The Last SAT Desmos Guide You'll Need

Main ideas and lessons

• Desmos is the key SAT advantage: the speaker claims scoring a perfect 800 and finishing early was largely due to using Desmos (the graphing calculator inside the SAT software, Bluebook).
• The video is a from-scratch Desmos guide, emphasizing:
  • basic graphing,
  • sliders/constants/function notation,
  • plotting points and using tables,
  • regression (core SAT strategy),
  • dealing with SAT-specific constraints (e.g., no copy/paste),
  • handling common pitfalls (inequality endpoints, radians vs degrees, regression limitations).
• Core SAT strategy: use Desmos to handle large chunks of the Math section by recognizing when a problem can be reduced to:
  • equations/expressions,
  • points,
  • systems,
  • or patterns solvable via regression and graphing.

What Desmos on the SAT is (and how it differs)

• The SAT version of Desmos is the same tool as in Bluebook.
• The main practical difference mentioned: the SAT version has extra options accessible via a plus button.
• The speaker downplays everything except needing the table feature later.

Fundamental graphing rules and syntax

Equations and inequalities

• Absolute value and standard function graphs are supported.
• Desmos reports:
  • intercepts,
  • intersections,
  • maximum/minimum (“vertex”),
  • and can handle inequalities by shading solution regions.
• Dashed vs solid boundaries (critical for inequalities):
  • Dashed line → points on the line are not included (open boundary).
  • Solid line → points on the line are included (closed boundary).
  • In practice: use ≤ vs < (speaker mentions typing the = form via keyboard).
• For multiple inequalities combined:
  • Example given: 2x < y results in a combined (double-shaded) solution region where both conditions hold.

Single-variable equations → vertical lines

• When you graph a single-variable equation (one variable only):
  • Desmos shows vertical lines.
  • The vertical line’s x-value(s) are the solutions for that equation.
• Then you can substitute the x-solution into whatever expression the question asks for.

Variable naming rules

• Lowercase x and y are special:
  • graphing generally expects lowercase x as the independent variable.
• If you use a letter to be graphed as a variable, Desmos might treat it as a variable unless it’s not x or y.
• Uppercase letters behave differently (can represent constants/parameters), but the speaker’s practical rule is:
  • use lowercase x for anything meant to be graphed as the variable.

Sliders and constants (parameter control)

Making a slider

• Set a letter equal to a number (where the letter is not lowercase x, y, or e).
• Desmos creates a slider automatically.
• Do not “define” with number on the left (e.g., 5 = a doesn’t create a slider).
• Slider default range: often -10 to 10, but you can adjust:
  • by typing a value outside the default, or
  • editing bounds and step size (e.g., change step to integers).

“Undefined constant” slider

• If you type a letter Desmos doesn’t know yet, you may see an “add slider” option.
• Even if it doesn’t appear, you can manually type Q = 1 (or any number) to define it.

Constants can represent more than numbers

• The speaker states constants can also be assigned to:
  • points and
  • lists (important for regression setups).

Practical advice

• The speaker often prefers manually typing constants rather than relying heavily on sliders for speed.

Function notation

• Desmos supports f(x)-style definitions:
  • define: f(x) = <expression>
  • reuse: type f(2) to evaluate.
• SAT copy/paste restriction:
  • SAT exam disables copy/paste, so defining f(x) helps avoid retyping expressions.
• Function transformations:
  • expressions like f(x) + 2 are handled automatically by Desmos.

Plotting points and using tables

Two ways to enter points

1. Directly type a point, e.g., (x, y)
2. Use a table:
   • add table,
   • enter x- and y-coordinates.

Zoom and corrections

• Zoom fit rescales the view to include plotted points.
• Default viewpoint resets to the initial perspective.
• Undo/redo and edit list mode are described as important tools.

Regression from a table (automated)

• Table regression:
  • After entering points, pressing “add regression” fits an equation through them.
  • Dropdown options exist (speaker says you mainly care about):
    • linear,
    • quadratic,
    • sometimes exponential.
• Export options:
  • can export the fitted equation,
  • or use “custom regression” to get constants in forms usable later.

Manual custom regression (when templates don’t fit)

• Some problems require an equation not matching standard templates (e.g., not simple y = mx + b).
• The speaker emphasizes knowing how to manually construct regression by typing the structure and constants.

Edit list mode (replacing copy/paste)

• Since SAT copy/paste is disabled:
  • the speaker uses edit list mode to duplicate equations and change properties like color.
• Edit list mode also provides:
  • delete all,
  • create table (with restrictions).

Create table restrictions

• “Create table” works only for normal equations with x on one side in a way Desmos expects.
• The speaker warns about equations that include y in a way that prevents “create table” from appearing.

Using create table to get values quickly

• Create table can transform a point into a table-like format.
• You can change x to instantly see corresponding y without manual reading from the graph.

Useful Desmos built-in functions (SAT-relevant + minor)

Commonly useful (speaker says these matter)

• mean(...)
• median(...)
• stdev(...) / standard deviation
• distance( (x1,y1), (x2,y2) )
• midpoint( (x1,y1), (x2,y2) )
• percent (e.g., 4% of 10)
• GCD(...) (greatest common divisor)

“Honorable mentions” (speaker says optional)

• mod, min, max, floor, ceil, round

Keypad vs keyboard

• Keyboard preferred; keypad is slower.
• The speaker includes examples of typing:
  • sqrt, cbrt, nth root,
  • pi,
  • operators like *, /, ^.

Operator/typing pitfalls (parentheses matter)

• Exponents with expressions:
  • if you add inside exponent without parentheses, Desmos may misinterpret it.
  • use parentheses to ensure the operation applies to the intended part.
• Fractions with multiple terms:
  • you must highlight/select the entire expression before dividing to make it the whole numerator.

