Desmos Basics for the ACT

Learn the on-screen Desmos calculator well enough to type any expression exactly, read graphs precisely, and dodge the mode and typing traps.

28 min
Desmos way

Ready when you are

7 steps · about 28 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

Expression list: The panel on the left where you type. Each line holds one expression, and Desmos prints its value at the right end of the line.
Point of interest: A gray dot Desmos shows when you click a graph, at intercepts, intersections, maxima, and minima. Click it to read its displayed coordinates.
Degree mode: The default on the ACT's Desmos. Angles are read in degrees unless you open the wrench and choose Radians.
Domain restriction: A condition in curly braces typed after an expression, like $y = 2x + 1 \; \{0 < x < 5\}$ , so Desmos draws only that part of the graph.

Step 2 of 7Try it firstDone

Try it first

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No pressure here. Use any method you like, even a guess. We’ll come back to this one at the end.

Step 3 of 7LearnDone

The big idea

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A calculator that draws

Think of Desmos as a calculator with a window attached. On the left is the expression list, where you type. On the right is the graph paper, where every equation you type gets drawn. Use the + and − buttons to zoom, and the home button (the little house) to snap back to the standard view. The keypad button opens on-screen keys, but typing on your keyboard is faster.

Click any curve and gray dots appear at its points of interest: intercepts, intersections, and turning points. Click a dot and Desmos shows its coordinates (which may be rounded).

Explore in Desmos

Find the gray dots

Click the curve. Then click each gray dot. Which dot is the minimum, and what are its coordinates?

The calculator is live: change anything and see what happens.

Find the gray dots

Degree mode

Loading Desmos…

Key idea

Click, don't estimate

Never eyeball a graph. Click the gray dot and read the displayed coordinates. If the choices are exact fractions or radicals, confirm algebraically; graph labels can be rounded.

Typing math exactly

Desmos reads what you type literally, so type like a programmer:

^ makes an exponent, and / starts a fraction.
sqrt becomes a square root sign, and pi becomes $\pi$ .
The | key makes absolute value bars.
To get a value, type the expression on its own line. Desmos prints the answer at the right.

Your cursor stays inside an exponent, a fraction, or a parenthesis until you press the right arrow. Whatever you type next lands wherever the cursor is.

Trap

The negative-square trap

Desmos reads $-3^2$ as $-(3^2) = -9$ . If you mean negative three squared, type (-3)^2 to get $9$ . Better yet, define the function and type f(-3), and Desmos adds the parentheses for you.

Degrees are the default

The ACT’s Desmos starts in degree mode, so sin(30) gives $0.5$ . You need the wrench, then Radians, only when a question gives angles in radians. The csc, sec, and cot functions are turned off, so rewrite them first.

Reciprocal trig

\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}

Type 1/sin(10), not csc(10).

Restrict, evaluate, explore

Add a condition in curly braces after an expression, like $y = 2x + 1 \; \{0 < x < 5\}$ , to draw only part of a graph. For a model that only applies for certain inputs, that keeps you from reading a point that doesn’t count.

Two more tools: the + menu at the top of the list adds a table, and typing a letter like $a$ into an equation offers a slider you can drag.

Explore in Desmos

Draw only part of a line

The segment starts at x = 0 and stops at x = 5. What y-value does it approach at the right end?

The calculator is live: change anything and see what happens.

Draw only part of a line

Degree mode

Loading Desmos…

Step 4 of 7Worked examplesDone

Worked examples

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Skip check · 2 questions

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Worked example

Example 1: Type a messy fraction exactly

Problem

What is the value of

\dfrac{3^4 - 11}{\sqrt{49}}

Click an empty line in the expression list. Type (3^4, then press the right arrow to leave the exponent.

Finish the top: type -11).

Predict the next move

Now type /. Desmos builds a fraction with

3^4 - 11

on top and the cursor in the denominator.

Type sqrt(49). Desmos prints $= 10$ .

Why does this step work?

Check:

3^4 = 81

81 - 11 = 70

, and

70 \div 7 = 10

Worked example

Example 2: Trig with a disabled key

Problem

What is the value of

\sec(60^\circ) + \cot(45^\circ)

The angles are in degrees, and the ACT’s Desmos is already in degree mode. Leave the wrench alone.

The sec and cot functions are disabled on the ACT.

Predict the next move

Type 1/cos(60), press the right arrow to leave the denominator, then type +1/tan(45).

Desmos prints $= 3$ , because $\sec(60^\circ) = 2$ and $\cot(45^\circ) = 1$ .

Why does this step work?

Faded example · you finish it

Example 3: You finish it

Problem

In the standard

(x,y)

coordinate plane, where does the line

y = 0.5x - 3

cross the

x

-axis?

Type y=0.5x-3 on an empty line.

Click the line so its gray points of interest appear.

The gray dot on the $x$ -axis shows

So the line crosses the $x$ -axis at $x =$

Desmos way

The Desmos way: a maximum in 10 seconds

Type y=-x^2+6x-5. Press the right arrow after the exponent 2.

Click the curve. Gray dots appear at both

x

-intercepts and at the top.

Click the top dot:

(3, 4)

. The maximum value is the

y

-coordinate,

4

, and it happens at

x = 3

The Desmos way: a maximum in 10 seconds

Degree mode

Loading Desmos…

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Practice

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Speed drills

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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