?

The Desmos Playbook

Shortcuts that turn Desmos from "just a calculator" into a solving engine. Every example below is a real SAT-style question, worked two ways.

Open Desmos โ†—
f(x) = xยฒ โˆ’ 6x + 13 โ€” the lowest point is the answer. Click it in Desmos and it labels the coordinates for you.
Jump to a trick
Vertex & minimum/maximum Solving by intersection Sliders for parameters Systems of equations Circles & distance Graph overlay matching Regression Test the answer choices Pick numbers Mean, median & SD Inequalities & shading Counting solutions Function notation Degree mode for trig Equations in other letters Points & distance Counting with lists Reading a model
Trick 01

Vertex & minimum/maximum

Skip completing the square. Graph the parabola and click its lowest (or highest) point โ€” Desmos labels the coordinates for you.

Open Desmos โ†—
If f(x) = xยฒ โˆ’ 6x + 13, what is the minimum value of f(x)?
  1. Type y = x^2 - 6x + 13 into Desmos
  2. Click the lowest point on the curve
  3. Desmos labels it (3, 4) โ€” the minimum value is 4
Try this graph โ†—
When to skip this: if the equation is already in vertex form (xโˆ’h)ยฒ+k, reading off the vertex by hand is just as fast.
Trick 02

Solving by intersection

For equations like โˆš(3x+16) = x+2, graph both sides as separate functions and find where the lines cross โ€” no squaring, no checking for extraneous roots.

Open Desmos โ†—
What is the set of all real solutions to โˆš(3x+16) = x+2?
  1. Type y1 = sqrt(3x+16)
  2. Type y2 = x+2
  3. Click each intersection point Desmos finds โ€” that's your solution set
Try this graph โ†—
Trick 03

Sliders for parameters

When a question asks "what's the max/min value of b such that...", add a slider for b and literally watch the answer happen.

Open Desmos โ†—
y = 2xยฒ โˆ’ 6x + 18
no real solutions
b = 6

2xยฒ โˆ’ bx + 18 = 0 has no real solutions and b is a positive integer. What's the max value of b?
  1. Type y = 2x^2 - bx + 18 โ€” Desmos offers to add a slider for b
  2. Drag the slider up until the parabola just touches the x-axis (b = 12)
  3. Since 12 gives exactly one solution, the max with no solutions is 11
Try this graph โ†—
Trick 04

Systems of equations

Graph both equations as lines (or curves) and read the intersection point directly โ€” skip substitution entirely.

Open Desmos โ†—
Tacos are $3.50, burritos are $8.00. 120 items sold for $615 total. How many burritos?
  1. Type y = 120 - x (total items)
  2. Type y = (615 - 3.5x)/8 (from the revenue equation)
  3. The intersection point gives tacos and burritos directly
Try this graph โ†—
Trick 05

Circles & distance

Forgot the distance formula? Type it straight into Desmos as a calculation โ€” no graph needed, just instant arithmetic.

Open Desmos โ†—
A circle has center (3, โˆ’4) and passes through (3, 2). What's its equation?
  1. Type sqrt((3-3)^2+(2-(-4))^2) directly into Desmos โ€” it returns 6
  2. That's your radius: r = 6, so rยฒ = 36
  3. Equation: (xโˆ’3)ยฒ + (y+4)ยฒ = 36
Try this calculation โ†—
Trick 06

Graph overlay matching

For "which expression is equivalent to..." questions, graph the original and every answer choice. The one that traces the exact same line is correct.

Open Desmos โ†—
Which expression is equivalent to (xยฒ โˆ’ 16)/(x โˆ’ 4) for x โ‰  4?
  1. Type y = (x^2-16)/(x-4) โ€” note the tiny hole in the graph at x = 4
  2. Type each answer choice as y2, y3, etc.
  3. Whichever line sits exactly on top of the original (minus the hole) is the answer: x + 4
Try this graph โ†—
Watch for: the "hole" at excluded values โ€” Desmos shows it as a small open circle, easy to miss if you're not looking for it.
Trick 07

Regression finds the equation for you

Put the points in a table and type a model with ~ โ€” Desmos solves for the constants. It works for a line of best fit, a quadratic through given points, or an exponential.

