How to use Desmos regression on the SAT

What the tilde does, and why it is the highest-leverage move

Type $7 \sim 2X + 1$ on an empty line, with a capital X. Desmos prints X = 3 underneath. It adjusted the undefined constant until both sides were equal, because 2 times 3 plus 1 is 7. That is all a regression does: the tilde asks Desmos to solve the equation for whatever letters are still undefined.

On a keyboard, the tilde is Shift plus the key directly under Escape, the backtick key in the top-left corner. On the on-screen keypad, open ABC and find the squiggly line in the bottom row. The Desmos Help Center article on regressions describes the same swap, and points at the same two places for the key: the row above the letters on a keyboard, or the bottom of the ABC keypad.

You take the SAT in Bluebook, the digital testing app. College Board's calculator policy page says Bluebook contains an embedded Desmos calculator with two options, graphing or scientific, and that you may use your own approved handheld calculator instead. The tilde lives in the graphing one, so any question with an equals sign in it is a candidate for it.

The Learn Desmos course gives regression its own section of eight chapters and calls it the single highest-leverage technique in the course. Together with the graphing section, the methods cover an estimated 53.55% of SAT Math question types (the curriculum's own estimate: Graphing 25.24% + Regression 28.31%).

$x_1$, $y_1$, and why a table is just two lists

Lowercase x and y belong to the graph. Type $7 \sim 2x + 1$ with a small x and Desmos throws an error, because it keeps those letters for drawing curves. A capital X works, since a capital letter is just another constant. The neater habit is a subscript: a letter followed by a number always becomes a subscript, so typing x then 1 gives $x_{1}$, a constant Desmos will solve for.

Subscripts also tie a regression to a table. Type table on an empty line and Desmos builds one with an $x_1$ column and a $y_1$ column. Enter moves down a column and Tab moves across to the next one.

Strip a table down and it is two lines of list. The course proves this by typing $x_{1} = \left[2, 5\right]$, $y_{1} = \left[8, 17\right]$ and $y_{1} \sim mx_{1} + b$ with no table anywhere. Desmos still returns m = 3 and b = 2. That is why the subscripts in a regression have to match the table's column headers. A second table gets $x_2$ and $y_2$, and a third gets $x_3$ and $y_3$.

Two rules about lines. Only one tilde per line: $12 \sim 3a \sim 4k$ is an error, so split it into $12 \sim 3a$ and $12 \sim 4k$. And a regression only solves for constants that are still undefined. If $b = 2$ sits on another line, $7 \sim 2m + b$ solves for m only and leaves b at 2. If you meant a different b, rename it to c. Never name a constant e, because Desmos keeps that letter for Euler's number.

Which regression form to type

Desmos can fit a form for you. Put your points in a table, open the edit list, and press Add regression on the table. Desmos fits a line straight away, and the drop-down beside it lists the other types. The course's advice is blunt: linear, quadratic, and occasionally exponential are the only ones you will need. Ignore everything else.

The three dots next to the drop-down offer Export as custom regression, which writes the regression out in full on its own line, $y_{1} \sim mx_{1} + b$, with the constants named so you can reuse m and b elsewhere. Typing that line yourself is the same thing, and faster once you know the forms.

Only two equations have to be memorized, according to the course. Circle standard form, $(x-h)^2+(y-k)^2=r^2$, with center $(h,k)$ and a radius equal to the square root of the right side. And quadratic vertex form, $y=a(x-h)^2+k$, where $(h,k)$ is the vertex, the maximum or minimum. Every other form is already in the drop-down.

Solving for an undefined constant

The first example worked because Desmos had one unknown. Ask for a constant instead and there are two unknowns on the line, so Desmos picks any pair that happens to work. Set the example below up the usual way and Desmos lists both k and $x_1$ as parameters; change $x_1$ by hand and the so-called constant changes with it. A constant has to stay constant while the variable varies, so make the variable vary.

• Any list works as long as it does not contradict the question. If the question says x is positive, keep the list positive.
• Use more numbers than constants you are solving for; two would do here. Ten is a safe habit, and $x_{1} = \operatorname{random}(10)$ picks ten for you.
• If the question already tells you what x equals, skip the list. For x = 2 and 2(x + 3) + kx = 16, type $x_{1} = 2$ and $2(x_{1} + 3) + kx_{1} \sim 16$.

A regression hands back exactly one solution. $x_{1}^{2} \sim 9$ reports 3 and never mentions −3, because Desmos favors whole numbers and the smaller absolute value. To force the other root, add a condition in curly brackets on the same line: $x_{1}^{2} \sim 9 \left\{x_{1} > 0\right\}$ gives 3, and $x_{2}^{2} \sim 9 \left\{x_{2} < 0\right\}$ on a separate line with a fresh subscript gives −3. The same brackets meet any stated condition: for $k^2=16$ where $k>0$, $16 \sim k^{2} \left\{k > 0\right\}$ gives 4.

Systems of equations: the bracket regression

Two separate regressions do not solve a system. Type 2x + 3y = 12 as one regression and x − y = 1 as another, and Desmos picks a pair for each line on its own. It never links the $x_1$ on one line to the $x_1$ on the other.

The fix is a bracket regression. Put every left side in one list, every right side in another list, and connect the two lists with a single tilde. Desmos matches the lists element by element: 12 has to equal 12 and 1 has to equal 1.

