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How to Use Desmos on the Digital SAT

The digital SAT gives you a Desmos graphing calculator for the entire Math section, built right into Bluebook. Used well, it turns long algebra into a picture you can read in seconds. Here are ten techniques that come up again and again, each with a worked example you can try yourself.

The basics

In Bluebook, open the calculator from the toolbar on any Math question. It is the same Desmos graphing calculator as on desmos.com: each row holds one expression, and anything with x and y is graphed.

  • Click a curve to reveal its points of interest (x-intercepts, y-intercept, maximum or minimum) as gray dots. Click a dot to see its coordinates.
  • Two curves that cross show their intersection as a gray dot too. Click it for the coordinates.
  • Type equations the way they are written. Desmos graphs 3x + 2y = 15 directly, with no need to solve for y first.
  • Use the functions you know. Typing sqrt makes a square root, ^ makes an exponent, and / makes a fraction.

When not to bother

A one-step equation, a percent or a unit conversion is faster by hand. Save Desmos for long algebra, counting solutions, anything graphical, and testing answer choices.

1. Solve an equation by graphing both sides

Put each side of the equation on its own line as y = …. Every intersection is a solution, and its x-coordinate is the value of x. This is also the fastest way to answer “how many solutions?” questions.

Worked example

How many real solutions?

How many real solutions does x² − 6x + 11 = 2x − 5 have?

Type into Desmos

  1. y = x^2 - 6x + 11
  2. y = 2x - 5
  1. The parabola and the line touch at exactly one point, (4, 3).
  2. One point of contact means one solution. (Algebra agrees: x² − 8x + 16 = 0 is (x − 4)² = 0.)

Answer: Exactly one real solution, x = 4.

2. Catch extraneous solutions

Squaring both sides of a radical equation can create answers that don’t actually work. The graph never shows those, so it settles the question at a glance.

Worked example

A radical equation

What value of x satisfies √(x + 7) = x − 5?

Type into Desmos

  1. y = sqrt(x + 7)
  2. y = x - 5
  1. The two graphs cross once, at (9, 4).
  2. Solving by squaring gives x² − 11x + 18 = 0, so x = 9 or x = 2. But at x = 2 the left side is √9 = 3 and the right side is −3, so 2 is extraneous.

Answer: x = 9.

3. Systems of equations

Type both equations exactly as given and click where they cross.

Worked example

Read the solution off the graph

If (x, y) is the solution to the system y = 2x − 3 and 3x + 2y = 15, what is x + y?

Type into Desmos

  1. y = 2x - 3
  2. 3x + 2y = 15
  1. Click the intersection: (3, 3).
  2. Add the coordinates.

Answer: x + y = 6.

Parallel lines (no intersection) mean no solution; lines that lie on top of each other mean infinitely many. That makes Desmos a quick check for “how many solutions does this system have?”

4. Quadratics: vertex and zeros

Graph the function and click the curve. The vertex and both x-intercepts appear as points you can click.

Worked example

Maximum value and zeros

The function f is defined by f(x) = −2x² + 12x − 7. What is the maximum value of f, and where are its zeros?

Type into Desmos

  1. f(x) = -2x^2 + 12x - 7
  1. Click the parabola. The top point is (3, 11), so the maximum value of f is 11 (it occurs at x = 3).
  2. The x-intercepts show as about 0.655 and 5.345.
  3. If the choices are exact, like 3 ± (√22)/2, convert them: √22 ≈ 4.690, so 3 ± 2.345 gives 0.655 and 5.345. That matches.

Answer: Maximum value 11; zeros x = 3 ± (√22)/2 (about 0.655 and 5.345).

5. Maximum and minimum in word problems

Desmos graphs in x and y, so rename the variable: h(t) becomes y with x standing for t. Zoom out if the interesting part of the graph is off screen.

Worked example

Projectile height

A ball’s height, in feet, t seconds after it is thrown is h(t) = −16t² + 48t + 6. What is the ball’s maximum height, and about when does it hit the ground?

Type into Desmos

  1. y = -16x^2 + 48x + 6
  1. Click the top of the parabola: (1.5, 42). The ball peaks at 42 feet after 1.5 seconds.
  2. Click the positive x-intercept: about (3.12, 0). Ignore the negative intercept, since time can’t be negative.

Answer: Maximum height 42 feet; it lands after about 3.12 seconds.

6. Inequalities and shading

Desmos shades the solution region of an inequality. A strict inequality (< or >) gets a dashed boundary, and ≤ or ≥ gets a solid one. With two inequalities, the solutions are where the shading overlaps. Type the answer-choice points as coordinates to see which one lands in the overlap.

Worked example

Which point is a solution?

Which point (x, y) is a solution to the system y > 2x − 1 and y ≤ −x + 5? Choices: (3, 4), (1, 5), (0, 0), (4, 0).

Type into Desmos

  1. y > 2x - 1
  2. y <= -x + 5
  3. (3, 4)
  4. (1, 5)
  5. (0, 0)
  6. (4, 0)
  1. Only (0, 0) sits in the region shaded by both inequalities.
  2. Check it: 0 > 2(0) − 1 = −1 is true, and 0 ≤ −0 + 5 is true.

Answer: (0, 0).

7. Sliders to test a constant

When an equation contains an unknown constant, type it as a letter. Desmos offers to add a slider for it, which you can drag or set to each answer choice in turn. This works well for “for what value of k…” questions.