Open Desmos โ†—
A quadratic function f has f(โˆ’1) = 8, f(1) = 2 and f(3) = 4. What is the value of f(5)?
  1. Click + โ†’ table and enter the points: xโ‚ = โˆ’1, 1, 3 and yโ‚ = 8, 2, 4
  2. Type y_1 ~ a*x_1^2 + b*x_1 + c โ€” the ~ means "fit this model to the table"
  3. Desmos reports a = 1, b = โˆ’3, c = 4 (three points fix a parabola, so the fit is exact): f(x) = xยฒ โˆ’ 3x + 4
  4. Type a*5^2 + b*5 + c โ†’ 14. Same idea for other models: y_1 ~ m*x_1 + b gives a line of best fit, y_1 ~ a*b^(x_1) an exponential
Try this graph โ†—
When to skip this: for a line through two given points, rise over run by hand is faster than building a table. And with real scatter-plot data the fit is only approximate, so choose the closest answer choice rather than expecting an exact match.
Trick 08

Test the answer choices

Multiple choice means the answer is already on the screen. Type each choice into Desmos (or plug it into the given equation) and see which one matches โ€” decimals make different-looking radicals and fractions easy to compare.

Open Desmos โ†—
What is the positive solution to xยฒ โˆ’ 6x + 4 = 0?
A) 3 + โˆš5   B) 3 + 2โˆš5   C) 6 + โˆš5   D) (3 + โˆš5)/2
  1. Type y = x^2 - 6x + 4 and click the right-hand x-intercept: (5.236, 0)
  2. Type each choice on its own line: 3+sqrt(5) โ†’ 5.236, 3+2sqrt(5) โ†’ 7.472, 6+sqrt(5) โ†’ 8.236, (3+sqrt(5))/2 โ†’ 2.618
  3. Only A matches the intercept, so the answer is 3 + โˆš5
  4. Or plug a choice straight in: (3+sqrt(5))^2 - 6(3+sqrt(5)) + 4 should give 0 (a tiny value like 10โปยนโต is rounding noise and still means 0)
Try this graph โ†—
When to skip this: if the choices contain letters rather than numbers, there's nothing to evaluate โ€” use "pick numbers" instead. And if one choice is obviously right after a quick estimate, don't type all four.
Trick 09

Pick numbers for the letters

When the question is full of letters, give each letter a simple value and let Desmos do the arithmetic. It works for equivalent expressions, rearranged formulas, and percent chains (start with 100).

Open Desmos โ†—
The formula T = km/dยฒ relates positive quantities T, k, m and d. Which expression gives d in terms of T, k and m?
A) โˆš(km/T)   B) โˆš(T/(km))   C) km/Tยฒ   D) โˆš(kmT)
  1. Pick easy values, avoiding 0 and 1 (they hide differences): type k = 3, m = 12 and, with d = 2, T = k*m/2^2 โ†’ T = 9
  2. Type each choice: sqrt(k*m/T) โ†’ 2, sqrt(T/(k*m)) โ†’ 0.5, k*m/T^2 โ†’ 0.444, sqrt(k*m*T) โ†’ 18
  3. Only A gives back d = 2, so the answer is A
  4. Percent chains work the same way: for "up 20%, then down 25%", type 100*1.2*0.75 โ†’ 90, so the final value is 90% of the original
Try this graph โ†—
When to skip this: if a rearrangement is one step by hand (like solving P = 2L + 2W for W), just do the algebra. And if two choices both match your numbers, pick a fresh set and test only those two again.
Trick 10

Lists: mean, median and standard deviation

Type the data as a list and Desmos computes the statistics instantly: mean(L), median(L), stdevp(L) (population) or stdev(L) (sample), plus total(L) and length(L).

Open Desmos โ†—
A data set consists of 4, 7, 7, 9 and 13. The value 20 is added to the data set. Which statement is true?
A) The mean and median increase by the same amount   B) The mean increases more than the median   C) The median increases more than the mean   D) Neither the mean nor the median changes
  1. Type L = [4, 7, 7, 9, 13], then mean(L) โ†’ 8 and median(L) โ†’ 7
  2. Add the new value with M = join(L, [20]), then mean(M) โ†’ 10 and median(M) โ†’ 8
  3. The mean rose by 2 and the median by 1, so the answer is B
  4. For spread questions, compare stdevp(L) โ‰ˆ 2.97 with stdevp(M) โ‰ˆ 5.23 โ€” adding a far-out value increases the standard deviation
Try this graph โ†—
When to skip this: many statistics questions are conceptual โ€” an outlier pulls the mean more than the median, and a tighter cluster has a smaller standard deviation. If you can answer from that idea alone, you don't need to type the list.
Trick 11

Inequalities and shaded regions

Type inequalities directly โ€” Desmos shades each one, and the region where the shading overlaps is the solution set. Drop the answer-choice points on top to see which lands inside.

