Free Cheat Sheet

When to Use Desmos
on the Digital SAT

The calculator is built into Bluebook. Know exactly when it saves you 2 minutes — and when it costs you 1.

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SAT Math • peezyprep.com
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The 10-Second Decision
Ask yourself these questions before touching the calculator

Desmos is always available on the Digital SAT — both modules. But opening it for every question is a trap. The fastest students know when Desmos gives them a shortcut and when algebra is quicker. Here's the framework.

The Golden Rule
If you can type it into Desmos and read the answer off the graph in under 15 seconds, use Desmos. If you'd need to rearrange, guess-and-check, or squint at a graph to find the answer, do algebra.
Open Desmos when...
• The question says "how many solutions"
• You see a system of 2 equations
• There's a square root or absolute value equation
• You need to find a parameter (slider!)
• You need the vertex, max, or min
• The equation looks ugly to solve by hand
Skip Desmos when...
• The question asks "which equation represents..."
• You can spot a factor trick in 5 seconds
• It's pure arithmetic (percents, ratios)
• You need to rearrange a literal equation
• The question asks for an expression, not a value
• It's a word-problem setup (reading, not math)
Trap Alert
Students who open Desmos for every question lose ~30 seconds per problem on setup alone. Over 44 questions, that's 22 minutes wasted. Be strategic.
Question Type Best Tool What to Type in Desmos
System of equations Desmos Type both equations → click intersection
How many solutions Desmos Graph both sides as y = ... → count intersections
Find the parameter Desmos Add a slider → adjust until condition is met
Extraneous solutions Desmos Graph both sides → real intersections only
Vertex / max / min Desmos Type the quadratic → click the vertex
Expression manipulation Algebra N/A — spot the factor trick
Literal equation Algebra N/A — just rearrange
Which equation represents Algebra N/A — reading comprehension
Percent / ratio arithmetic Algebra N/A — no variables to graph
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Use Desmos: 5 Question Types
These are your guaranteed time-savers
1
Systems of Equations
Saves 30-60 sec
Example Problem
\(y = 3x - 1\)
\(2x + y = 14\)

What is the value of \(x + y\)?
A) 3
B) 8
C) 11
D) 14
In Desmos
Type y = 3x - 1
Type 2x + y = 14
Click the intersection point → you get \((3, 8)\)
Add: \(3 + 8 = 11\). Done. Answer: C
2
How Many Solutions?
Saves 45-90 sec
Example Problem
How many real solutions does \(x^4 + x^2 - 12 = 0\) have?
A) 0
B) 2
C) 3
D) 4
In Desmos
Type y = x^4 + x^2 - 12
Count where the curve crosses the x-axis → 2 crossings
Done in 10 seconds. Answer: B
Why this is faster
By hand, you'd need to recognize the "disguised quadratic" (\(u = x^2\)), factor into \((u+4)(u-3) = 0\), reject \(u = -4\), solve \(x^2 = 3\), and count two roots. That's 60+ seconds of careful algebra. Desmos: instant.
3
Find the Parameter (Slider Problems)
Saves 30-60 sec
Example Problem
For which positive value of \(b\) does \(2x^2 + bx + 18 = 0\) have exactly one real solution?
A) 6
B) 9
C) 12
D) 18
In Desmos
Type y = 2x^2 + bx + 18 — Desmos creates a slider for \(b\)
Drag the slider until the parabola just touches the x-axis (one solution = tangent)
The slider lands on \(b = 12\). Answer: C
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Extraneous Solutions (Radicals & Rationals)
Saves 45-90 sec
Example Problem
\(\sqrt{5x + 11} = x + 1\)

