Core Desmos Skills for the Digital SAT Math Section

Desmos Version & Environment

  • Always open the College Board-locked version (URL in video description).
    • Banner reads “Desmos College Board Digital PSAT/SAT.”
    • The public calculator contains features (folders, image uploads, function inspector, etc.) that are disabled in Bluebook.
  • Video demonstrations are done in the web version, but everything port-identically to the in-test app.
  • Josh’s downloadable PDF “Desmos Operations” worksheet + 100-problem practice set are linked under the video and may update over time.

Fractions, Decimals & Mixed Numbers

  • Typing a fraction auto-displays a decimal preview. Click the grey preview bubble to toggle fraction ⇄ decimal.
    • e.g.
    • Input 114/9 → preview 12.666\ldots → click → \frac{38}{3} (reduced).
  • Mixed numbers (rare on SAT): type
    • whole part → numerator → highlight numerator → / → denominator
    • 2 3 / 4 (with 3 highlighted) → 2\tfrac{3}{4}
    • Click bubble to convert to improper form \frac{11}{4}.

Percentages

  • You can calculate directly with %.
    • 60% of 36 → 60%*36 → 21.6.
    • Equations such as x%*50 = 35 solve exactly once you invoke Desmos’ equation-solver (see “Solving Linear Equations”).

Storing Exact Values in Variables (Carry-Forward Technique)

  • Any expression can be stored: a = sqrt(21^2 – 14^2) keeps the unrounded radical for later work.
  • Works for trig outputs, radicals, decimals, etc. Avoid x, y, r (reserved).
  • Preserves hidden digits and eliminates re-typing.
    • Ex:
      a = acos(24/25) // angle in ° if mode set to Degree b = 90 – a sin(b) // returns 0.96 exactly, not 0.959…

Sliders (Dynamic Parameters)

  • Desmos auto-offers a slider when it sees an undefined letter (except x/y).
  • Manual creation: type b = 10, then click gear ⚙️ to set
    • min & max (e.g. 10 → 30)
    • step (e.g. 1 or 0.5).
  • Use to explore maxima/minima, discriminant cut-offs, parameter sweeps in quadratics, etc.

Distance & Midpoint

  • Distance: distance((x1,y1),(x2,y2)) or define points A,B & use distance(A,B).
    • Formula d=\sqrt{(x2-x1)^2+(y2-y1)^2} appears in preview.
    • Common use: diameter → divide by 2 to obtain circle radius.
  • Midpoint: midpoint((x1,y1),(x2,y2)) OR midpoint(A,B) → shows point; click “Label” to see coordinates.
    • Vital for finding a circle’s center from endpoints of a diameter.

Mean & Median (Small Data Sets ≤ ~20 values)

  • mean( …comma-separated list… )
    • \bar{x}=\dfrac{\sum x_i}{n}
  • median( … ) picks middle value (or average of two middles).
  • Both live in keyboard ▶ Functions → Statistics but typing is faster.

Detecting # of Solutions to a Linear Equation

  • Graph method (Y-Y):
    1. Enter y = (left side)
    2. Enter y = (right side)
    3. Inspect:
    • Same line → infinitely many solutions.
    • Parallel distinct lines → no solution.
    • Intersecting once → one solution (x-coordinate of intersection).
  • Directly typing the whole equation reveals nothing when solutions = ∞ or 0, so split is safer.

Systems of Equations / Inequalities

  • Type each equation/inequality exactly; intersection(s) or overlapping shading give solutions.
  • To find which system has no solution, graph all answer-choice pairs and look for parallel/non-overlapping graphs.
  • For inequality multiple-choice, graph candidate region & test whether provided point lies in overlapping shading.

Verifying Whether Given Points Are Solutions

  • Plot candidate point(s) via a table or (x,y) list.
  • Graph the function(s) or inequality system.
  • A point is a solution iff it lies on the curve (equation) or in the shaded region (inequality system).

Linear Regression (Finding m & b Quickly)

  • Create table (⛭ gear → “Add Table”) and enter ≥2 known points.
  • Regression syntax: y1 ~ m x1 + b
    • Returns m (slope) and b (y-intercept).
    • If slope known (parallel/ perpendicular prompt), substitute that value → regression solves for b only.
  • Typical Uses:
    • Find equation from two points, or point + slope.
    • Match an answer choice, then evaluate at new x or y values.
    • Rapidly derive line for later intersection work.