Open Desmos โ†—
Which point (x, y) is a solution to the system y โ‰ค โˆ’2x + 8 and y > x โˆ’ 1?
A) (1, 5)   B) (3, 1)   C) (4, 2)   D) (2, 0)
  1. Type y <= -2x + 8 (Desmos turns <= into โ‰ค) โ€” a solid boundary, shaded below
  2. Type y > x - 1 โ€” a dashed boundary, because points on that line don't count; shaded above
  3. Type each choice as a point: (1,5), (3,1), (4,2), (2,0)
  4. Only (1, 5) sits in the area shaded by both, so the answer is A
Try this graph โ†—
When to skip this: with only one or two points to check, plugging them into the inequalities by hand is just as quick. And if a point lands right on a boundary line, don't trust your eyes โ€” check whether the sign is โ‰ค (solid, counts) or < (dashed, doesn't).
Trick 12

Counting solutions and reading zeros

Graph y = the expression and count where it meets the x-axis. If a parabola never meets the axis, there are no real solutions.

Open Desmos โ†—
How many distinct real solutions does the equation xยณ โˆ’ 4xยฒ + 4x = 0 have?
  1. Type y = x^3 - 4x^2 + 4x
  2. Count the x-axis contacts: the curve crosses at (0, 0) and just touches at (2, 0) โ€” a touch counts once
  3. Two contact points, so there are 2 distinct real solutions (x = 0 and x = 2)
  4. To list solutions instead, type an equation in x alone: x^2 - 5x + 6 = 0 draws vertical lines at x = 2 and x = 3
Try this graph โ†—
When to skip this: if the equation is already factored, count the distinct factors by hand. For a quadratic, the discriminant bยฒ โˆ’ 4ac is often just as fast. If you do graph it, zoom out once so no zeros are hiding off-screen.
Trick 13

Function notation, composition and shifts

Define the function once and Desmos evaluates f(3), f(g(2)) or a shifted graph for you. Changes inside the brackets move the graph sideways, the opposite way to the sign.

Open Desmos โ†—
The functions f and g are defined by f(x) = xยฒ โˆ’ 3 and g(x) = 2x + 1. What is the value of f(g(2))?
  1. Type f(x) = x^2 - 3 and g(x) = 2x + 1
  2. Type f(g(2)) โ†’ Desmos shows 22. (Order matters: g(f(2)) gives 3.)
  3. For shift questions, type y = f(x - 3) + 2: the vertex moves from (0, โˆ’3) to (3, โˆ’1) โ€” right 3 (opposite to the minus sign) and up 2
Try this graph โ†—
When to skip this: a single substitution like g(2) = 5 is quicker in your head. And if the function is given only as a table or a graph, Desmos doesn't know its rule โ€” read the values off the question instead.
Trick 14

Degree mode for trig

Desmos starts in radians, so sin(30) gives a strange answer. Switch to degrees first, then evaluate or graph trig expressions to check identities.

Open Desmos โ†—
If sin(xยฐ) = cos(32ยฐ) and 0 < x < 90, what is the value of x?
  1. Click the wrench (Graph Settings) and choose Degrees
  2. Check it worked: sin(30) should give 0.5 (if you see โˆ’0.988, you're still in radians)
  3. Type y = sin(x) and y = cos(32), zoom out to show x from 0 to 90, and click the intersection: (58, 0.848), so x = 58
  4. To see the identity behind it, graph y = cos(90 - x) โ€” it lies exactly on top of y = sin(x)
Try this graph โ†—
When to skip this: once you know the rule that sin(xยฐ) = cos(yยฐ) when x + y = 90, this question takes seconds by hand (90 โˆ’ 32 = 58). And switch back to radians for any question written with ฯ€.
Trick 15

Equations in other letters

Desmos only graphs x and y, so an equation in t, a or w needs a quick rename. Swap the letter for x and Desmos draws a vertical line at the solution.

Open Desmos โ†—
If 3(w โˆ’ 4) + 2w = 18, what is the value of w?
  1. Typed as-is, w is treated as an unknown constant and Desmos offers a slider instead of solving
  2. Rewrite it with x: 3(x - 4) + 2x = 18
  3. Desmos draws the vertical line x = 6, so w = 6
  4. Check: 3(6-4) + 2*6 โ†’ 18
Try this graph โ†—
When to skip this: a two-step linear equation like this one is quicker by hand โ€” save the trick for messy fractions or radicals. Watch two traps: r alone means the polar radius (r = 5 draws a circle), and if the equation already contains x, renaming another letter to x merges the two variables.
Trick 16

Points and distance

Type points like (2, 3) and Desmos plots them. distance() gives the length between two points โ€” the Pythagorean theorem without the algebra โ€” and polygon() draws the shape so you can check a figure.

