Guide

SAT Desmos cheat sheet

Everything the Learn Desmos course teaches about the Desmos graphing calculator on SAT Math, on one page. Keep it beside the calculator while you practice, then read it again the night before the test. Every keystroke below is drilled somewhere in the 27 chapters, so the course explains anything here that looks new.

Last updated: September 5, 2026

The calculator you get on test day

Everything on this page assumes the graphing calculator built into Bluebook. Per the calculator policy College Board publishes for the SAT Suite, that embedded Desmos calculator comes in two forms, graphing or scientific, a handheld calculator you own and that is on the approved list is allowed as an alternative, and nothing at all is allowed during Reading and Writing. Chapter 1 of the course shows you how to open the graphing form in a browser so you can practice on it for free.

One thing to be clear about before the tables start. This is an independent study page. It describes a calculator someone else built, for a test someone else writes, and nobody at College Board or Desmos reviewed a word of it.

The one rule behind every trick

Only lowercase x and y graph. Every other letter, including capital X, is a constant that Desmos will ask you to define or, with a tilde, solve for.

The Graphing and Regression sections cover an estimated 53.55% of SAT Math question types (the curriculum’s own estimate: Graphing 25.24% + Regression 28.31%). The Prerequisites section teaches every keystroke below first.

Typing shortcuts, key by key

Every key on the on-screen keypad is faster from the keyboard, so keep the keypad closed and type. These are the keys taught in the first Prerequisites chapter, Using Desmos as a Calculator.

Operators and symbols
You want	Press	What happens
Add, subtract	`+` and `-` (the hyphen next to the 0)	The answer appears at the right end of the row.
Multiply	`*` (Shift + 8)	The asterisk becomes a dot on screen, so $5\cdot 2$ reads 5·2 and returns 10. The letter x is never a times sign here; Desmos hears the variable.
Divide	`/`	Whatever sits immediately left of the cursor is all that jumps up as the numerator, which makes $\frac{5+3}{2}$ equal 6.5. Want the whole sum on top? Select $5+3$ , then press the key.
Exponent	`^` (Shift + 6)	Type the parentheses first: $2^{(1+2)}$ is 8, but $2^{1}+2$ is 4. The right arrow comes back down out of the exponent.
Brackets	`(` and `)`	They decide what gets raised or rooted: $(2+3)^{2}$ is 25, and $\sqrt{9+16}$ is 5.
Square root, cube root	Type `sqrt` or `cbrt`	The letters become a root sign with the cursor inside it. `cbrt` then 27 gives 3.
nth root	Type `nthroot`	Degree first, right arrow to move under the root, then the number: 4, right arrow, 16 gives 2.
Pi	Type $\pi$	Becomes the π symbol.
Percent	`%` (Shift + 5), then `of`	$50\% of 40$ gives 20. Desmos needs the word `of` after the sign.
Absolute value	`\|` (Shift + backslash)	Bar, `x`, bar: $\|x\|$ .
Less than, greater than	Shift + comma, Shift + period	Press `=` right after `<` or `>` and Desmos joins the two into a single ≤ or ≥.
Tilde (regression)	`~` (Shift + the backtick key under Escape)	On the keypad it is under ABC. One tilde per line.
Subscript	A letter followed by a number	$x_{1}$ becomes x₁ by itself; the underscore, $x_{1}$ , gets you the same thing. The right arrow leaves the subscript.
Curly brackets (conditions)	Shift + `[` and Shift + `]`	$y = x \left\{x > 3\right\}$ draws the line only to the right of 3.
Square brackets (lists)	`[` and `]`	$\left[1, 2, 3\right]$ is one object Desmos treats like a number.