Constraints and domain control

• Desmos supports constraints using curly braces:
  • e.g., specify x > 3 to restrict where the graph exists.
• This is important for cases where SAT questions specify domain conditions.

Trigonometry setting pitfall: radians vs degrees

• Desmos may default to radians.
• If the question is in degrees, you must switch graph settings to degrees, otherwise answers are wrong.

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Regression (the core methodology)

What regression does

• Regression uses the squiggly line ~ to estimate unknown constants so two sides become equal.
• Only constants can be solved for in regression:
  • you cannot directly use free variables like x or y as regression unknowns.

Regression requirements / conversions

Turn a variable into a constant parameter

• Use subscripts (speaker’s rule):
  • replace x with something like x_1 (subscript) to allow regression to solve for it.
• Generally:
  • constants should be symbols Desmos can treat as unknowns.

Regression format logic

• Must be an equation-like structure:
  • essentially “expression = expression” replaced by regression expression1 ~ expression2.
• If the problem only provides one side (e.g., “find f(2)” without another equality), the speaker says regression may not apply; instead you graph/evaluate directly.

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Regression patterns for SAT question types (detailed)

1) “Expression equals expression” (classic regression)

• When you see:
  • the question contains an equality between two expressions (including “is equivalent to”, “is equal to”, etc.)
• Steps:
  • Type the left expression.
  • Type the right expression.
  • Convert = to the regression squiggly ~ form so Desmos estimates the unknown constants.

2) Solving for a constant that must not depend on the variable

• Problem type:
  • the question asks for constants like a or b, while the equations include variables.
• Key challenge:
  • regression might give wrong constants if you only solve for one value of the variable.
• Method:
  • simulate multiple values of the regression variable using a list.
  • Example pattern described:
    • set x_1 = 1, 2, 3, ... (or another set of values)
    • regression then tries to satisfy equality for all those x values simultaneously.
• Guideline:
  • if you’re finding constant values, you “shouldn’t see X” as a regression unknown in a way that implies it can vary.
  • ensure the constants truly remain constant while variables vary.

3) Multiple solutions (regression limitation)

• Regression gives one answer; it can miss multiple roots.
• Rule stated:
  • if the question asks for sums of solutions, greatest/least possible values involving multiple intersections, or number of solutions:
    • graph instead of relying on regression.

4) Systems of equations (multiple equalities)

• Regression works only if everything fits on one squiggly line.
• Method:
  • Put all left-hand expressions into one list.
  • Put all right-hand expressions into another list.
  • Use regression on those lists together.
• Extra caution:
  • keep the order consistent (speaker warns not to swap items between lists).

5) Percent chain problems (regression via “is =” substitution)

• Pattern:
  • convert word “is” into =.
  • convert “X is 300% of Y” into X = 300% * Y.
• Then:
  • build the chain of equalities and solve via regression/constants.
• Speaker notes typical chain structure like:
  • X = (p%)Y
  • Y = (q%)Z
  • then solve for the unknown percentage factor.

6) Hard “constant under constraints” with inequalities/conditions

• The speaker emphasizes meeting conditions using:
  • constraints via curly braces (domain restrictions),
  • and ensuring constants truly stay constant across allowed variable changes.
• Verification:
  • vary variables and check whether the supposed constant changes.

7) Regression with tables (point-based regression)

• Rule:
  • if points appear in the problem (including from graphs), use table.
• Two approaches:
  • use add regression dropdown from the table (linear/quadratic/exponential),
  • or manually type custom regression when the equation form isn’t one of the templates.

8) When you must memorize only a couple SAT forms for non-template fits

• Speaker says two key forms to memorize:
  • Circle equation (standard form): identifies center and radius.
  • Quadratic vertex form: gives maximum/minimum efficiently.

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Handling SAT Math “non-regression” question types (graphing first)

The speaker includes guidelines for when regression isn’t worth it:

• Circle equation questions:
  • often easier to graph and use:
    • max/min y-values to identify diameter,
    • distance between extremes (and divide by 2 for radius),
    • midpoint to find center.
• Equivalent expression questions:
  • graph each candidate expression and find which matches.
• Factor/divisibility-style questions:
  • if expressions share factors, they share x-intercepts → graph and compare intercepts.
• Inequalities:
  • handled by shading regions.
• No solutions / number of solutions:
  • must graph and count intersections; ensure you plot both sides as separate lines (two equations), not one combined line that obscures “no solution” vs “infinite solutions”.

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Graphing vs regression choice framework (flowchart described)

The speaker references a flowchart (linked in the video) and summarizes decision logic:

• Ask whether Desmos is usable/appropriate.
• Then branch by question type, roughly:
  • Isolation problem? → use isolation method.
  • Can you find points? → use table regression.
  • Multiple equations? → use regression with lists/brackets.
  • Otherwise decide whether to:
    • treat unknown as a constant (use lists of variable values), or
    • treat unknown as a variable (don’t predefine it as a constant).
• Regression is powerful but not always the best use of time; some problems are simpler by direct graph reading.

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“Desmos theory” (speaker’s mental model)

• Regardless of recognition of pattern, the speaker’s framework is:
  • meet every condition the question states:
    • function definition,
    • constant relationships,
    • positivity/inequality constraints,
    • evaluation at specific inputs,
    • domain restrictions.
  • Then use regression or graphing to compute what’s asked.
• For tricky cases where regression isn’t obvious:
  • ensure the equality structure exists,
  • satisfy conditions via constraints,
  • define constants on the left side and values on the right.

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Main speakers / sources featured

• Primary speaker: Unnamed instructor/YouTube creator (speaks throughout; no other named individual identified).
• Software/tool source: Desmos graphing calculator inside SAT testing software (Bluebook).