Open Desmos โ†—
In the xy-plane, a rectangle has sides parallel to the axes and opposite vertices at (โˆ’2, 1) and (6, 7). What is the length of a diagonal of the rectangle?
  1. Type (-2, 1) and (6, 7) โ€” Desmos plots both corners
  2. Type distance((-2, 1), (6, 7)) โ†’ 10, so the diagonal is 10
  3. To see the figure, type polygon((-2, 1), (6, 1), (6, 7), (-2, 7)): the sides are 8 and 6, so if the question asked for the perimeter, 2*(8 + 6) โ†’ 28
  4. For a missing side of a right triangle (hypotenuse 13, leg 5), put the right angle at the origin: type (5, 0), find the other leg with sqrt(13^2 - 5^2) โ†’ 12, and check with distance((5, 0), (0, 12)) โ†’ 13
Try this graph โ†—
When to skip this: if the sides form a familiar triple (3-4-5, 5-12-13, 8-15-17) or the two points share an x- or y-coordinate, the length is quicker by hand. And Desmos only draws the coordinates you type โ€” it can't tell you anything a test figure "not drawn to scale" hides, so trust the calculation, not the picture.
Trick 17

Counting with lists

For "how many numbers from 1 to nโ€ฆ" and probability questions, let Desmos build the list and filter it. [1...120] makes every whole number from 1 to 120, square brackets keep only the ones that pass a test, and length() counts them.

Open Desmos โ†—
A whole number from 1 to 150 is chosen at random. What is the probability that the number is even and greater than 100?
  1. Type L = [1...150] โ€” Desmos turns the three dots into an ellipsis and makes the list 1, 2, 3, โ€ฆ, 150
  2. Type M = L[mod(L, 2) = 0] to keep only the even numbers; length(M) โ†’ 75
  3. Chain the second condition: length(M[M > 100]) โ†’ 25 (the evens 102 to 150)
  4. Type length(M[M > 100])/length(L) โ†’ 0.1667, which is 25/150 = 1/6. To add numbers instead of counting them, swap in total(): total(M[M > 100]) โ†’ 3150
Try this graph โ†—
When to skip this: with one simple condition, arithmetic is faster โ€” there are 150 รท 2 = 75 evens up to 150. And read the wording carefully before you type: "greater than 100" is M > 100, but "at least 100" is M >= 100, and the wrong one shifts the count by one.
Trick 18

Graph the model to read what its numbers mean

For "what does 20 represent?" questions, graph the model and look: the y-intercept is the starting value, and the slope is how much y changes each time x goes up by 1.

Open Desmos โ†—
The height h, in centimeters, of a sunflower w weeks after it was moved to a garden is modeled by h = 4.5w + 20. What is the best interpretation of 20 in this context?
A) The sunflower grows 20 cm each week   B) The sunflower was 20 cm tall when it was moved to the garden   C) The sunflower reaches 20 cm after 4.5 weeks   D) The sunflower's maximum height is 20 cm
  1. Type y = 4.5x + 20 {x >= 0} โ€” the braces restrict the domain, because weeks can't be negative
  2. Click where the line starts on the y-axis: (0, 20). At 0 weeks the height is already 20 cm, so the answer is B
  3. Click the gear above the expression list, then the table icon beside the line (if it doesn't appear, retype the line without the braces). The table shows 0 โ†’ 20, 1 โ†’ 24.5, 2 โ†’ 29, 3 โ†’ 33.5: y goes up by 4.5 each week, so 4.5 is the growth per week, not 20
  4. If the model is written like y โˆ’ 5x = 6, rewrite it as y = 5x + 6 first, so the slope and starting value sit where you expect them
Try this graph โ†—
When to skip this: once you know that in y = mx + b the lone number b is the starting value and m is the change per unit, a linear model like this one takes seconds by hand. Graphing earns its keep for less familiar forms โ€” in 20(1.05)x, the 20 is still the starting value (try x = 0), and 1.05 means growth of 5% per unit.

Ready to drill these for real?

Every math question in the bank comes with both the formula method and the Desmos method side by side.

Go to Question Bank Quick practice