Digital SAT Math — cheat sheet

Everything to know cold, then the 24 question shapes that use it. Downloaded from iwillget800.com on 2026-09-24.

Blue the formula itself Green what's true or given Amber the trap Teal type this into Desmos

Know these cold

None of this is printed for you on test day.

Lines

Slope from two points	$m = \dfrac{y_2 - y_1}{x_2 - x_1}$
Slope-intercept form	$y = mx + b$	b is the value when x = 0.
Point-slope form	$y - y_1 = m(x - x_1)$	Fastest when you have a point and a slope.
Standard form	$ax + by = c$	Slope is $-a/b$ — worth knowing, it saves rearranging.
Parallel	$m_1 = m_2$
Perpendicular	$m_1 m_2 = -1$	Negative reciprocal — both parts, not one.
Midpoint	$\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$
Distance	$\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Quadratics

Quadratic formula	$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant	$b^2 - 4ac$	$>0$ two solutions, $=0$ one, $<0$ none.
Vertex (axis of symmetry)	$x = -\dfrac{b}{2a}$
Vertex form	$y = a(x-h)^2 + k$	Vertex is $(h, k)$ — signs flip.
Factored form	$y = a(x-r_1)(x-r_2)$	$r_1, r_2$ are the x-intercepts.
Sum and product of roots	$r_1 + r_2 = -\dfrac{b}{a}, \quad r_1 r_2 = \dfrac{c}{a}$

Exponents and radicals

Multiplying	$x^a \cdot x^b = x^{a+b}$
Dividing	$\dfrac{x^a}{x^b} = x^{a-b}$
Power of a power	$(x^a)^b = x^{ab}$
Power of a product	$(xy)^a = x^a y^a$	The coefficient takes the power too.
Negative exponent	$x^{-a} = \dfrac{1}{x^a}$
Fractional exponent	$x^{a/b} = \sqrt[b]{x^a}$
Zero exponent	$x^0 = 1$	For any $x \ne 0$ .

Exponential models

General form	$y = ab^x$	Growth if $b>1$ , decay if $0<b<1$ .
Percent growth or decay	$y = a(1 \pm r)^t$	$r$ as a decimal: 7% is 0.07.
Multiplies every T units	$y = ab^{t/T}$	Divide the exponent by $T$ , never multiply.
Check any model		At $t = 0$ it must give the starting value.

Percentages

Percent change	$\dfrac{\text{new} - \text{old}}{\text{old}} \times 100$	Always over the original.
Increase by p%	$\times \left(1 + \dfrac{p}{100}\right)$
Decrease by p%	$\times \left(1 - \dfrac{p}{100}\right)$
Two changes in a row		Multiply the factors. Percentages never add.
Undo a percentage		Divide by the factor. Adding it back lands short.

Statistics

Mean	$\text{total} = \text{mean} \times \text{count}$	Turn every mean question into a total.
Median		Sort the list first. Always.
Outliers		Drag the mean a long way, the median barely at all.
Standard deviation		Compare how tightly values cluster. Never compute it.
Margin of error	$\text{estimate} \pm \text{MOE}$	A plausible range for the population, not the sample.
Bigger sample		MOE shrinks with $\sqrt{n}$ — 4x the people halves the margin.
Random selection		Lets you generalise to the population sampled from.
Random assignment		Lets you claim cause. Different word, different job.
Residual	$\text{actual} - \text{predicted}$	Positive means the point sits above the line of best fit.

Probability

Basic probability	$\dfrac{\text{favourable}}{\text{total}}$	Over the total, not over the unfavourable.
Conditional probability		"Given" names your new denominator — that row or column only.
Reversed conditionals		$P(A \mid B) \ne P(B \mid A)$ . Check which is asked.

