The built-in Desmos calculator: when it saves time and when it does not
A graphing calculator sits on screen for all 70 minutes. Knowing the four things it does in ten seconds — and the four it does slower than a pencil — is worth more than any single formula.
What is the value of the y-coordinate?
In the xy-plane, the graph of y = x² − 4x + 1 and the line y = x − 3 intersect at two points. What is the y-coordinate of the intersection point with the greater x-coordinate?
Set them equal: x² − 4x + 1 = x − 3 gives x² − 5x + 4 = 0, so (x − 1)(x − 4) = 0 and x = 1 or x = 4. The greater x is 4, and y = 4 − 3 = 1. Typed into the calculator, the two curves cross at (1, −2) and (4, 1), and clicking the right-hand crossing reports exactly that. A is the y-coordinate at the other point, x = 1. C is 4 — the x-coordinate of the point you were asked about, which is the single commonest slip once the calculator hands you an ordered pair. D adds the two x-values, 1 + 4, which is not a coordinate of anything.
1
The calculator is on screen for the whole maths section, and it is Desmos.
Both modules, all 44 questions, no “calculator section” and no “no-calculator section”. You may also bring your own, but the built-in one is the one worth practising with.
It opens as a panel beside the question: an expression list on the left where you type, and a graphing window on the right. Anything you type in the list is drawn instantly, and the panel remembers your expressions until you clear them.
graphs anything you typearithmetic keypadtables of valuessliders for constantsa line of best fita separate reference sheet
What it does not do is read the question. It will not decide which quantity you were asked for, will not turn “increases by 15% each year” into a multiplier, and will not notice that you typed the stem's 4x as 4 + x. Every mistake you can make on paper you can also make in the expression list, only faster.
Clear the list between questions. A leftover equation from the previous item still draws a curve, and a screen with three curves on it is how a correct intersection gets read off the wrong pair.
2
Type the equation, then read the root or the crossing off the picture.
This is the opening problem. Two expressions, no algebra, and the answer is a point you can click.
type y = x² − 4x + 1, then y = x − 3
two lines in the expression list, about 12 seconds
click the right-hand crossing
Desmos prints (4, 1)
the question asked for y, so the answer is 1
the second number, not the first
The same trick finds a root: type y = 2x² − 7x − 9 and the curve's two x-intercepts are (−1, 0) and (4.5, 0), so the solutions are x = −1 and x = 4.5. A root is just a crossing with the x-axis, and Desmos marks those too.
Desmos hands you a point; the question asked for a number
Choice C in the opening problem is 4 — the x-coordinate of the very point the calculator just showed you. The screen says (4, 1) and the stem says “y-coordinate”, and under time pressure the eye takes whichever number it saw first. Before you click a crossing, say out loud which of the two numbers you are going to keep.
3
A system of two equations is two curves and one crossing.
Elimination on decimal coefficients is where careful students lose ninety seconds. Graphing skips the arithmetic entirely.
0.6x + 0.8y = 5 1.4x − 0.5y = 2.2
type both equations exactly as printed
Desmos accepts x and y in any arrangement
click the crossing → (3, 4)
so x = 3 and y = 4
check: 0.6(3) + 0.8(4) = 1.8 + 3.2 = 5
and 1.4(3) − 0.5(4) = 4.2 − 2 = 2.2
Two lines that never cross means no solution; two lines that lie on top of each other means infinitely many. The graph answers those questions faster than comparing slopes, because you see it rather than compute it.
If the stem asks for x + y or for 2y rather than for x or y, the crossing still does the work — but do the last step on paper. Read (3, 4) off the screen, write it down, then form what was asked. Answering the wrong quantity is the failure mode this whole lesson keeps returning to.
4
Vertex, intercepts and an unknown constant on a dial.
Desmos marks the interesting points of a curve for you: the top or bottom, the x-intercepts, the y-intercept. Click one and it prints the coordinates.
f(x) = −2x² + 12x − 13
an SAT “what is the maximum value” stem
x = −b2a = −122 × −2 = 3
by hand: find the axis of symmetry
f(3) = −18 + 36 − 13 = 5
and substitute back — two steps, about 35 seconds
on Desmos: click the top of the curve → (3, 5)
about 20 seconds, and the maximum is 5, not 3
top or bottom point → vertexcrosses the x-axis → solutionscrosses the y-axis → the constant termcurves cross → a system
The same curve's x-intercepts are (1.419, 0) and (4.581, 0), from 2x² − 12x + 13 = 0 and the quadratic formula. Those are numbers you would not want to produce by hand under time pressure, and they are one click away.
