The only SAT Math Desmos Guide you'll ever need

Main Ideas

The video presents Desmos as a powerful tool for solving many SAT Math problems, often by graphing or regression instead of doing lengthy algebra. The central lesson is to choose a Desmos method that matches the problem—and to know when the calculator is not enough. The presenter repeatedly recommends practicing the techniques on actual problems rather than simply watching the tutorial.

Methods and Techniques

Functions and Intercepts

• Enter a function in Desmos to graph it and inspect its intercepts.
• For a free-response decimal answer, enter as many decimal digits as the answer field permits rather than worrying about whether to truncate, round, or convert the answer to a fraction.

Tables and Regression

• Create a table, enter the given (x)- and (y)-values, and use regression to find a function that fits them.
• With two points, a linear regression gives the line through them. With enough points, polynomial regressions can identify the corresponding function.
• The presenter emphasizes using regression on the SAT not just for statistical trends, but also to determine unknown coefficients in a function.
• For a custom function form, use Desmos’s regression operator (~) and specify the desired expression. Table variables such as x1 and y1 refer to the table’s columns.
• To fit a function to an expression rather than a supplied table, provide a list of (x)-values and have Desmos match the chosen function form at those values.

Systems of Equations

• Graph both equations and identify their intersection points; each intersection represents a solution.
• The number of solutions corresponds to the number of intersections: one, two, none, or infinitely many if the graphs coincide.
• If an equation includes an unknown constant, create a slider for it and adjust the value to see when the required number of intersections occurs.
• Slider exploration may be insufficient when the correct value is an awkward fraction; such cases may require algebraic methods.

Single-Variable Equations

• A reliable approach is to graph the left-hand side as one function and the right-hand side as another. Their intersections give the solutions.
• This is often more dependable than graphing the equation in a way that makes it hard to read precise values.
• For equations involving a parameter, use a slider for the parameter and compare the resulting solution values with the answer choices.

Checking Answers Expressed in Terms of a Parameter

When answer choices describe a point or value in terms of a variable such as (r), enter an answer choice and vary (r) with a slider. Check whether the points generated by that choice lie on the graph required by the problem. This can make tedious algebraic comparisons quick.

Systems of Inequalities

• Desmos shades the region satisfying each inequality. For a system, solutions are points in the overlapping shaded regions.
• To test answer choices, substitute their coordinates and see whether each point lies in the solution region.
• For a maximum or minimum, inspect the relevant extreme point in the feasible region. Do not assume the answer is always an intersection; other constraints or boundaries may determine the extreme.

Equivalent Expressions and Finding Coefficients

• To compare expressions, graph the original and each answer choice; equivalent expressions produce the same graph.
• When asked to produce an equivalent form with unknown coefficients, use custom regression to fit the desired form to points generated from the original expression.
• Include all problem constraints in the regression. The presenter cautions that a regression may return only one possible set of coefficients, even when multiple valid sets exist.

Avoiding Missed Solutions

• Regression is not a general-purpose substitute for solving equations: it may show only one solution when multiple solutions exist.
• For equations with possible multiple roots, such as those involving a squared variable, absolute values, or plus/minus alternatives, graph both sides and inspect all intersections.
• If unsure whether there is one solution or more, the two-graph intersection method is the safer approach.

Lists of Values

• Desmos lists can be used to impose several related equations at once, keeping shared variables consistent across the equations.
• This can help with systems involving function values, vertex information, or relationships between quantities.
• Lists can also encode percentage statements as equations—for example, translating “a certain percentage greater than” or “a percentage less than” into arithmetic expressions.

Statistics

• Desmos can calculate a list’s mean and median. Its range can be found by subtracting the minimum from the maximum.
• It can also calculate standard deviation, but the presenter says SAT standard-deviation questions are generally conceptual rather than requiring a direct calculation.
• An unknown data value can be included in a list and solved for by setting the list’s mean equal to the target average.

Geometry

• Desmos can evaluate trigonometric expressions. Check whether it is in radians or degrees before entering an angle.
• For circle questions, graph the equation to inspect the circle and its key points rather than completing the square by hand.
• Desmos can calculate the distance between two points and their midpoint, which can help find a diameter, radius, or center.
• Do not mistake a circle’s left and right (x)-intercepts for its endpoints; the graph may show the relevant endpoints elsewhere.

Limitations and Cautions

• Desmos is useful for a substantial share of SAT Math, but it does not replace understanding the mathematics or translating word problems into equations.
• The presenter estimates that, on one Bluebook practice test, about 43% of problems could be solved almost entirely with Desmos and about 75% could use it in some way. These figures are the presenter’s estimate for that test, not a guarantee about every SAT.
• Geometry and statistics questions may rely on concepts that graphing alone will not explain.
• Some problems ask about the absence or number of real solutions. A regression that tries to make two expressions match at selected points may not answer those questions; understanding conditions for zero, one, or infinitely many solutions may be necessary.
• The video’s overall advice is to use Desmos strategically, check that all problem constraints are represented, and practice the techniques on SAT-style questions.

Speakers and Sources Featured

• Eric — The video’s presenter and SAT Math tutor; the only substantive speaker.
• Desmos — The calculator demonstrated throughout the tutorial.
• The College Board / SAT — Discussed as the source of the exam and its question styles.
• Dr. Phil — Mentioned as a joke, but does not appear as a speaker.