Circles

Equation of a circle	$(x-h)^2 + (y-k)^2 = r^2$	Centre $(h,k)$ ; the right side is $r^2$ .
Completing the square		Halve the linear coefficient, square it, add to both sides.
Arc length	$\dfrac{\theta}{360} \times 2\pi r$
Sector area	$\dfrac{\theta}{360} \times \pi r^2$
Radians	$\pi \text{ rad} = 180^\circ$
Chord		A perpendicular from the centre bisects it — then use Pythagoras.

Triangles and trigonometry

SOH-CAH-TOA	$\sin = \dfrac{O}{H}, \ \cos = \dfrac{A}{H}, \ \tan = \dfrac{O}{A}$
Complementary angles	$\sin x^\circ = \cos(90 - x)^\circ$	Equal, not reciprocal. Comes up nearly every test.
Pythagorean identity	$\sin^2\theta + \cos^2\theta = 1$
Triples worth knowing	$3\text{-}4\text{-}5, \ 5\text{-}12\text{-}13, \ 8\text{-}15\text{-}17, \ 7\text{-}24\text{-}25$
Similar triangles		One scale factor across every pair of sides.
Scaling	$\text{length } k, \ \text{area } k^2, \ \text{volume } k^3$
Exterior angle		Equals the sum of the two remote interior angles.

Already on the reference sheet

Printed on the reference sheet in the exam. Don't spend memory here.

Given to you

Circle	$A = \pi r^2, \quad C = 2\pi r$
Rectangle and triangle	$A = \ell w, \quad A = \tfrac{1}{2}bh$
Pythagorean theorem	$a^2 + b^2 = c^2$
Special right triangles	$x, x\sqrt{3}, 2x \quad\text{and}\quad s, s, s\sqrt{2}$
Rectangular prism	$V = \ell wh$
Cylinder	$V = \pi r^2 h$
Sphere	$V = \tfrac{4}{3}\pi r^3$
Cone	$V = \tfrac{1}{3}\pi r^2 h$
Pyramid	$V = \tfrac{1}{3}\ell wh$
Angles	$360^\circ \text{ in a circle}, \quad 180^\circ \text{ in a triangle}$

The question shapes

Algebra

Lines, systems, and inequalities — the largest single slice of the section.

Line of best fit from a tableMust Know

$y_1 \sim mx_1 + b$

A table of x and y values, and the question wants the linear model, the slope, the intercept, or a prediction at some x.

In Desmos

table: paste x values into x_1, y values into y_1
y_1 ~ mx_1 + b

Desmos returns m and b in the regression panel. Read them straight off — no arithmetic.

Trap: Choosing the answer that swaps m and b. If the question asks for the value of the model at x = 0, that's b, not m.

System of two linear equationsHigh Frequency

$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$

Two equations, and the question wants x, y, x + y, or how many solutions exist.

In Desmos

type both equations on separate lines
click the intersection point

Desmos labels the intersection with exact coordinates. For x + y, add them in a third line.

Trap: Solving correctly for x and then answering with x when the question asked for x + y or for y. Underline what's being asked before you type.

No solution / infinitely many solutionsMust Know

$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$

A system with an unknown coefficient, asking for the value that makes it have no solution or infinitely many.

In Desmos

type both equations using a slider for the unknown
drag the slider and watch the lines

Parallel and distinct = no solution. Perfectly overlapping = infinitely many.

Trap: Mixing up the two cases. Same slope + different intercept = no solution; same slope + same intercept = infinitely many. The wrong answer is always the other one.

Which point satisfies the system of inequalities

$y > mx + b$

Two or more inequalities, and four candidate points — which one lies in the shaded region.

In Desmos

type each inequality with > or <
type the candidate point as (a, b)

Desmos shades the feasible region automatically. A point inside all overlaps is the answer.

Trap: Ignoring strict vs. non-strict inequality. A point exactly on a dashed boundary line does not satisfy > or <.

Slope from two points

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Two points given in any form — coordinates, a table, or a word problem about rate of change.