Calculator controls and table tricks
Task	How	Why it matters
New table	Type the word `table` on an empty line	Enter moves down a column, Tab moves across.
Undo, redo	Ctrl + Z, Ctrl + Shift + Z (Cmd on a Mac)	Same as the two arrows at the top of the list.
Reuse a line	Edit list, duplicate, Done	The app switches Ctrl + C and Ctrl + V off, so duplicating a row is the only shortcut left for getting a second copy.
Clear everything	Edit list, Delete All, Done	Leftover constants from the last question error out or quietly change this one.
Switch a graph off and on	Click the colored circle at the left of the line	Reveals a graph hiding exactly under another one.
Export a point	Click the graph, click a gray dot, click the arrow beside its label	Intercepts, intersections, maximums and minimums land in the list as real points.
Decimal to fraction	Click the icon at the left of a decimal result	It only appears when the decimal converts cleanly.
Radians or degrees	Wrench at the top right, then Degrees at the bottom	The default is radians. Set it before any trig expression.
Zoom	`+` and `-` buttons, scrolling, the home button, zoom fit on a table	The default view hides small gaps: a parabola that looks like it touches the axis often does not.
Slider bounds and step	Click an end value, type the minimum, Tab to the maximum, Tab to the step	A value outside the range expands it. Type small decimals like $0.03$ instead of dragging.
Equation to table	Edit list, table icon on the line, Done	The icon only shows up when the equation is solved for y, so $x + y = 5$ does not get one.
Read y at any x	Turn the equation into a table, click the x column, type a value	For $y = (x - 1)(x - 3)$ , typing 7 returns 24. You can only edit x; y is computed.
Factor pairs	Type $\frac{24}{x}$ , table icon, Done, then hold Enter on the last x	Each row adds 1 to x. Read the rows where both numbers are whole: 1 and 24, 2 and 12, 3 and 8, 4 and 6.

Syntax: what to type so Desmos understands you

Desmos has no solve button. It reads exactly what you type, so a handful of notation rules decide whether you get an answer or an error. This is the notation the course uses on every question.

Notation the course uses
Feature	Type this	Rule
Graph an equation	$y = 2x - 3$	Anything with both x and y needs an equals sign. $y =$ in front of an expression makes it graph every time.
Single-variable equation	$2x + 3 = 7$	One vertical line comes up per solution. The x-coordinate where you click is the answer; whatever y comes back means nothing.
Define a constant	$a = 3$	Letter first. $3 = a$ is an error. Never define a letter twice, and never use `e`, which Desmos keeps for Euler’s number.
Constant as a point or list	$p = (2, 5)$ and $L = \left[1, 2, 3\right]$	Use `p` or `L` on later lines like a number.
Function notation	$f(x) = 2x + 1$ , then $f(5)$	Define first, then evaluate. $g(f(3))$ works inside out. $f(x) + 3$ shifts up, $f(x - 2)$ shifts right. Any letter works.
Absolute value	$\|x\|$	Graphs like any other expression.
Inequality	$y < 2x + 1$ and $y \ge -x - 2$	Strict is dashed and excludes the line; non-strict is solid and includes it. Two inequalities: the answer is the double-shaded region.
Point	$(1, 3)$	A typed point like `(a, b)` can be dragged and updates its constants. Table points cannot.
Regression (tilde)	$7 \sim 2x_{1} + 1$	Desmos adjusts every still-undefined constant until both sides match, and gives x₁ = 3. Lowercase x and y throw an error, so use a subscript or a capital letter. One tilde per line.
Subscripts	$x_{1}$ and $y_{1}$	The column names of your first table; a second table gives $x_{2}$ and $y_{2}$ . A table is just $x_{1} = \left[list\right]$ and $y_{1} = \left[list\right]$ .
Table regression	$y_{1} \sim mx_{1} + b$	Word for word what Export as custom regression writes, under the three dots beside the regression drop-down. Typing it yourself does exactly the same thing, which is how you use a form the drop-down does not offer, such as $y_{1} \sim a(x_{1} - h)$ .
Bracket regression	$\left[2x_{1} + 3y_{1}, x_{1} - y_{1}\right] \sim \left[12, 1\right]$	Every left side in one list, every right side in the other, same order. Solves the whole system at once.
Right sides in their own list	$r = \left[12, 1\right]$ , then $\left[2x_{1} + 3y_{1}, x_{1} - y_{1}\right] \sim r$	Naming the number list separately gives you one spot to proofread when a system runs long. Only that side can move out: a list containing unknown letters will not accept a definition of its own.
List	$\left[4, 8, 15, 16\right]$	Brackets are required outside a function; inside $\operatorname{mean}(\dots )$ they are optional. A list cannot hold another list.
Integer range	$\left[1\dots 10\right]$	Three dots between the numbers gives every integer from 1 to 10. The Desmos help pages add that $\left[1,3\dots 11\right]$ steps by 2, so you get the odd numbers.
Random list	$\operatorname{random}(10)$	Ten random numbers between 0 and 1, all new every time you edit the line.
Restriction on a graph	$y = x \left\{x > 3\right\}$	Curly brackets at the end limit the domain or range. The Desmos help pages also allow $\left\{x > 0 and y > 0\right\}$ for two conditions at once.
Restriction on a constant	$9 \sim k^{2} \left\{k > 0\right\}$	Forces the root you want. Right arrow out of the exponent before typing the bracket.
Chained condition	$\left\{a > b > 87\right\}$	One pair of brackets, several conditions, all true at the same time.

