Exhaustive Guide to Desmos for SAT Math Mastery of the SAT Math Section

Transitioning from Traditional SAT Math to the Desmos Era

• Historical Context: Around $2 billion$ years ago (metaphorically, in $2023$), students took the SAT on paper using pencils.
• The Shift: David Coleman introduced a change where math on the SAT no longer required manual calculation for many problems because every student received a Desmos calculator within the testing software.
• The Paradigm Shift: Using traditional methods (sticks and stones) like those taught on Khan Academy or in many prep books is depicted as inefficient compared to using the "nuclear bomb" capability of Desmos.
• The Instructor's Background: The lecturer is an SAT math tutor and creator of the most-watched SAT math video on YouTube.

Basic Functionality in Desmos

• Graphing and Evaluation: Desmos is a graphing calculator; users can type in any function and it will render visually.
• Point Evaluation: Functions can be solved for specific points using $f(x)$ notation (e.g., $f(3)$).
• Transformations: The tool easily handles vertical and horizontal translations.
• Intersects: Clicking on a function allows users to find the $x$-intercept and $y$-intercept immediately.
• BlueBook Decimal Input Guidelines: If a problem requires a decimal answer (e.g., the $x$-intercept is $-2.6666...$), BlueBook accepts three formats:
  • Truncated: $-2.666$
  • Rounded: $-2.667$
  • Fractional: $-8/3$
  • Pro Tip: Type the decimal into the input field until the software stops accepting digits to ensure the highest degree of precision and guaranteed correctness.

Mastering Regressions for Linear and Quadratic Models

• The Tablet Method: Instead of using "rise over run" to find slope ($m$) and then solving for the $y$-intercept ($b$) manually, Desmos can automate the entire process.
• Step-by-Step Table Regression:
  1. Type the word table into a cell.
  2. Input known points (e.g., $(2, 5)$ and $(4, 11)$).
  3. Click the "add regression" button to generate the equation.
• Defining Regressions: In statistics, a regression finds the line of best fit for a scatter plot (e.g., X-axis: minutes watched; Y-axis: SAT score). On the SAT, regressions are used to solve for the specific values that define a function (e.g., slope, intercept, or the $a$, $b$, and $c$ terms in a quadratic).
• Types of Regressions: Users can select linear, exponential, quadratic, or cubic regressions from a drop-down menu.
• Applying Regressions to SAT Problems: This method removes the need for algebra in most sections of the test, including exponent rules and converting quadratic forms.

Systems of Equations and Solution Types

• Definition of Solution: In SAT math, a "solution" is synonymous with a "point of intersection."
• Solution Counts:
  • One Solution: Two functions intersect at a single point.
  • Two Solutions: Often seen with a quadratic and a line intersecting twice.
  • No Solution: Two lines are parallel and never touch.
  • Infinite Solutions: Two equations represent the exact same line, touching at every point.
• Using Sliders: For problems where an equation has an unknown constant (e.g., finding $B$ so that a system has exactly one solution), users can type the equations into Desmos and create a slider for $B$.
• Slider Bounds: It may be necessary to change the bounds of the slider (e.g., from $10$ up to $50$) to find the correct intersection point (e.g., $B = 24$).
• Discriminant Caveat: If the answer is a complex fraction, sliders might not be precise enough. In these cases (scores above $700$), students should know the discriminant formula ($b^2 - 4ac$). For scores below $700$, this is rarely necessary.

Single Variable Equations and Substitution

• The System Method: The most reliable way to solve a single variable equation is to split it into two equations:
  • $y = \text{Left Hand Side}$
  • $y = \text{Right Hand Side}$
• Why Verbatim Entry Fails: Entering an equation like $|x^2 - 4| = 12$ directly can lead to precision errors where Desmos cannot pinpoint the exact decimal during zooming.
• Solving for X in Terms of K: For complex algebraic problems where students must find $x$ relative to a constant $K$:
  • Graph $y = \text{LHS}$ and $y = \text{RHS}$.
  • Add a slider for $K$.
  • Pick an arbitrary value for $K$ (e.g., $K = 10$).
  • Check the intersection point ($x$-value).
  • Plug the same $K$ value into the answer choices. The choice that produces the matching $x$-value is correct.