In Desmos

(x_1, y_1)
(x_2, y_2)
then y_1 ~ mx_1 + b

Trap: Flipping the subtraction order in only one of the two differences, which gives you the negative of the right slope — and that value is always one of the choices.

Advanced Math

Quadratics, exponentials, and anything with a variable in an exponent.

Solutions / zeros of a quadraticMust Know

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Any question asking for the roots, the x-intercepts, the zeros, or the solutions of a quadratic.

In Desmos

type the equation set equal to 0
click each x-intercept

You almost never need the quadratic formula on the digital SAT — graph it and click.

Trap: Giving only the positive root. If the parabola crosses twice, both values are solutions, and the answer choices will include each one alone.

How many real solutionsHigh Frequency

$b^2 - 4ac$

A quadratic with an unknown coefficient, asking for the value that gives exactly one real solution, two, or none.

In Desmos

type the quadratic with a slider for the unknown
drag until the graph just touches the x-axis

Touching at exactly one point means the discriminant is 0 — that's the 'exactly one real solution' case.

Trap: Reading 'no real solutions' as 'no solutions'. The discriminant being negative means the parabola never crosses the x-axis — it doesn't mean the equation is broken.

Vertex, minimum, or maximumMust Know

$y = a(x - h)^2 + k \quad \text{vertex } (h, k)$

Asking for the minimum value, maximum value, or the x that produces it.

In Desmos

type the function
click the turning point

Desmos labels the vertex directly. Also: x = -b/(2a) if you'd rather compute it.

Trap: Answering with h when the question asked for the minimum value, which is k. 'At what x' wants h; 'what is the minimum' wants k.

Exponential growth and decayHigh Frequency

$y = a(1 + r)^t$

Something grows or shrinks by a percentage each period — population, investment, bacteria, depreciation.

In Desmos

y = a(1 + r)^x with sliders for a and r
or type the given function and read values

Decay is the same formula with a negative r, i.e. (1 − r).

Trap: Using the percentage as-is instead of converting it. A 3% increase makes the base 1.03, not 3 or 0.03.

Evaluating f(g(x))

$f(g(x))$

Two functions defined, asking for a composed value at a specific input.

In Desmos

f(x) = ...
g(x) = ...
f(g(3))

Define both functions by name and Desmos evaluates the composition directly.

Trap: Composing in the wrong order. f(g(x)) means g runs first — the inner function is the one that touches x.

Which expression is equivalent

A messy expression and four candidate rewrites, asking which one is equivalent for all x.

In Desmos

y = original expression
y = candidate expression

If the two graphs sit exactly on top of each other, they're equivalent. Test each choice in turn.

Trap: A choice that matches everywhere except one excluded value. Check where the original has a denominator of zero — the graphs will differ there.

Problem-Solving & Data

Percentages, rates, statistics, and reading a chart under time pressure.

Percent increase or decreaseMust Know

$\frac{\text{new} - \text{old}}{\text{old}} \times 100$

Two values, asking by what percent one changed into the other.

In Desmos

(new - old)/old * 100

Trap: Dividing by the new value instead of the original. The denominator is always where you started, and the answer choices include both versions.

Successive percent changesHigh Frequency

$P \times (1 + r_1)(1 + r_2)$

A price rises 20% then falls 20%, or similar — asking for the net result.

In Desmos

100 * 1.2 * 0.8

Multiply the factors; never add the percentages.

Trap: Concluding it returns to the original. Up 20% then down 20% lands at 96%, not 100% — and 'no change' is always offered.

Effect of an outlier on mean vs. medianMust Know

A data set changes — one value is added, removed, or altered — asking what happens to the mean and the median.

In Desmos

mean(L_1)
median(L_1)
then edit the list and compare

Put the data in a list L_1 and recompute after the change.

Trap: Assuming both move together. The median often doesn't move at all when an extreme value changes, because it only depends on position.

Margin of error and confidence intervals

A survey result with a margin of error, asking which conclusion is supported.