From question pattern to Desmos move

One row per chapter of the Graphing and Regression sections. Spot the pattern, make the move. Each chapter drills it in a live calculator, then tests it on four hard SAT-style questions.

Graphing section: type the equation and look at the picture
Chapter	Pattern to spot	Desmos move
Basic Graphing	An equation, plus one value the picture already contains: an intercept, or y at a given x	Type it with $y =$ in front and click the gray dot on the feature you want, then export it. For y at x = 2, put $x = 2$ on a second line and click where the two meet.
Functions	The only equals sign in the question is the one defining the function	Copy the definition across unchanged, $f(x) = 2x^{2} - 3x + 1$ , and then ask for the one thing named: $f(4)$ , or $f(g(1))$ when two functions are nested. Should a second equals sign turn up, as in f(a) = 21, the pattern has changed to regression: $f(a) \sim 21$ .
Inequalities	A `<` or `>` sign, usually two of them	Graph them, type each answer-choice point on its own line, and pick the dot sitting in the double-shaded region.
Circle Equations	Any circle equation: radius, center, or which point is inside	Graph it as written. Click the top and bottom points, not the left and right ones, which often sit on the x-axis where the dot and the axis overlap and are hard to click cleanly. Export both, name them $p =$ and $q =$ . Then $\operatorname{distance}(p, q)/2$ is the radius and $\operatorname{midpoint}(p, q)$ is the center. For inside or outside, plot the choices and look.
Multiple Solutions	Anything about the solutions themselves: add them, pick the biggest or smallest, or count them	Always graph. $x^{2} - 4x - 5 = 0$ gives one vertical line per solution. For a system, split at the equals sign into two $y =$ lines and count the intersections. Zoom in first: parallel lines look like one line.
Equivalent Expressions	"Equivalent to", or which choice equals the given expression	Put $y =$ in front of the given expression and every choice. The match sits exactly on top; switch the original off and on to be sure. If a constant is involved, set it to 2, never the slider default of 1.
Factoring	"Is a factor of" or "is divisible by"	A shared factor means a shared x-intercept. Graph both with $y =$ , add a slider, and type the choices in until the intercepts line up. Or substitute the root instead: $3^{2} + 3m - 12 \sim 0$ .

