Reference sheet

Desmos cheat sheet

24 tested patterns, the exact Desmos input for each one, and the specific mistake every wrong answer is engineered around.

Know these cold

None of this is printed for you on test day. Everything below it — the question shapes — is built out of these.

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: $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

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: $r_1 + r_2 = -\dfrac{b}{a}, \quad r_1 r_2 = \dfrac{c}{a}$

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.

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.

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.

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.

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.

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.

These come printed in the exam. The point of listing them is knowing you can stop revising them.

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: $360^\circ \text{ in a circle}, \quad 180^\circ \text{ in a triangle}$

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

5 patterns

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

6 patterns

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

7 patterns

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