What is the positive value of \(x\) that satisfies the equation above?
A) −2
B) 1
C) 5
D) −2 and 5
In Desmos
Type y = sqrt(5x + 11)
Type y = x + 1
Look at where the curves actually cross — only at \(x = 5\)
The "solution" at \(x = -2\) is extraneous (doesn't appear as an intersection). Answer: C
Common Trap
If you solve algebraically by squaring both sides, you get \(x = 5\) and \(x = -2\). But \(x = -2\) fails the check: \(\sqrt{5(-2)+11} = 1\) but \(-2 + 1 = -1 \neq 1\). Desmos shows you only the real intersection instantly — no checking needed.
5
Vertex, Maximum, or Minimum
Saves 20-40 sec
Example Problem
A ball is launched upward with height modeled by \(h(t) = -16t^2 + 48t + 4\), where \(t\) is time in seconds. What is the maximum height, in feet, that the ball reaches?
A) 36
B) 40
C) 48
D) 52
In Desmos
Type y = -16x^2 + 48x + 4 (use \(x\) instead of \(t\))
Click the peak of the parabola — Desmos labels it \((1.5, 40)\)
Maximum height = 40. Answer: B
Pro Move
For any quadratic, Desmos automatically labels the vertex when you click near it. No formula needed — no \(-b/2a\), no completing the square. Just click.
Bonus: Systems with Word Problems
"Company A charges $50 per month plus $0.10 per text. Company B charges $30 per month plus $0.20 per text. At how many texts do they cost the same?"

Type y = 50 + 0.1x and y = 30 + 0.2x. Click the intersection: \(x = 200\). Done in 15 seconds.
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Skip Desmos: 5 Question Types
Opening the calculator here actually slows you down
1
Expression Manipulation / Factor Tricks
Desmos adds 20+ sec
Example Problem
If \(4x + 2 = 12\), what is the value of \(16x + 8\)?
A) 40
B) 48
C) 56
D) 60
Why Algebra is Faster
Notice: \(16x + 8 = 4(4x + 2) = 4(12) = 48\). Done in 5 seconds. No solving for \(x\) needed.

Desmos can't help — there's nothing to graph. You'd have to solve for \(x\) first, then compute \(16x + 8\). That's slower and more error-prone.
2
"Solve for the Expression" Questions
Desmos adds 15+ sec
Example Problem
If \(7x + 10 = 44\), what is the value of \(7x - 10\)?
A) 14
B) 24
C) 34
D) 44
Why Algebra is Faster
From \(7x + 10 = 44\), subtract 20 from both sides: \(7x - 10 = 24\). One step. Answer: B.

Pattern to spot: When the question asks for a different expression than what's given, look for a shortcut that transforms one into the other.
3
"Which Equation Represents..." Setup Questions
Desmos adds 30+ sec
Example Problem
A plumber charges a $50 visit fee plus $25 per hour of labor. The total bill for a job is $T. Which equation gives the number of hours \(h\) in terms of the total bill?
A) \(h = \frac{T - 50}{25}\)
B) \(h = \frac{T}{25} - 50\)
C) \(h = \frac{T + 50}{25}\)
D) \(h = \frac{T}{75}\)
Why Algebra is Faster
This is a reading comprehension problem. Total = 50 + 25h, so \(h = (T - 50)/25\). Answer: A.

There's nothing to graph — you need to translate words into math, and Desmos can't do that for you.
4
Percent / Ratio Arithmetic
Desmos adds 20+ sec
Example Problem
An $80 jacket is marked up 50%, then immediately discounted 50%. What is the final price?
A) $80
B) $60
C) $40
D) $70
Why Algebra is Faster
\(80 \times 1.5 = 120\), then \(120 \times 0.5 = 60\). Answer: B.

The trap: "+50% then −50%" does NOT get you back to the original price. There are no variables here — just multiply.
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Literal Equations (Rearranging Formulas)
Desmos adds 15+ sec
Example Problem
The formula \(T = 2d + 3h + 50\) gives the total cost for a job requiring \(d\) parts and \(h\) hours. Which correctly expresses \(h\) in terms of \(T\) and \(d\)?
A) \(h = \frac{T - 50 - 2d}{3}\)
B) \(h = \frac{T - 50}{3 + 2d}\)
C) \(h = \frac{T}{3} - 50 - 2d\)
D) \(h = \frac{T + 50 - 2d}{3}\)
Why Algebra is Faster
Subtract 50 and 2d from both sides: \(T - 50 - 2d = 3h\). Divide by 3. Answer: A.