Worked example

No solution

In the system 3x + ky = 7 and 6x + 8y = 5, k is a constant. For what value of k does the system have no solution?

Type into Desmos

  1. 3x + ky = 7
  2. 6x + 8y = 5
  1. Accept the slider for k, then set k to each answer choice (or drag it).
  2. At k = 4 the lines are parallel and never meet: no solution. (Check: 6x + 8y = 5 is the same as 3x + 4y = 2.5, which has the same slope as 3x + 4y = 7 but a different constant.)

Answer: k = 4.

8. Regression with a table

Add a table (type table on a new line, or use the add menu). Its columns are x1 and y1. Then type a model with a tilde (~), and Desmos fits it and reports the parameters. Type y_1 to get the subscript.

Worked example

Line of best fit

The table shows x = 1, 2, 3, 4, 5 and y = 3.1, 4.9, 7.2, 8.8, 11.0. Using a linear model, predict y when x = 8.

Type into Desmos

  1. table: x₁ = 1, 2, 3, 4, 5 and y₁ = 3.1, 4.9, 7.2, 8.8, 11.0
  2. y_1 ~ m x_1 + b
  1. Desmos reports m ≈ 1.97 and b ≈ 1.09, with r² ≈ 0.998: a very strong linear fit.
  2. Predict: 1.97(8) + 1.09 ≈ 16.85.

Answer: About 16.9.

Other models work the same way: y_1 ~ a x_1^2 + b x_1 + c for a quadratic and y_1 ~ a b^(x_1) for an exponential.

9. Check equivalent expressions

Graph the original expression and each answer choice as y = …. The equivalent choice lies exactly on top of the original everywhere; the others separate somewhere. Click a choice’s colored circle to hide or show it.

Worked example

Which expression is equivalent?

Which expression is equivalent to (2x + 3)² − (x + 1)²? Choices: 3x² + 10x + 8, 3x² + 14x + 10, 3x² + 8, 5x² + 14x + 10.

Type into Desmos

  1. y = (2x + 3)^2 - (x + 1)^2
  2. y = 3x^2 + 10x + 8
  1. The second graph sits exactly on the first; the other choices pull away.
  2. Spot-check at x = 2: the original gives 49 − 9 = 40, and 3(4) + 20 + 8 = 40.

Answer: 3x² + 10x + 8.

Watch the domain

Graphs that match on screen only match where both are defined. (x² − 9)/(x + 3) and x − 3 look identical, but the first is undefined at x = −3. Check any restriction the question mentions.

10. Functions and lists

Define a function once and evaluate it as often as you like: type f(x) = x^2 - 3x, then f(5) shows 10. Compositions like f(g(2)) work too once g is defined.

For data questions, lists save arithmetic: mean([12, 15, 15, 18, 40]) gives 20 and median([12, 15, 15, 18, 40]) gives 15. That shows at a glance how the outlier, 40, pulls the mean above the median.

Common pitfalls

  • The answer is off screen. If you expect an intersection and don’t see one, zoom out before deciding there isn’t one.
  • Missing parentheses. 1/2x and 1/(2x) are different functions. Check the typeset expression before you trust the graph.
  • Angles. Desmos treats angles as radians by default, so sin(30) is not 0.5. Most SAT trigonometry is right-triangle reasoning anyway; if you do compute, convert first: sin(30π/180) = 0.5.
  • Decimals against exact answers. Convert the answer choices to decimals and compare, rather than trying to recognize a decimal on sight.
  • Over-using it. Typing takes time. If you can do it in your head in ten seconds, do.

How to practice

Practice with the calculator you will actually use. Bluebook’s full-length practice tests include the same built-in Desmos, so use it there under timed conditions. Between tests, keep desmos.com/calculator open while you work StudyMondo’s Advanced Math practice sets and grid-in problems, and try each technique on purpose until it is automatic.

Frequently asked questions

Is the Desmos calculator available on the whole SAT Math section?
Yes. Bluebook, the app you take the digital SAT in, has a Desmos graphing calculator built in for both Math modules. It is not available in the Reading and Writing section. You may also bring your own approved calculator.
Do I need Desmos to get a high Math score?
No. Every question can be solved by hand. Desmos saves time on certain question types (systems, quadratics, counting solutions, regression, checking answer choices) and protects against algebra slips, so it is worth practicing until using it is automatic.
Does Desmos give exact answers?
It shows decimals. When the answer choices are exact values such as 3 ± (√22)/2, convert the choices to decimals and match them. For a question where you type your own answer, read College Board’s entry directions once in a Bluebook practice test so you know how fractions and long decimals are accepted.
Should I graph every Math question?
No. A one-step equation, a percent or a unit conversion is faster by hand. Reach for Desmos when the algebra is long, when you need to see how many solutions there are, or when testing answer choices is quicker than solving.

Free tools to use next

Free SAT diagnostic36 questions, about 30 minutes: an estimated score range and a study plan.Digital SAT strategy guideFormat, adaptive modules, pacing and strategy by domain.Bluebook companionLog a Bluebook score report and get a plan for your weakest domains.SAT practice testsFull-length, adaptive Digital SAT practice with a score range.

SAT® and Bluebook™ are trademarks of College Board, and PSAT/NMSQT® is a registered trademark of College Board and the National Merit Scholarship Corporation. Neither is affiliated with or endorses StudyMondo. Desmos is a trademark of Desmos Studio PBC.