Regression section: state every condition, swap equals for a tilde
Chapter	Pattern to spot	Desmos move
Single Variable Equations	One expression set equal to another, with one letter in it	Every `x` becomes $x_{1}$ , `=` becomes `~`: $3(x_{1} + 2) - 5 \sim 2x_{1} + 7$ . Then type what the question actually asks for, $4x_{1} - 1$ . The words "is equivalent to" count as an equals sign. Regression hands back one solution, so add $\left\{x_{1} > 0\right\}$ to choose which, or graph it instead.
Constants	The question asks for a constant, not for the variable	Set it up as usual, $2(x_{1} + 3) + kx_{1} \sim 5x_{1} + 6$ , then force the variable to move: $x_{1} = \left[1\dots 10\right]$ on its own line. The fit now has to hold for all ten values at once, which only the true constant does. If $x_{1}$ appears under regression parameters, you forgot the list.
Infinite Solutions	The words "infinitely many solutions" are in the question, and a constant is what it wants	$3(x_{1} + 2) \sim ax_{1} + 6$ with $x_{1} = \left[1\dots 10\right]$ underneath. Use it only when infinitely many is stated as fact, never when it is one of the answer choices.
Systems of Equations	Two or more equations at once	Bracket regression, $\left[2x_{1} + 3y_{1}, x_{1} - y_{1}\right] \sim \left[12, 1\right]$ , same order in both lists, then ask for $x_{1} + y_{1}$ . Percent word problems come out the same way once "is" and "was" become equals signs: $\left[p, p\right] \sim \left[1.2c, 60\right]$ .
Given Points	Points written out, read off a graph in the question, or hidden in function notation: f(2) = 8 is the point (2, 8)	Type the known points into a table, then fit $y_{1} \sim mx_{1} + b$ , or whichever form the question uses, such as $y_{1} \sim a(x_{1} - h)$ . Circle standard form and quadratic vertex form are the only two worth memorizing. Check $R^2$ under statistics: it has to be exactly 1, or the model is the wrong shape.
Isolation	"Expresses R in terms of P, Q and F"	Type it as $F \sim (P - Q)/R$ . Leave R undefined and pin every other letter to a different value, never 0 or 1: $P = 11$ , $Q = 3$ , $F = 2$ . Desmos returns R = 4. Now type each answer choice and keep the one that also gives 4.
Choosing Your Method	Every question, before you type anything	If there is an equation in it, Desmos can do it. Then pick the tool: a tilde returns one unknown number, while the graph shows shape, count and comparison.
Desmos Theory	Several stated conditions in one question	Turn the question into a checklist and meet every item: $f(x) = (x - a)^{2} + b$ , then $\left[f(a), f(100)\right] \sim \left[90, 190\right] \left\{a > b > 87\right\}$ , then $a + b$ . When the answer is an expression rather than a number, pick constants that fit the constraint, read the value, then move them and check it did not change.

Statistics and number functions

Type the name, open a bracket, separate the numbers with commas. Store a list in a constant to reuse it: $L = \left[4, 8, 15, 16\right]$ , then $\operatorname{mean}(L)$ , $\operatorname{median}(L)$ and $\operatorname{stdev}(L)$ each work in one go. All of these are taught in the Desmos & Statistics chapter.

Functions the course teaches
Function	Type this	Result
Mean	$\operatorname{mean}(4, 8, 15, 1)$	7
Median	$\operatorname{median}(4, 8, 15, 1)$	6, halfway between 4 and 8.
Standard deviation	$\operatorname{stdev}(2, 4, 4, 6)$	1.63. It is `stdev`, not `stdevp`.
Distance	$\operatorname{distance}((1, 2), (4, 6))$	5. Each point in its own brackets.
Midpoint	$\operatorname{midpoint}((1, 2), (4, 6))$	(2.5, 4), and Desmos plots it for you.
Percent of	$15\% of 80$	12
Greatest common divisor	$\gcd (36, 24)$	12
Remainder	$\operatorname{mod}(17, 5)$	2
Smallest, largest	$\min (4, 8, 15, 1)$ , $\max (4, 8, 15, 1)$	1 and 15
Round down, up, nearest	$\operatorname{floor}(4.6)$ , $\operatorname{ceil}(4.6)$ , $\operatorname{round}(4.6)$	4, 5 and 5. It is `ceil`, four letters, not `ceiling`.
Step graph	$\operatorname{floor}(x)$	Put x inside instead of a number and the graph becomes a staircase; $\operatorname{ceil}(x)$ does the same one step higher.

Test-day settings and habits

Put $y =$ in front of anything you want drawn. Without it some expressions return a number, some error, and some graph anyway, so make it a habit rather than a decision.
Split every equation at the equals sign into two $y =$ lines. On one line, no solutions and infinitely many both draw nothing, and you cannot tell them apart.
Zoom in before you commit to what the picture shows, and zoom out before answering a greatest-possible-value question. The largest value is often off the default screen.
When you pick values for constants yourself, never 0 or 1, never two the same, never e. Those choices hide mistakes instead of exposing them.
Trust whole numbers and tidy fractions. Something like 2.333 coming back is nearly always a badly typed setup rather than an ugly answer, so go back to the question.
Regression returns one solution and hides the rest. Counting solutions or adding them is a graph job.
Regression only solves for letters that are still undefined. If b is already set on another line, $7 \sim 2m + b$ solves for m alone. Rename the letter when you meant a new one.
Do not over-constrain. Pin both variables and the constant, and the system has more conditions than unknowns, so Desmos solves nothing at all.
Once you are using bracket regression you cannot also set a variable equal to a list. Vary that variable by hand instead: set it to 2, read the constant, set it to 5, check the constant did not move.
You need at least as many points as constants you are solving for: two for a line, three for a parabola in standard form. Vertex form needs only one, if the question hands you the vertex.
Only graph the points you know, never the one you are solving for.
No fraction icon beside a result means the answer is probably the decimal.
A question with no equation and no points anywhere in it is the one kind that stays on paper.