Many systems arrive as word problems. Replace 'is' and 'was' with an equals sign and type the sentence almost verbatim, capital letters and all. The price of a jacket was 20% more than its cost, and the price was $60: that is p = 1.2c and p = 60. As a bracket regression, $\left[p, p\right] \sim \left[1.2c, 60\right]$, and the cost comes out as 50.

When a system gets long, keep the right sides in their own list: $r = \left[12, 1\right]$ on one line, then $\left[2x_{1} + 3y_{1}, x_{1} - y_{1}\right] \sim r$. Now the numbers you might have mistyped all sit in one place. The left sides have to stay inside the regression, because a list with unknown letters in it cannot be defined on its own line.

When the constant will not stay constant, collapse the system by substitution. If y = 2x + n and y = 5x − 4 meet where x = 2, both right sides equal y, so type $2x_{1} + n \sim 5x_{1} - 4$ and $x_{1} = 2$, and n comes out as 2. If even that stalls, test the answer choices one at a time. Or graph it: $n = 2$, $y = 2x + n$ and $y = 5x - 4$, then click the intersection at (2, 6).

Infinitely many solutions, and when the tilde is the wrong tool

This pattern is easy to recognize. The question states that an equation has infinitely many solutions and asks for a constant. Regression works for one reason: infinitely many solutions means the two sides are literally the same graph, the same line drawn twice. So the sides are equal for every x, which is exactly what a regression with a list checks.

Given-points questions: table, equation, tilde

Any time a question gives you points, it is this pattern. The points might be written out, read off a graph in the question, or hidden in function notation: f(2) = 8 is the point (2, 8). Into the table go the points you already know, and nothing else. If the question asks for f(10), leave 10 out. A point with a constant in it, like (3, k), is fine unless k is what you want.

1. Type the points into a table. Type table on an empty line, then enter each x under $x_1$ and each y under $y_1$.
2. Type the equation that goes through them. A straight line is y = mx + b. Use the form the question or the answer choices use.
3. Convert it into regression format. y becomes $y_1$, the equals sign becomes a tilde, and x becomes $x_1$: $y_{1} \sim mx_{1} + b$. That is exactly what Add regression on the table writes for you.

Method 1 is the drop-down: open the edit list and press Add regression on the table. Method 2 is typing the custom regression yourself, for when the drop-down lacks the form the question uses. For y = a(x − h), type $y_{1} \sim a(x_{1} - h)$: a = 3 and h = −2/3 for the same two points. A regression only draws a curve when it is in y = form, so to see this one, duplicate the line in the edit list and strip the regression off the copy: $y_{1} \sim$ becomes $y =$ and every $x_{1}$ becomes x. Keep the regression line above it, because that is what defines a and h.

You need at least as many points as constants: two for a line, three for a parabola in standard form. With too few, infinitely many curves fit and Desmos just picks one. Vertex form is the way around that: given the vertex (1, −4) and the point (3, 8), put 3 and 8 in the table and type $y_{1} \sim a(x_{1} - 1)^{2} - 4$. The vertex used up h and k, so one point finds a = 3.

If the regression refuses to pass through the points, check $R^2$ under Statistics. It has to be exactly 1. Fit $y_{1} \sim mx_{1} + b$ through (1, 2), (2, 8) and (3, 18) and $R^2$ is below 1: wrong shape. Change it to $y_{1} \sim ax_{1}^{2} + bx_{1} + c$ and Desmos returns a = 2, b = 0, c = 0 with $R^2$ exactly 1. When the shape is right and the fit still misses, the course adds a tiny number to the model, or subtracts one, so Desmos starts its search somewhere else. And when a question asks for the greatest possible value, zoom out first; the largest value may sit off screen.

One exception: the standard form of a line, Ax + By = C. Points can never pin it down, because multiplying all three by any number gives the same line. Read the form out of the answer choices, then force one value yourself: $a = 2$, then $ax_{1} + by_{1} \sim c$, gives b = −2/3 and c = −4/3 for (2, 8) and (5, 17). Prove the 2 was harmless by moving it. Change it to $a = 4$ and b and c double, to −4/3 and −8/3, but $\frac{c}{a}$ still reads −2/3, so the line never changed.

Isolation: rearrange a formula without doing the algebra

The isolation pattern uses the phrase 'expresses R in terms of P, Q, and F' or 'which equation gives R in terms of the others'. There is no algebra to do. Type the formula, give every other letter a number, and let Desmos find the one you left alone.

• Never use 0 or 1. Squaring or square-rooting leaves them unchanged, so a wrong choice with an extra square in it would still look right.
• Never give two constants the same value. With Q = F = 2, choices A and C both give 4.5 and you cannot tell them apart.

Choosing regression or graphing: a decision table

First ask whether the question can be done in Desmos at all. If there is an equation in it, and especially if there are points, the answer is yes. A question built only out of words, with nothing set equal to anything, is the one kind you still work on paper. Then pick the tool: a regression hands back a number, and the graph hands back a picture you can count and compare.

Ask those questions in order and the method picks itself. The subscripts climb through the regression rows because each earlier line already owns its subscript and Desmos refuses to solve for the same letter twice; on a real question, press Delete All, then Done, and start clean.

Common errors and how to spot them

Every one of these has a matching guided task in the course, where you type the broken line into a live calculator, watch it break, and fix it. Start with the Regression section, or step back to the full guide to using Desmos on the SAT if the graphing half is new to you.