In Desmos

estimate - margin
estimate + margin

The plausible range is estimate ± margin. Any conclusion outside it is unsupported.

Trap: Choosing a statement about the sample when the question asks about the population, or one that claims certainty rather than plausibility.

Can you conclude causationHigh Frequency

A study is described, and four conclusions are offered — which one the design actually supports.

In Desmos

Trap: Accepting a causal claim from an observational study. Only random assignment to groups licenses cause; random selection only licenses generalising to the population.

Rates and unit conversion

$\frac{\text{units}}{\text{time}} \times \text{conversion factor}$

A rate in one unit, asking for the equivalent in another — miles per hour to feet per second, and so on.

In Desmos

60 * 5280 / 3600

Chain the factors in one line so the units visibly cancel.

Trap: Inverting one conversion factor. Write the units next to each number and confirm the ones you don't want cancel diagonally.

Probability from a two-way tableMust Know

$P(A \mid B) = \frac{n(A \cap B)}{n(B)}$

A table of counts, asking for a probability — often restricted to one row or column.

In Desmos

count / total

Trap: Using the grand total as the denominator on a conditional question. 'Given that the student is a senior' means the denominator is the senior total, not everyone.

Geometry & Trigonometry

Angles, circles, right triangles, and the formulas that aren't on the reference sheet.

30-60-90 and 45-45-90 trianglesMust Know

$30\text{-}60\text{-}90: x, x\sqrt{3}, 2x \quad 45\text{-}45\text{-}90: x, x, x\sqrt{2}$

A right triangle with a 30°, 60°, or 45° angle, asking for a missing side.

In Desmos

x * sqrt(3)
x * sqrt(2)

Trap: Putting the √3 on the hypotenuse. In a 30-60-90 the hypotenuse is 2x — the √3 belongs to the side opposite the 60° angle.

Equation of a circleHigh Frequency

$(x - h)^2 + (y - k)^2 = r^2$

An equation of a circle, possibly needing completing the square, asking for the center or radius.

In Desmos

type the equation exactly as given

Desmos graphs it and you can read the center and radius off the axes.

Trap: Sign errors on the center. (x − 3)² means h = 3, but (x + 3)² means h = −3. And the right side is r², so r = √(right side), not the right side itself.

Arc length and sector area

$s = r\theta \quad A = \tfrac{1}{2}r^2\theta$

A circle with a central angle, asking for arc length or the area of the slice.

In Desmos

r * theta (theta in radians)
0.5 * r^2 * theta

Degrees → radians: multiply by π/180.

Trap: Using degrees in a radian formula. s = rθ is only true in radians; in degrees it's (θ/360) × 2πr.

Right triangle trigonometryMust Know

$\sin\theta = \frac{\text{opp}}{\text{hyp}}, \; \cos\theta = \frac{\text{adj}}{\text{hyp}}, \; \tan\theta = \frac{\text{opp}}{\text{adj}}$

A right triangle with one angle and one side, asking for another side.

In Desmos

sin(30 deg)
arcsin(0.5)

Type 'deg' after the number, or Desmos assumes radians.

Trap: Leaving Desmos in radians. sin(30) in radians is −0.988, not 0.5 — and the resulting wrong answer is always a listed choice.

Similar triangles and proportions

$\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$

Two triangles sharing angles, or a triangle cut by a parallel line, asking for a missing length.

In Desmos

solve the proportion: a/b = c/x

Trap: Pairing sides that don't correspond. Match sides by the angles they sit opposite, not by their position in the picture.

What happens when a dimension changesHigh Frequency

$\text{area} \propto k^2 \quad \text{volume} \propto k^3$

A radius or side is doubled or tripled, asking what happens to area or volume.

In Desmos

(2)^2
(2)^3

Trap: Scaling volume linearly. Doubling the radius multiplies volume by 8, not 2 — and 2 is always offered.