When not to use the tilde

A tilde balances two sides that are supposed to be equal to each other. An inequality describes a whole region rather than a value, and counting or summing solutions needs the whole picture at once. All three of those are graphed, every time.

Guide

SAT Desmos cheat sheet

Last updated: September 5, 2026

The calculator you get on test day

The one rule behind every trick

Only lowercase x and y graph. Every other letter, including capital X, is a constant that Desmos will ask you to define or, with a tilde, solve for.

Typing shortcuts, key by key

Every key on the on-screen keypad is faster from the keyboard, so keep the keypad closed and type. These are the keys taught in the first Prerequisites chapter, Using Desmos as a Calculator.

Operators and symbols
You want	Press	What happens
Add, subtract	`+` and `-` (the hyphen next to the 0)	The answer appears at the right end of the row.
Multiply	`*` (Shift + 8)	The asterisk becomes a dot on screen, so $5\cdot 2$ reads 5·2 and returns 10. The letter x is never a times sign here; Desmos hears the variable.
Divide	`/`	Whatever sits immediately left of the cursor is all that jumps up as the numerator, which makes $\frac{5+3}{2}$ equal 6.5. Want the whole sum on top? Select $5+3$ , then press the key.
Exponent	`^` (Shift + 6)	Type the parentheses first: $2^{(1+2)}$ is 8, but $2^{1}+2$ is 4. The right arrow comes back down out of the exponent.
Brackets	`(` and `)`	They decide what gets raised or rooted: $(2+3)^{2}$ is 25, and $\sqrt{9+16}$ is 5.
Square root, cube root	Type `sqrt` or `cbrt`	The letters become a root sign with the cursor inside it. `cbrt` then 27 gives 3.
nth root	Type `nthroot`	Degree first, right arrow to move under the root, then the number: 4, right arrow, 16 gives 2.
Pi	Type $\pi$	Becomes the π symbol.
Percent	`%` (Shift + 5), then `of`	$50\% of 40$ gives 20. Desmos needs the word `of` after the sign.
Absolute value	`\|` (Shift + backslash)	Bar, `x`, bar: $\|x\|$ .
Less than, greater than	Shift + comma, Shift + period	Press `=` right after `<` or `>` and Desmos joins the two into a single ≤ or ≥.
Tilde (regression)	`~` (Shift + the backtick key under Escape)	On the keypad it is under ABC. One tilde per line.
Subscript	A letter followed by a number	$x_{1}$ becomes x₁ by itself; the underscore, $x_{1}$ , gets you the same thing. The right arrow leaves the subscript.
Curly brackets (conditions)	Shift + `[` and Shift + `]`	$y = x \left\{x > 3\right\}$ draws the line only to the right of 3.
Square brackets (lists)	`[` and `]`	$\left[1, 2, 3\right]$ is one object Desmos treats like a number.

Calculator controls and table tricks
Task	How	Why it matters
New table	Type the word `table` on an empty line	Enter moves down a column, Tab moves across.
Undo, redo	Ctrl + Z, Ctrl + Shift + Z (Cmd on a Mac)	Same as the two arrows at the top of the list.
Reuse a line	Edit list, duplicate, Done	The app switches Ctrl + C and Ctrl + V off, so duplicating a row is the only shortcut left for getting a second copy.
Clear everything	Edit list, Delete All, Done	Leftover constants from the last question error out or quietly change this one.
Switch a graph off and on	Click the colored circle at the left of the line	Reveals a graph hiding exactly under another one.
Export a point	Click the graph, click a gray dot, click the arrow beside its label	Intercepts, intersections, maximums and minimums land in the list as real points.
Decimal to fraction	Click the icon at the left of a decimal result	It only appears when the decimal converts cleanly.
Radians or degrees	Wrench at the top right, then Degrees at the bottom	The default is radians. Set it before any trig expression.
Zoom	`+` and `-` buttons, scrolling, the home button, zoom fit on a table	The default view hides small gaps: a parabola that looks like it touches the axis often does not.
Slider bounds and step	Click an end value, type the minimum, Tab to the maximum, Tab to the step	A value outside the range expands it. Type small decimals like $0.03$ instead of dragging.
Equation to table	Edit list, table icon on the line, Done	The icon only shows up when the equation is solved for y, so $x + y = 5$ does not get one.
Read y at any x	Turn the equation into a table, click the x column, type a value	For $y = (x - 1)(x - 3)$ , typing 7 returns 24. You can only edit x; y is computed.
Factor pairs	Type $\frac{24}{x}$ , table icon, Done, then hold Enter on the last x	Each row adds 1 to x. Read the rows where both numbers are whole: 1 and 24, 2 and 12, 3 and 8, 4 and 6.

