DDesmos SAT Prep

Reference

Desmos Cheat Sheet

Every general-purpose Desmos trick used across the practice categories, in one place. Skim this before a timed practice set.

Typing math

Desmos accepts math notation almost exactly as you'd write it by hand — no special syntax to memorize for the basics.

  • ^ for exponents: x^2
  • sqrt(...) for square roots, or type \sqrt and press tab
  • abs(...) or the | | keys for absolute value
  • π is typed as 'pi'; it auto-converts to the symbol
  • Fractions: typing / automatically creates a stacked fraction

Solving equations & systems

The single biggest Desmos trick for the SAT: graph both sides and click where they meet.

  • Graph each side of an equation as its own line: y = left side, y = right side
  • Click directly on the intersection point — Desmos snaps to and labels the exact coordinates
  • Works for linear, quadratic, absolute value, exponential, and radical equations alike
  • If a line and curve are tangent (touch once), that's your 'exactly one solution' case

Finding roots / zeros

Reading x-intercepts replaces factoring, the quadratic formula, or synthetic division.

  • Graph the polynomial as given — no need to set it equal to zero first
  • Click near each x-intercept; Desmos snaps to the exact root and shows (x, 0)
  • Each root r corresponds to a factor (x − r)

Sliders

Turn an unknown constant into something you can drag and visually solve for.

  • Type an expression with an undefined letter, e.g. y = a*x^2 + 3
  • Click the letter in the expression list, then 'add slider'
  • Drag the slider until the graph matches the condition in the problem (passes through a point, has a given root, etc.)
  • Adjust the slider's min/max/step from the small gear icon next to it for finer control

Tables

Evaluate a function at specific inputs, or hold a small data set, without a separate calculator.

  • Click '+' → 'table' to add one
  • Define your function first (f(x) = ...), then type x-values into the table's first column
  • The table's second column automatically fills in f(x) for each input

Regression (line / curve of best fit)

Fit a model to a table of data using a tilde (~) instead of an equals sign.

  • Linear: y1 ~ m*x1 + b
  • Quadratic: y1 ~ a*x1^2 + b*x1 + c
  • Exponential: y1 ~ a*b^x1
  • The solved parameter values (m, b, a, c, ...) appear listed right below the regression expression

Inequalities & shading

See an entire solution region at once instead of testing points one by one.

  • Replace = with >, <, >=, or <= to get automatic shading
  • Graph two inequalities in the same calculator to see the overlapping region of a system
  • Plot a candidate point as an ordered pair, e.g. (2, 5), to check if it falls in the shaded region

Distance & midpoint

Built-in functions mean you never need to recall the distance or midpoint formulas.

  • distance((x1,y1),(x2,y2)) returns the exact distance
  • midpoint((x1,y1),(x2,y2)) returns the exact midpoint

Circles & conics

Center-radius form graphs directly — Desmos handles implicit equations natively.

  • Type (x-h)^2+(y-k)^2=r^2 exactly as given
  • Plot a point to check visually whether it's inside, on, or outside the circle

Trig mode

Degrees vs. radians is the #1 silent killer on trig problems — always check the mode first.

  • Click the wrench/settings icon in the top-right of the graph to toggle Degrees ↔ Radians
  • Trace along a sine/cosine curve to read amplitude (max height) and period (distance between peaks)

Backsolving answer choices

When algebra gets messy, let Desmos test the answer choices for you.

  • Graph the left-hand side and right-hand side of an equation as two separate functions and find their intersection
  • Watch for extraneous solutions on radical/absolute-value equations — always verify by substituting back in

Checking equivalent expressions

One of the single highest-leverage Desmos tricks on the whole test.

  • Graph both expressions as separate functions — if only one curve is visible, they're equivalent
  • Fastest check: graph (expression 1) - (expression 2); a flat line at y = 0 confirms equivalence
  • Use it to test 'which choice is equivalent to...' answer options one at a time instead of expanding by hand

Counting solutions to a system

0, 1, 2, or infinite — read it straight off the graph instead of the algebra.

  • Count the visible intersection points between the two graphs
  • Parallel lines that never touch mean 0 solutions
  • If the two graphs completely overlap into a single visible curve, there are infinitely many solutions

Equations not solved for y

Desmos doesn't need an equation rearranged before it can graph it.

  • Type an equation exactly as given, e.g. 2x + 5y = 20 — Desmos graphs it directly
  • To find x given y (or y given x), graph a horizontal or vertical line at the known value and click the intersection

Lists & statistics

Skip the formulas for mean, median, and range entirely.

  • Type a data set as a list using curly braces: L = {4, 8, 2, 9, 5}
  • mean(L), median(L), and stdev(L) are all built in
  • There's no range() function — type max(L) - min(L) directly

Percent word problems

Type the whole calculation in one line instead of converting by hand.

  • A percent of a number: 0.15*240 for '15% of 240'
  • Percent change in one expression: (new - old)/old * 100
  • Chain multiple percent changes together: 500*1.08*0.95