How to use Desmos on the SAT: techniques with worked examples

Bluebook has the Desmos calculator built into the Math section, and College Board's calculator policy says you can switch between its graphing and scientific versions at any point. Used well, it turns a messy algebra question into a graph you can read in 20 seconds. Used badly, it eats time on questions you could solve in your head. Below are the techniques that save the most time, with the exact input for each.

Set up before you need it

A few habits make Desmos faster on test day:

Open it at the start of Math Module 1. Expand the calculator window so graphs are easier to read.
Stay in the right calculator. Desmos notes that work in the scientific calculator doesn't carry over to the graphing calculator, so check which one is open before you type.
Delete old expressions between questions. A leftover line from the last question can create an intersection that has nothing to do with the current one.
Check degrees or radians in the wrench menu before any trig question.
Practice on the test version. Desmos posts the SAT-configured calculators on its Testing page (desmos.com/testing). According to Desmos, the test version turns off images, folders, and notes. Graphing, tables, and regressions all work.

Technique 1: graph both sides of an equation

Any equation in one variable can be split into two graphs. Type each side as its own line, starting with y=. The x-coordinates of the points where they cross are the solutions. Click either graph and Desmos marks intersections, intercepts, and turning points with gray dots. Click a dot to see its coordinates.

Worked example 1. How many solutions does $x^2 - 6x + 5 = 2x - 7$ have, and what is their product?

Type these two lines:

y=x^2-6x+5
y=2x-7

The graphs cross at $(2, -3)$ and $(6, 5)$ , so there are two solutions, $x = 2$ and $x = 6$ . Their product is $12$ . The algebra agrees: moving everything to one side gives $x^2 - 8x + 12 = 0$ , which factors as $(x - 2)(x - 6) = 0$ .

A shortcut: type the equation exactly as written, x^2-6x+5=2x-7. Because there's no $y$ , Desmos draws a vertical line at each solution.

Technique 2: read systems off the intersection

Desmos graphs equations in standard form, so you don't need to solve for $y$ first.

Worked example 2. If $3x + 2y = 19$ and $x - 4y = -17$ , what is the value of $x + y$ ?

Type 3x+2y=19 and x-4y=-17. Click the point where the lines cross: $(3, 5)$ . So $x + y = 8$ .

Graphs also answer "how many solutions" questions. Two lines that cross once give one solution. Parallel lines give no solution. If you only see one line, the second is sitting on top of it, and the system has infinitely many solutions.

Technique 3: solve quadratics with x-intercepts

To solve $3x^2 - 5x - 2 = 0$ , type y=3x^2-5x-2 and click the parabola. The gray dots on the x-axis are at $(-0.333, 0)$ and $(2, 0)$ , so the solutions are $x = -\frac{1}{3}$ and $x = 2$ . The same click shows the vertex and the y-intercept, which covers most questions about a minimum or maximum value.

Desmos shows decimals, so learn the common ones: $0.333$ is $\frac{1}{3}$ , $0.667$ is $\frac{2}{3}$ , $1.414$ is $\sqrt{2}$ , and $1.732$ is $\sqrt{3}$ . When the answer choices are exact, match your decimal to the choice.

Technique 4: use a slider for an unknown constant

When an equation contains a constant like $k$ or $a$ , type it with the letter in place. Desmos offers to add a slider. Drag the slider and watch the graph change until it matches the condition in the question.

Worked example 3. In the system $y = 2x^2 + 3$ and $y = kx + 1$ , $k$ is a positive constant. The system has exactly one solution. What is the value of $k$ ?

Type y=2x^2+3 and y=kx+1.
Desmos shows an "add slider" button under the second line. Click $k$ to create the slider.
Drag $k$ up from 0. At first the line passes under the parabola and never touches it. Near $k = 4$ the line just touches the parabola. Past that, it crosses twice.
Click the slider's value and type 4. The line now touches the parabola at a single point, $(1, 5)$ .

So $k = 4$ . To confirm with algebra, set the two equations equal: $2x^2 - kx + 2 = 0$ . Exactly one solution means the discriminant is zero, so $k^2 - 4(2)(2) = 0$ , which gives $k^2 = 16$ and $k = 4$ .

Sliders run from $-10$ to $10$ by default. If the answer might be outside that range, click the limits and change them.

A related trick works when there is only one unknown: rename it $x$ . If $f(x) = a(x - 3)^2 + 5$ and $f(1) = 13$ , type y=x(1-3)^2+5 and y=13. The lines cross at $x = 2$ , so $a = 2$ .

Technique 5: fit a regression to a table

When a question gives a table and asks for a value that isn't in it, let Desmos find the equation.

Worked example 4. The table shows values of a quadratic function $f$ . What is $f(5)$ ?

$x$	0	1	2	3
$f(x)$	5	8	15	26

Click the + button in the top left and choose "table." Enter 0, 1, 2, 3 in the $x_1$ column and 5, 8, 15, 26 in the $y_1$ column.
On a new line, type y1~ax1^2+bx1+c. Desmos turns y1 and x1 into $y_1$ and $x_1$ for you.
Desmos reports $a = 2$ , $b = 1$ , $c = 5$ , and $R^2 = 1$ . An $R^2$ of 1 means the curve passes through every point, so $f(x) = 2x^2 + x + 5$ .
Type f(x)=2x^2+x+5 and then f(5). Desmos shows $60$ .

Use y1~mx1+b for a linear table and y1~ab^x1 for an exponential one. If $R^2$ is not 1 on a table that should be exact, check for a typo in your entries.

Technique 6: check answer choices by graphing

For "which expression is equivalent" questions, graph the original and then each choice. The equivalent choice lands exactly on top of the original. For example, y=(x+3)^2-2(x+3) and y=x^2+4x+3 draw the same parabola, so the expressions are equivalent. If two graphs separate anywhere, they aren't.

The same idea works for inequalities. Graph each inequality in the system, then type each answer choice as a point, such as (3,4). The correct point sits in the region where the shading overlaps.

When not to use Desmos

Desmos is not a computer algebra system, as Desmos itself points out. It graphs and evaluates, but it doesn't rearrange formulas or explain what a number means. Skip it in these cases:

Interpretation questions. If a question asks what $1.05$ represents in $P = 200(1.05)^t$ , a graph won't tell you. You need to know that it means growth of 5% per time period.
Solving a formula for one variable. "Solve $V = \pi r^2 h$ for $h$ " is an algebra question. Desmos can't rewrite it.
One-step equations. Solving $3x + 7 = 22$ on paper takes less time than typing it.
Intersections far off the screen. If two lines cross at $x = 4{,}000$ , zooming out takes longer than the algebra.

A good rule: if you can see the algebra and it will take under 30 seconds, do it by hand and use Desmos to check. Reach for Desmos first when the algebra is messy, such as a quadratic that doesn't factor, a system with fractions, or a line meeting a curve.

How to practice

Pick 10 questions from the free nonlinear functions questions or systems of linear equations questions and solve each one twice, once by hand and once in Desmos. Time both. After a few sets you'll know which question types are faster with a graph.

To find out which math skills cost you the most points, take SAT Owl's free 24-question diagnostic. A Desmos calculator is available on every Math question, so you can try these techniques as you go.