Syntax: what to type so Desmos understands you

Desmos has no solve button. It reads exactly what you type, so a handful of notation rules decide whether you get an answer or an error. This is the notation the course uses on every question.

Notation the course uses
Feature	Type this	Rule
Graph an equation	$y = 2x - 3$	Anything with both x and y needs an equals sign. $y =$ in front of an expression makes it graph every time.
Single-variable equation	$2x + 3 = 7$	One vertical line comes up per solution. The x-coordinate where you click is the answer; whatever y comes back means nothing.
Define a constant	$a = 3$	Letter first. $3 = a$ is an error. Never define a letter twice, and never use `e`, which Desmos keeps for Euler’s number.
Constant as a point or list	$p = (2, 5)$ and $L = \left[1, 2, 3\right]$	Use `p` or `L` on later lines like a number.
Function notation	$f(x) = 2x + 1$ , then $f(5)$	Define first, then evaluate. $g(f(3))$ works inside out. $f(x) + 3$ shifts up, $f(x - 2)$ shifts right. Any letter works.
Absolute value	$\|x\|$	Graphs like any other expression.
Inequality	$y < 2x + 1$ and $y \ge -x - 2$	Strict is dashed and excludes the line; non-strict is solid and includes it. Two inequalities: the answer is the double-shaded region.
Point	$(1, 3)$	A typed point like `(a, b)` can be dragged and updates its constants. Table points cannot.
Regression (tilde)	$7 \sim 2x_{1} + 1$	Desmos adjusts every still-undefined constant until both sides match, and gives x₁ = 3. Lowercase x and y throw an error, so use a subscript or a capital letter. One tilde per line.
Subscripts	$x_{1}$ and $y_{1}$	The column names of your first table; a second table gives $x_{2}$ and $y_{2}$ . A table is just $x_{1} = \left[list\right]$ and $y_{1} = \left[list\right]$ .
Table regression	$y_{1} \sim mx_{1} + b$	Word for word what Export as custom regression writes, under the three dots beside the regression drop-down. Typing it yourself does exactly the same thing, which is how you use a form the drop-down does not offer, such as $y_{1} \sim a(x_{1} - h)$ .
Bracket regression	$\left[2x_{1} + 3y_{1}, x_{1} - y_{1}\right] \sim \left[12, 1\right]$	Every left side in one list, every right side in the other, same order. Solves the whole system at once.
Right sides in their own list	$r = \left[12, 1\right]$ , then $\left[2x_{1} + 3y_{1}, x_{1} - y_{1}\right] \sim r$	Naming the number list separately gives you one spot to proofread when a system runs long. Only that side can move out: a list containing unknown letters will not accept a definition of its own.
List	$\left[4, 8, 15, 16\right]$	Brackets are required outside a function; inside $\operatorname{mean}(\dots )$ they are optional. A list cannot hold another list.
Integer range	$\left[1\dots 10\right]$	Three dots between the numbers gives every integer from 1 to 10. The Desmos help pages add that $\left[1,3\dots 11\right]$ steps by 2, so you get the odd numbers.
Random list	$\operatorname{random}(10)$	Ten random numbers between 0 and 1, all new every time you edit the line.
Restriction on a graph	$y = x \left\{x > 3\right\}$	Curly brackets at the end limit the domain or range. The Desmos help pages also allow $\left\{x > 0 and y > 0\right\}$ for two conditions at once.
Restriction on a constant	$9 \sim k^{2} \left\{k > 0\right\}$	Forces the root you want. Right arrow out of the exponent before typing the bracket.
Chained condition	$\left\{a > b > 87\right\}$	One pair of brackets, several conditions, all true at the same time.