Classic trap: Choice B puts \(3 + 2d\) in the denominator — students who "combine" additive terms into a single division.
⚖️
Either Works: Your Call
These problems can go either way — pick your stronger method
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Pattern Recognition + Algebraic Identities
Speed depends on you
Example Problem
If \(x^2 - 8x = 15\), what is the value of \((x - 4)^2\)?
A) 15
B) 23
C) 31
D) 49
Algebra Route (~15 sec)
Expand: \((x-4)^2 = x^2 - 8x + 16\).
Substitute: \(15 + 16 = 31\).
Answer: C
Desmos Route (~20 sec)
Type x^2 - 8x = 15.
Read x-values, compute \((x-4)^2\).
Same answer, slightly slower.
How to Decide
If you instantly see that \((x-4)^2 = x^2 - 8x + 16\), algebra is faster. If you don't spot it within 10 seconds, switch to Desmos and solve for \(x\) directly. Don't waste time staring — commit to a method.
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Quadratic: Find the Zeros
Speed depends on you
Example Problem
What are the solutions to \(x^2 - 5x - 14 = 0\)?
If you can factor fast
\((x-7)(x+2) = 0\)
\(x = 7\) or \(x = -2\).
~10 seconds if you see it.
If factoring isn't clicking
Type y = x^2 - 5x - 14.
Click both x-intercepts.
~15 seconds, guaranteed.
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Desmos Power Tips
Tricks that most students don't know
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Deep Dive: "Number of Solutions"
The #1 question type where Desmos dominates

"How many solutions" appears 3-5 times per test across both modules. Here's every variant and what to type.

What the Question Looks Like What to Type What to Count
How many solutions does
\(f(x) = 0\) have?		 y = f(x) x-intercepts
(where curve hits y = 0)
How many values of \(x\) satisfy
\(f(x) = g(x)\)?		 y = f(x) and
y = g(x)		 Intersection points
(where curves cross)
For what value of \(k\) does
the system have no solution?		 y = f(x) with
slider for \(k\)		 Adjust until the curves
never touch (parallel)
How many real solutions does
\(x^4 - 5x^2 + 4 = 0\) have?		 y = x^4 - 5x^2 + 4 x-intercepts
(4 in this case)
For which \(b\) does
\(ax^2 + bx + c = 0\) have one solution?		 y = ax^2 + bx + c
with slider for \(b\)		 Parabola tangent to x-axis
(just barely touches)
Speed Benchmark
By algebra, "number of solutions" problems take 45-90 seconds (discriminant, factoring, case analysis). By Desmos, they take 10-20 seconds. On a timed test, that's the difference between finishing with 5 minutes to spare or running out of time.
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Quick-Reference: Every SAT Math Topic
Tear this page out and keep it next to you during practice
Topic Verdict Notes
Linear equations (basic solve) Algebra Faster to isolate than to graph
Expression manipulation Algebra Look for factor tricks — nothing to graph
Systems of equations Desmos Type both → click intersection
Linear inequalities Either Desmos shades regions; algebra is fast too
Linear function graphs Desmos Match equation to graph instantly
Quadratic: solve for zeros Either Factor if obvious; graph if not
Quadratic: vertex/max/min Desmos Click the peak — no formula needed
Quadratic: discriminant / # of solutions Desmos Graph + count x-intercepts or use slider
Polynomial equations (higher degree) Desmos Graph → count real roots
Radical / square root equations Desmos Shows extraneous solutions visually
Rational equations Desmos Graph both sides; avoid algebraic errors
Exponential growth/decay Either Useful for "when does it reach X?" questions
Absolute value equations Desmos Graph abs(expression) — see both branches
Literal equations / rearranging Algebra Just rearrange — nothing to graph
"Which equation represents..." Algebra Reading comprehension, not calculation
Percents, ratios, proportions Algebra Pure arithmetic — no variables to graph
Statistics (mean, median) Algebra Conceptual / arithmetic
Probability / two-way tables Algebra Read the table — no graphing needed
Geometry (area, volume, triangles) Algebra Formula + plug in. Desmos can help with
coordinate geometry (distance, midpoint)
Trig (right triangle / unit circle) Either Desmos can graph trig functions; basic
SOHCAHTOA is faster by hand
Data & scatterplots Desmos Enter points in table → fit regression line
Want the Full Course?
This cheat sheet covers when to use Desmos. Our full SAT Math course teaches you how to solve every question type — with video walkthroughs, trap analysis, and 200+ practice problems.
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