Type a letter you have not defined and Desmos offers a slider
Enter y = 2x² + bx + 7 and a button appears: “add slider: b”. Drag b and the parabola moves in real time. That turns a whole category of SAT parameter questions — for what value of the constant does this graph touch that line, or cross the x-axis exactly once — into something you can watch happen. Find the value on the slider, then confirm it with the discriminant so you are not trusting a drag.
5
A table turns a model into numbers, and a list of points into a model.
The expression list has a table option. It works in both directions, and the second direction is the one almost nobody uses.
Forwards: define the function, then let the table evaluate it.
Backwards: type the data as a table, then ask for the line through it.
Months after purchase, x
1
2
3
4
5
Value, y (dollars)
480
445
410
375
340
enter the two columns as x₁ and y₁
a table, not five separate points
type y₁ ~ mx₁ + b
the tilde asks for a line of best fit
m = −35, b = 515
so y = 515 − 35x, and 515 − 35(1) = 480 ✓
For a table this clean you would get there faster by subtracting: 480 − 445 = 35 down per month, so the slope is −35, and one month before the first row the value was 480 + 35 = 515. Keep the regression for scatterplots, where the points do not sit on a line and the question says “line of best fit”.
6
Half the time, one line of algebra beats the whole calculator.
Typing costs about 6 seconds a line, plus the time to find the window and click the point. A question you can finish in two written steps is a question you should not be typing.
The question
By hand
On Desmos
Use
Solve 5x − 12 = 3x + 8
two lines → x = 10, 15 s
type two equations, find the crossing, 35 s
by hand
What is 12% of 148?
0.12 × 148 in your head, 20 s
keypad, 5 s
Desmos
Solve x² − 5x + 4 = 0
factors to (x − 1)(x − 4), 20 s
graph and read two intercepts, 25 s
by hand
Solve x² + x − 14 = 0
quadratic formula, √57, 70 s
graph and click, 25 s
Desmos
Where y = x² − 4x + 1 meets y = x − 3
rearrange, factor, substitute, 90 s
two lines typed, 30 s
Desmos
What does 90 mean in P(n) = 35n + 90?
read the sentence, 10 s
nothing to type
by hand
Ugly numbers and crossings → the calculator. Clean numbers and meanings → the pencil.
The real cost is not the seconds
It is the switch. Every time you move to the expression list you re-read the stem, retype it, and come back having lost the thread of the algebra you had started. That is where a 4x becomes 4 + x and a 2(x + 3) loses its bracket. Decide before you touch the panel: is there a number here I cannot get in two written steps? If not, do not open it.
interpretation → no calculatorequivalent expressions → no calculatorugly roots → calculatortwo curves crossing → calculatora constant to tune → slider
✓
Check: did the screen answer the question that was asked?
the crossing(4, 1)y was asked, so 1 ✓
the system(3, 4)0.6(3) + 0.8(4) = 5 ✓
the vertex(3, 5)the maximum is 5, not 3 ✓
the tableC(21) = 197.25under 200, and C(22) is not ✓
Remember the flow
Type it exactly · Click the point · Keep the right coordinate
Two written steps or fewer? Do not open the panel.
Now you try
Practice problem
What is the value of b?
In the xy-plane, the graph of y = 2x² + bx + 7 and the line y = 3x − 1 intersect at exactly one point, where b is a constant and b > 0.
Set them equal: 2x² + bx + 7 = 3x − 1 becomes 2x² + (b − 3)x + 8 = 0. Exactly one intersection means exactly one solution, so the discriminant is 0: (b − 3)² − 4(2)(8) = 0, giving (b − 3)² = 64 and b − 3 = ±8. That is b = 11 or b = −5, and b > 0 leaves 11. On the calculator, type both equations, accept the slider for b and drag it: the parabola drops onto the line at exactly b = 11. B is 8, the square root of 64, with the +3 never added back. D is 3 + 16, from square-rooting 4ac as 4√ac instead of 2√ac: 4 × 4 = 16 rather than 2 × 4 = 8. A is the 3 on its own, from stopping at b − 3 = 0.
Hint: “intersect at exactly one point” is a discriminant sentence. Move everything to one side first, then decide whether you would rather drag a slider or set b² − 4ac to zero — and notice that the slider tells you which of the two roots the condition b > 0 keeps.