From question pattern to Desmos move

One row per chapter of the Graphing and Regression sections. Spot the pattern, make the move. Each chapter drills it in a live calculator, then tests it on four hard SAT-style questions.

Graphing section: type the equation and look at the picture
Chapter	Pattern to spot	Desmos move
Basic Graphing	An equation, plus one value the picture already contains: an intercept, or y at a given x	Type it with $y =$ in front and click the gray dot on the feature you want, then export it. For y at x = 2, put $x = 2$ on a second line and click where the two meet.
Functions	The only equals sign in the question is the one defining the function	Copy the definition across unchanged, $f(x) = 2x^{2} - 3x + 1$ , and then ask for the one thing named: $f(4)$ , or $f(g(1))$ when two functions are nested. Should a second equals sign turn up, as in f(a) = 21, the pattern has changed to regression: $f(a) \sim 21$ .
Inequalities	A `<` or `>` sign, usually two of them	Graph them, type each answer-choice point on its own line, and pick the dot sitting in the double-shaded region.
Circle Equations	Any circle equation: radius, center, or which point is inside	Graph it as written. Click the top and bottom points, not the left and right ones, which often sit on the x-axis where the dot and the axis overlap and are hard to click cleanly. Export both, name them $p =$ and $q =$ . Then $\operatorname{distance}(p, q)/2$ is the radius and $\operatorname{midpoint}(p, q)$ is the center. For inside or outside, plot the choices and look.
Multiple Solutions	Anything about the solutions themselves: add them, pick the biggest or smallest, or count them	Always graph. $x^{2} - 4x - 5 = 0$ gives one vertical line per solution. For a system, split at the equals sign into two $y =$ lines and count the intersections. Zoom in first: parallel lines look like one line.
Equivalent Expressions	"Equivalent to", or which choice equals the given expression	Put $y =$ in front of the given expression and every choice. The match sits exactly on top; switch the original off and on to be sure. If a constant is involved, set it to 2, never the slider default of 1.
Factoring	"Is a factor of" or "is divisible by"	A shared factor means a shared x-intercept. Graph both with $y =$ , add a slider, and type the choices in until the intercepts line up. Or substitute the root instead: $3^{2} + 3m - 12 \sim 0$ .

Regression section: state every condition, swap equals for a tilde
Chapter	Pattern to spot	Desmos move
Single Variable Equations	One expression set equal to another, with one letter in it	Every `x` becomes $x_{1}$ , `=` becomes `~`: $3(x_{1} + 2) - 5 \sim 2x_{1} + 7$ . Then type what the question actually asks for, $4x_{1} - 1$ . The words "is equivalent to" count as an equals sign. Regression hands back one solution, so add $\left\{x_{1} > 0\right\}$ to choose which, or graph it instead.
Constants	The question asks for a constant, not for the variable	Set it up as usual, $2(x_{1} + 3) + kx_{1} \sim 5x_{1} + 6$ , then force the variable to move: $x_{1} = \left[1\dots 10\right]$ on its own line. The fit now has to hold for all ten values at once, which only the true constant does. If $x_{1}$ appears under regression parameters, you forgot the list.
Infinite Solutions	The words "infinitely many solutions" are in the question, and a constant is what it wants	$3(x_{1} + 2) \sim ax_{1} + 6$ with $x_{1} = \left[1\dots 10\right]$ underneath. Use it only when infinitely many is stated as fact, never when it is one of the answer choices.
Systems of Equations	Two or more equations at once	Bracket regression, $\left[2x_{1} + 3y_{1}, x_{1} - y_{1}\right] \sim \left[12, 1\right]$ , same order in both lists, then ask for $x_{1} + y_{1}$ . Percent word problems come out the same way once "is" and "was" become equals signs: $\left[p, p\right] \sim \left[1.2c, 60\right]$ .
Given Points	Points written out, read off a graph in the question, or hidden in function notation: f(2) = 8 is the point (2, 8)	Type the known points into a table, then fit $y_{1} \sim mx_{1} + b$ , or whichever form the question uses, such as $y_{1} \sim a(x_{1} - h)$ . Circle standard form and quadratic vertex form are the only two worth memorizing. Check $R^2$ under statistics: it has to be exactly 1, or the model is the wrong shape.
Isolation	"Expresses R in terms of P, Q and F"	Type it as $F \sim (P - Q)/R$ . Leave R undefined and pin every other letter to a different value, never 0 or 1: $P = 11$ , $Q = 3$ , $F = 2$ . Desmos returns R = 4. Now type each answer choice and keep the one that also gives 4.
Choosing Your Method	Every question, before you type anything	If there is an equation in it, Desmos can do it. Then pick the tool: a tilde returns one unknown number, while the graph shows shape, count and comparison.
Desmos Theory	Several stated conditions in one question	Turn the question into a checklist and meet every item: $f(x) = (x - a)^{2} + b$ , then $\left[f(a), f(100)\right] \sim \left[90, 190\right] \left\{a > b > 87\right\}$ , then $a + b$ . When the answer is an expression rather than a number, pick constants that fit the constraint, read the value, then move them and check it did not change.