Advanced Desmos Tricks and Custom Regressions

• The "Infinite Point" Variable Trick: If a problem asks which point in terms of a constant $R$ (e.g., $(R, 2R+5)$) lies on a specific line:
  1. Graph the original line.
  2. Type the answer choice coordinates into Desmos with a slider for $R$.
  3. Move the slider. The correct answer choice will "trace" the original line exactly as $R$ changes.
• Inequalities: Desmos shades regions representing solutions. For systems of inequalities, the solution is the overlapping shaded area.
  • Syntax: To type $\le$, type < followed by =.
  • Max/Min Values: Graph the system and click the highest or lowest points within the shaded region to find maximum or minimum $x$ or $y$ values.
• Equivalent Expressions: To check if two complex expressions are equivalent, graph the first as a function and then graph each answer choice. The correct answer will overlap perfectly.
• Custom Regressions with Tildes ($\sim$): The squiggly line (tilde) is used for custom regressions when a standard table option isn't available.
  • Rational Regression: $y_1 \sim \frac{a}{x_1 - b}$
  • Polynomial Regression: For zeros at $-5, 6, 7$, use the form provided in the problem with the tilde to find coefficients $a$, $b$, and $c$.
• The Constant Substitution Regression: To avoid algebra on problems converting forms:
  1. Type $\text{LHS} \sim \text{RHS}$.
  2. Use $x_1$ instead of $x$.
  3. Define the points: $x_1 = [1, 2, 3, 4, 5]$.

Specialized Math Tools and Statistics

• Exponents and Radicals: Desmos handles complex roots via specific commands:
  • Square root: sqrt ($\sqrt{x}$).
  • Cube root: cbrt ($\sqrt[3]{x}$).
  • Any other root: nthroot ($\sqrt[n]{x}$).
• Element Lists: Used for solving systems with multiple constraints or percentages. For example, if $C$ is $150\%$ greater than $D$, it can be represented as $C \sim 1.5D + D$.
• Statistical Functions:
  • mean(list): Calculates average.
  • median(list): Calculates the middle value.
  • max(list) - min(list): Calculates the range.
  • stdev(list): Standard deviation (Calculation usually not required on SAT, only conceptual understanding).
• Find Missing Value for Mean: If searching for an exam score to reach a specific average, include the unknown as $x_1$ in the list and set the regression: mean([80, 85, 92, x1]) ~ 90.

Geometry and Trigonometry

• Trigonometry: Desmos handles sin, cos, and tan calculations.
  • Crucial Setting: Desmos defaults to Radians. For SAT problems involving degrees (e.g., $\sin(60^{\circ})$), users must switch to Degree mode in the settings.
• Circles: Users can find the radius and center without completing the square.
  • Plot the circle equation.
  • Identify grey dots (endpoints) at the top and bottom of the circle to determine diameter.
  • Distance/Midpoint: Points can be assigned names (e.g., $A$, $B$) and functions like distance(A, B) or midpoint(A, B) can be used.

Limitations of Desmos

• Non-Desmosable Content: Approximately $25\%$ of the SAT math section cannot be solved using Desmos. This includes:
  • Word Problem Setup: Converting a scenario into a mathematical equation.
  • Conceptual Geometry and Statistics: Understanding properties rather than calculating values.
  • "No Real Solution" Logic: Regressions assume equality across points. To find where no solutions exist, students must understanding properties of parallel lines or discriminants.
• Efficiency Gap: For about $32\%$ of problems, Desmos is possible but slower than mental math.
• Regression Sensitivity: Regressions typically only provide one solution (usually the positive one). For problems with multiple possible solutions (like absolute value or squared variables), the system of equations method ($y = \text{LHS}$, $y = \text{RHS}$) is superior.

Resources and Recommendations

• Practice Platform: savedesmos.com (created for this curriculum) contains $27$ practice problems corresponding to the video chapters.
• Advanced Study: Students aiming for a perfect score ($800$) should consult learnsatmath.com for the SAT Math Masterclass or bedrockprep.com for non-Desmosable material.
• Cultural Reference: The instructor repeatedly mentions a Japanese $1984$ Dune poster featuring Sting in his background as a running bit.