Evaluating Functions & Combinations

  1. Define the function: f(x) = ….
  2. Three evaluation methods:
    • Method 1: type f(value) (fastest for 1-off).
    • Method 2: ⛭ gear → “Create Table” → type the x-values; f(x) auto-fills.
    • Method 3: graph x = value; click intersection with curve.
  • Combinations: e.g. 2f(1) – g(–4) or f(2) + g(7).
  • Useful when tasks ask for f(4), g(3) after doing regression or translations.

Finding x for a Given y (Inverse Lookup)

  • Graph original function.
  • Also graph y = (target value).
  • Intersection point(s) → read x-coordinate(s).
    • Example: For 6x^2-5x+1=24, graph y=24; positive intersection at x≈2.48.

Solving Equations

Two Main Graphical Methods

  1. Split (Preferred): graph y = LHS and y = RHS; intersection x-coords = solutions.
    • Reveals 0, 1, or ∞ solutions instantly.
    • Works for linear, absolute, radical, rational, quadratic, etc.
  2. Whole Equation: type exactly; vertical lines appear at roots.
    • Sometimes line is un-clickable for irrational roots; can miss extraneous or hidden roots in abs/radical contexts.

Categories & Tips

  • Linear: parallel vs overlapping logic (see earlier section).
  • Absolute‐value: expect two symmetric roots; find both intersections with y = constant.
  • Radical: may produce single root; use split method to avoid non-clickable line.
  • Rational: intersection often yields one positive & one negative root—filter as question dictates.
  • Quadratic: two roots (possibly irrational); add, multiply, or choose smallest/largest as prompted.

Solving with Regression (Advanced Shortcut)

  • Replace every x with x_1 and the equal sign with ~ (tilde).
    • Example solving x^2+4x-6=0:
      x1^2 + 4x1 - 6 ~ 0
      → Desmos returns one root in x1.
  • Add domain constraint in braces to find additional roots:
    • x2^2 + 4x2 - 6 ~ 0 {x2 < -2} finds the left root.
  • Extremely handy for Pythagorean-type questions:
    a^2 + 7^2 ~ 25^2 instantly gives a=24.

Translations & Shifts of Graphs

  • Start with original function f(x) (must be in y=… or f(x)=… form).
  • Vertical shift:
    • Up k → g(x) = f(x) + k
    • Down k → g(x) = f(x) - k
  • Horizontal shift:
    • Right h → g(x) = f(x - h)
    • Left h → g(x) = f(x + h)
  • Combine for circles/other conics: examine center displacement:
    • Original (x-1)^2 + (y-3)^2 = 16
    • Shift left 2 & up 5 → replace with (x+1)^2 + (y-8)^2 = 16.
  • Typical tasks:
    • Find g(1) after shifting f(x) down 5.
    • Determine new x-intercept once a given line moves up 7.
    • Identify which answer-choice circle shows the required left/right & up/down shift.
  • If starting line is not in slope-intercept form, either:
    1. Rearrange manually, OR
    2. Enter two easy points, run linear regression to recover mx+b before translating.

Practical & Ethical Notes

  • Use only the sanctioned College Board Desmos during the test; extra features are disallowed.
  • Know when to abandon manual algebra for graphing (time management) but still understand underlying math to avoid mis-reporting (e.g.
    • Whole-animal context ⇒ integer answer, no 6.857 pets!).
  • Store exact values to prevent rounding errors in multi-step questions.

Quick Reference of Key Syntax

  • Fractions ↔ decimals: click preview bubble.
  • Mixed number: whole numerator ▢ ÷ denominator.
  • Percentage: x%.
  • Variable store: A = (expression).
  • Slider: declare undefined letter or click “Add Slider.”
  • Distance: distance((a,b),(c,d)).
  • Midpoint: midpoint( … ).
  • Mean / Median: mean(…), median(…).
  • Regression: y1 ~ m x1 + b.
  • Evaluate: f(3), or table, or x=3 intersection.
  • Split-graph equation solving: y = …, y = ….
  • Constrained regression root: expression ~ 0 {x1>0}.

Concept Connections & Significance

  • Nearly every SAT Digital math topic (linear models, systems, quadratics, statistics, geometry, trigonometry) can be attacked graphically in Desmos.
  • Mastery of these “core skills” allows:
    • Faster answer confirmation vs. algebra alone.
    • Reduced computational error via exact-value storage.
    • Strategic skipping of tedious manipulation (e.g. Pythagorean, regression for slope).
  • Video assumes completion of Josh White’s “Intro to Desmos” lesson; upcoming Intermediate & Advanced videos build on these fundamentals.