DESMOS & MATH REFERENCE

Desmos & Formula Cheatsheet

High-frequency algebraic formulas, regression syntax, circle equations, and grammar rules for the adaptive Digital SAT. Click to copy Desmos commands directly into your calculator.

Desmos Hacks

Linear & Quadratic Regression (Instant Solver)

y_1 ~ m x_1 + b \text{or} y_1 ~ a x_1^2 + b x_1 + c

Given any table of coordinates (x, y) or two points, create a table in Desmos (type table). Desmos instantly computes the exact coefficients m, b, a, c without any manual algebra.

Example: Points (2, 7), (5, 22), (8, 43) -> Desmos yields m = 6, b = -5.

💡Rule: Never solve 3-variable quadratic systems manually. Always feed the points into a Desmos table and use ~.

Desmos Hacks

Slider Root & Tangency Detector

f(x) = ax^2 + bx + k

When an equation contains unknown constant k and states "intersects at one point" or "has no real solutions", add a slider for k and observe the exact boundary value on the graph.

Example: Adjust k slider until the curve just grazes the line at a single tangent point.

💡Rule: Zoom in near the intersection to read coordinates to 3 decimal places.

Desmos Hacks

System of Equations Intersections

\text{Graph Equation 1 and Equation 2}

Type any two equations directly. Click on the gray intersection dot. If lines overlap entirely, infinitely many solutions. If lines are parallel, no solution.

Example: Find where y = 2x^2 - 3x + 1 meets y = x + 7. Click the gray dot to get (3, 10) and (-1, 6).

💡Rule: No need to isolate y first. Desmos parses implicit equations natively.

Algebra & Functions

Quadratic Discriminant Matrix

\Delta = b^2 - 4ac

Determines the exact number of real solutions for ax^2 + bx + c = 0.

Example: \Delta > 0: 2 real solutions (2 x-intercepts) \Delta = 0: 1 real solution / tangent (1 x-intercept) \Delta < 0: 0 real solutions / floating (no x-intercepts)

💡Rule: Whenever the prompt says "tangent", "one solution", or "touches once", set b^2 - 4ac = 0 immediately.

Algebra & Functions

Sum & Product of Quadratic Roots (Vieta’s Formulas)

x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}

Solves "What is the sum of the solutions to 3x^2 - 15x + 7 = 0?" in 2 seconds without quadratic formula.

Example: Sum = -(-15) / 3 = 5. Product = 7 / 3.

💡Rule: Watch the sign: sum is negative b over a, product is positive c over a.

Algebra & Functions

Vertex Form of a Parabola

y = a(x - h)^2 + k

Reveals the vertex (h, k) as constants in the equation without calculation. Maximum occurs at k if a < 0, minimum at k if a > 0.

Example: y = -3(x - 5)^2 + 18 has maximum value 18 at x = 5.

💡Rule: When asked "Which equation displays the minimum/maximum as constants?", pick vertex form.

Geometry & Trig

Standard Circle Equation & Center-Radius

(x - h)^2 + (y - k)^2 = r^2

Center is (h, k) and radius is r. To convert x^2 + ax + y^2 + by = c into standard form, add (a/2)^2 and (b/2)^2 to both sides.

Example: x^2 + 8x + y^2 - 6y = 24 => (x+4)^2 + (y-3)^2 = 24 + 16 + 9 = 49 => center (-4, 3), radius 7.

💡Rule: In Desmos, you can type the expanded circle directly. No completing the square needed.

Geometry & Trig

Sine-Cosine Co-Function Complementary Angles

\sin(A) = \cos(B) \iff A + B = 90^\circ \; (\text{or } \frac{\pi}{2})

Solves SAT problems like "sin(4x - 10°) = cos(3x + 16°)". Set (4x - 10) + (3x + 16) = 90.

Example: 7x + 6 = 90 => 7x = 84 => x = 12°.

💡Rule: The angles inside sin and cos must add up to 90 degrees.

EBRW Grammar Algorithms

The Semicolon Algorithm

[\text{Independent Clause}] \; ; \; [\text{Independent Clause}]

A semicolon connects two complete thoughts that could each stand alone as separate sentences without a coordinating conjunction.

Example: The rocket launched successfully; telemetry data confirmed orbital insertion.

💡Rule: If you can replace the semicolon with a period and both sides make sense, the semicolon is valid.

EBRW Grammar Algorithms

The Colon Rule (100% Tested)

[\text{MUST be an Independent Clause}] \; : \; [\text{List, Explanation, or Clause}]

Everything BEFORE the colon must be a grammatically complete sentence. Everything after can be a list, fragment, word, or clause explaining the first part.

Example: She carried three essentials: water, compass, and notebook.

💡Rule: Never put a colon after a verb like "are:" or "including:". The clause before must be independent.