SAT Math · formula sheet

Digital SAT Math: formulas and Desmos

The algebra, geometry and trigonometry formulas the test expects you to know, and four things worth being able to do in the built-in Desmos calculator.

21 formulas and facts in 6 sections

A 30-60-90 triangle with sides 1, root 3 and 2, and a 45-45-90 triangle with sides 1, 1 and root 2. — The two special right triangles and their side ratios.

Lines

Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$
Slope-intercept and point-slope form: $y = mx + b \qquad y - y_1 = m(x - x_1)$
Parallel and perpendicular lines: $m_1 = m_2 \qquad m_1 m_2 = -1$

Quadratics

Quadratic formula: $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$
Discriminant: $D = b^{2} - 4ac$
D > 0: two real solutions. D = 0: one. D < 0: none.
Vertex: $x = -\frac{b}{2a} \qquad y = a(x-h)^{2} + k \text{ has vertex } (h, k)$
Sum and product of the solutions: $x_1 + x_2 = -\frac{b}{a} \qquad x_1 x_2 = \frac{c}{a}$

Exponents, growth and percentages

Exponential growth or decay: $y = a(1 + r)^{t}$
r is the rate per period as a decimal; negative for decay.
Percent change: $\frac{\text{new} - \text{old}}{\text{old}} \times 100\%$
Exponent rules: $a^{m}a^{n} = a^{m+n} \qquad (a^{m})^{n} = a^{mn} \qquad a^{-n} = \frac{1}{a^{n}} \qquad a^{1/n} = \sqrt[n]{a}$

Circles

Equation of a circle: $(x-h)^{2} + (y-k)^{2} = r^{2}$
Centre (h, k), radius r. Complete the square to reach this form.
Arc length and sector area (angle in degrees): $\text{arc} = \frac{\theta}{360}\cdot 2\pi r \qquad \text{sector} = \frac{\theta}{360}\cdot \pi r^{2}$
Degrees and radians: $\pi \text{ radians} = 180^{\circ}$

Triangles and trigonometry

Pythagorean theorem: $a^{2} + b^{2} = c^{2}$
Special right triangles: $45^{\circ}\text{-}45^{\circ}\text{-}90^{\circ}:\; x,\; x,\; x\sqrt{2} \qquad 30^{\circ}\text{-}60^{\circ}\text{-}90^{\circ}:\; x,\; x\sqrt{3},\; 2x$
Sine, cosine, tangent: $\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}$
Complementary angles: $\sin x^{\circ} = \cos(90^{\circ} - x^{\circ})$

In the Desmos calculator

Solve a system: Type both equations. Click the point where the graphs cross to read the solution.
Find solutions and the vertex: Graph the expression as y = …; click the x-intercepts for the solutions and the turning point for the vertex.
Line of best fit from a table: $y_1 \sim m x_1 + b$
Enter the data in a table (columns x₁ and y₁), then type this with the ~ key. Desmos reports m and b.
Quadratic fit from a table: $y_1 \sim a x_1^{2} + b x_1 + c$

A reference for revision. In an exam, use the formula sheet your exam board provides and the notation your course uses.