Statistics and number functions

Functions the course teaches
Function	Type this	Result
Mean	$\operatorname{mean}(4, 8, 15, 1)$	7
Median	$\operatorname{median}(4, 8, 15, 1)$	6, halfway between 4 and 8.
Standard deviation	$\operatorname{stdev}(2, 4, 4, 6)$	1.63. It is `stdev`, not `stdevp`.
Distance	$\operatorname{distance}((1, 2), (4, 6))$	5. Each point in its own brackets.
Midpoint	$\operatorname{midpoint}((1, 2), (4, 6))$	(2.5, 4), and Desmos plots it for you.
Percent of	$15\% of 80$	12
Greatest common divisor	$\gcd (36, 24)$	12
Remainder	$\operatorname{mod}(17, 5)$	2
Smallest, largest	$\min (4, 8, 15, 1)$ , $\max (4, 8, 15, 1)$	1 and 15
Round down, up, nearest	$\operatorname{floor}(4.6)$ , $\operatorname{ceil}(4.6)$ , $\operatorname{round}(4.6)$	4, 5 and 5. It is `ceil`, four letters, not `ceiling`.
Step graph	$\operatorname{floor}(x)$	Put x inside instead of a number and the graph becomes a staircase; $\operatorname{ceil}(x)$ does the same one step higher.

Test-day settings and habits

Put $y =$ in front of anything you want drawn. Without it some expressions return a number, some error, and some graph anyway, so make it a habit rather than a decision.
Split every equation at the equals sign into two $y =$ lines. On one line, no solutions and infinitely many both draw nothing, and you cannot tell them apart.
Zoom in before you commit to what the picture shows, and zoom out before answering a greatest-possible-value question. The largest value is often off the default screen.
When you pick values for constants yourself, never 0 or 1, never two the same, never e. Those choices hide mistakes instead of exposing them.
Trust whole numbers and tidy fractions. Something like 2.333 coming back is nearly always a badly typed setup rather than an ugly answer, so go back to the question.
Regression returns one solution and hides the rest. Counting solutions or adding them is a graph job.
Regression only solves for letters that are still undefined. If b is already set on another line, $7 \sim 2m + b$ solves for m alone. Rename the letter when you meant a new one.
Do not over-constrain. Pin both variables and the constant, and the system has more conditions than unknowns, so Desmos solves nothing at all.
Once you are using bracket regression you cannot also set a variable equal to a list. Vary that variable by hand instead: set it to 2, read the constant, set it to 5, check the constant did not move.
You need at least as many points as constants you are solving for: two for a line, three for a parabola in standard form. Vertex form needs only one, if the question hands you the vertex.
Only graph the points you know, never the one you are solving for.
No fraction icon beside a result means the answer is probably the decimal.
A question with no equation and no points anywhere in it is the one kind that stays on paper.

When not to use the tilde

The calculator you get on test day

Typing shortcuts, key by key

Syntax: what to type so Desmos understands you

From question pattern to Desmos move

Statistics and number functions

Test-day settings and habits

Where to go next

The calculator you get on test day

Typing shortcuts, key by key

Syntax: what to type so Desmos understands you

From question pattern to Desmos move

Statistics and number functions

Test-day settings and habits

Where to go next