flat charge | Build the line from what you are given, then add x = 0 | Click the crossing on the vertical axis and read the y-coordinate. |
| The graph is shifted up or down and you need an x-intercept | Add or subtract the shift from the constant only, graph the new line, then add y = 0 | Click the crossing on the horizontal axis and read the x-coordinate. |
| The question asks where f(x) - g(x) equals some number | y = (the f rule) - (the g rule), brackets round the second one, then y = that number | Click the crossing and read the x-coordinate. |
Before you start, you should be able to
- Type an equation into a Desmos expression line and see the graph appear
- Read a coordinate off a labeled point, x first, then y
- Recognize y = mx + b as a line with slope m and y-intercept b
- Work out rise over run between two points, even if you are slow at it
01f(5) is not an instruction to do arithmetic
f(5) means one thing: go along to x = 5, and read how high the line is there. That is it. The number in the brackets is a position on the horizontal axis. The value of f(5) is the height above that position.
So f(x) = 8x - 3 is a line, and f(5) is a point on it. You do not have to multiply anything. You have to find a point.
Once you see it that way, this whole skill collapses into one move with two directions. Either you are given the x and asked for the height, or you are given the height and asked for the x. The graph answers both, and it answers them the same way: put two things on screen that cross, click the crossing, read the coordinate you were asked for.
One rule for this app before you start. Type the rule as y = , not as f(x) = . The step checklist watches for a graph, and f(x) = 8x - 3 is a definition, not a graph. Same math, but the checklist will sit there unticked and you will not know why.
02Given x, pin it with a vertical line
You could graph the line and try to trace along it to x = 5 with the arrow keys. Do not. It is slow and you can land on 4.98.
Instead, put a vertical line exactly where you need it. In Desmos, x = 5 is a legal expression on its own. It draws a vertical line straight up through 5. Where it hits your function is a real intersection point, and Desmos labels it with exact coordinates the moment you click it.
So it is always two lines and a click. One line for the rule. One line for the input. Click the dot.
That costs you one extra expression and saves you every negative-times-negative slip you were going to make.
The rest of this lesson is in DesPrep
Linear Functions continues with 4 more sections, and a worked example on every one of them:
- Given the output, read the x instead
- When you get points instead of a rule
- Intercepts are just crossings with an axis
- Two lines, one question, one coordinate
Then 65 practice problems on this skill alone, solved inside a live Desmos calculator with step feedback and an expert replay on every one.
Mistakes that cost marks on linear functions
Reading the wrong coordinate off the crossing
The intersection label gives you two numbers and the question wants one. On the drone problems the time in seconds is sitting in the answer choices right next to the height. On any question that starts for what value of x, the y-coordinate is a listed choice too.
FixBefore you click, say what unit the answer is in. Metres means y. Seconds means x. Value of x means x. Value of the function means y.
Typing a slope as a fraction and losing everything after it into the denominator
In Desmos, / opens a fraction and the cursor stays in the denominator until you press the right-arrow key. Typing -9/3(x - 1) - 8 straight through builds -9 over 3(x - 1) - 8. The graph is wrong, the step will not tick, and nothing on screen tells you why.
FixDivide in your head and type the whole number. Every slope in this skill divides evenly. If you do type the fraction, press the right arrow the moment the denominator is finished, and check that the fraction bar on screen covers only the number you meant.
Subtracting the second function without brackets
f(x) - g(x) with g(x) = 3x + 10 means you subtract the 10 as well. Typing -3x + 10 - 3x + 10 leaves that constant positive, which gives a line that crosses in the wrong place. The distractors on those questions are built from this exact slip, usually as the right answer with the sign flipped.
FixPut the second rule in brackets every time and let Desmos do the subtraction. If your answer and one of the choices are the same number with opposite signs, this is what happened.
Shifting the graph by changing x instead of the constant
Shifted down 12 units means the whole line drops, which is subtracting 12 from the constant term. Students who touch the x term instead get a line with the wrong steepness, and the shift amount itself is usually one of the answer choices waiting for them.
FixDown k means minus k on the end. Up k means plus k on the end. The slope never changes in a vertical shift, so check the new line looks parallel to the original.
Answering with a number the question already handed you
The input, the y-value of a given point, the target height, the shift amount: every one of these turns up in the answer choices somewhere in this set. They are there because a student who runs out of time writes down the last number they read.
FixIf your answer is a number you can find by glancing back at the question text, treat that as a warning and re-read the crossing.
Typing f(x) = 8x - 3 instead of y = 8x - 3
Desmos accepts both, but f(x) = 8x - 3 defines a function without drawing it, and the step checklist here is watching for a graph. Your math is fine and the step stays unticked.
FixEnter the rule as y = when you practice. If you want the function name too, define it and then graph y = f(x) on a second line.
Check yourself on linear functions
4 questions that are not in the practice bank. Work each one the Desmos way before you open the answer.
- 1
The function f is defined by f(x) = -4x + 9. What is the value of f(3)?
Show the answer
Answer: -3
f(3) is the height of the line at x = 3. Type y = -4x + 9, then x = 3 on its own line, and click the intersection: (3, -3). The answer is the y-coordinate, -3. Doing it by hand risks losing the sign on -4 times 3.
- 2
The function f is defined by f(x) = 3x - 8. For what value of x is f(x) = 13?
Show the answer
Answer: 7
This one runs backwards: you are given the output and asked for the input. Type y = 3x - 8 and y = 13, then click the crossing at (7, 13). Read the x-coordinate this time, not the y.
- 3
The graph of y = -3x + 15 in the xy-plane is shifted down 6 units. What is the x-coordinate of the x-intercept of the new graph?
Show the answer
Answer: 3
Shifting down 6 subtracts 6 from the constant, so the new line is y = -3x + 9. Type that, add y = 0, and click the crossing at (3, 0). Shift the equation before you graph it, rather than graphing the old line and trying to picture the new one.
- 4
A linear function passes through the points (0, -4) and (2, 6). What is the value of the function when x = 5?
Show the answer
Answer: 21
The slope is (6 - (-4))/(2 - 0) = 5, and (0, -4) hands you the y-intercept, so the line is y = 5x - 4. Type it with x = 5 and click: (5, 21). You can also type y = 5(x - 0) - 4 without simplifying — the checklist compares graphs, not spellings.
More Algebra guides
- Linear Equations in One VariableSolve a linear equation by graphing each side as its own function and reading the x-coordinate where they meet — no algebra, no sign errors.
- Linear Equations in Two VariablesRead intercepts and paired values straight off the line, including equations given in standard form.
- Systems of Linear EquationsGraph both equations and click where they cross — the fastest route to a solution, and it never makes an elimination error.
- Linear InequalitiesGraph the boundary line and let the picture tell you which values satisfy the inequality.