ACT and Algebra Math Formulas

A collection of 50 essential math formulas and concepts for Algebra, Geometry, Trigonometry, and Statistics based on lecture notes for exam preparation.

Slope (m)

$m = \frac{y_2 - y_1}{x_2 - x_1}$ used for lines, rate of change, and parallel or perpendicular lines.

Slope-Intercept Form

$y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

Point-Slope Form

$$y - y_1 = m(x - x_1)$$

Distance Formula

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

Midpoint Formula

$$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for the equation $ax^2 + bx + c = 0$

Vertex Formula (x-coordinate)

$x = \frac{-b}{2a}$ then plug this value into the equation to find the y-coordinate.

Multiplication Exponent Rule

$$a^m \cdot a^n = a^{m+n}$$

Division Exponent Rule

$$\frac{a^m}{a^n} = a^{m-n}$$

Power Exponent Rule

$$(a^m)^n = a^{mn}$$

Negative Exponent Rule

$$a^{-n} = \frac{1}{a^n}$$

Zero Exponent Rule

$$a^0 = 1$$

Difference of Squares

$$a^2 - b^2 = (a+b)(a-b)$$

Perfect Square Trinomial (Sum)

$$(a + b)^2 = a^2 + 2ab + b^2$$

Perfect Square Trinomial (Difference)

$$(a - b)^2 = a^2 - 2ab + b^2$$

Area of a Rectangle

$$A = lw$$

Area of a Triangle

$$A = \frac{1}{2}bh$$

Area of a Parallelogram

$$A = bh$$

Area of a Trapezoid

$$A = \frac{1}{2}(b_1 + b_2)h$$

Area of a Circle

$$A = \pi r^2$$

Circumference of a Circle

$C = 2\pi r$ or $C = \pi d$

Arc Length

$$\frac{\theta}{360}(2\pi r)$$

Sector Area

$$\frac{\theta}{360} \pi r^2$$

Pythagorean Theorem

$$a^2 + b^2 = c^2$$

Volume of a Rectangular Prism

$$V = lwh$$

Volume of a Cylinder

$$V = \pi r^2 h$$

Volume of a Cone

$$V = \frac{1}{3} \pi r^2 h$$

Volume of a Pyramid

$V = \frac{1}{3}Bh$ where $B$ is the area of the base.

Volume of a Sphere

$$V = \frac{4}{3} \pi r^3$$

Surface Area of a Cylinder

$$SA = 2\pi r^2 + 2\pi rh$$

Surface Area of a Sphere

$$SA = 4\pi r^2$$

45-45-90 Special Triangle Ratio

$x$, $x$, $x\sqrt{2}$

30-60-90 Special Triangle Ratio

$x$, $x\sqrt{3}$, $2x$

Sine (sin)

$$\sin = \frac{\text{opposite}}{\text{hypotenuse}}$$

Cosine (cos)

$$\cos = \frac{\text{adjacent}}{\text{hypotenuse}}$$

Tangent (tan)

$$\tan = \frac{\text{opposite}}{\text{adjacent}}$$

Pythagorean Identity

$$\sin^2(x) + \cos^2(x) = 1$$

Mean

$$\frac{\text{sum of values}}{\text{number of values}}$$

Probability (P)

$$P = \frac{\text{desired outcomes}}{\text{total outcomes}}$$

Complement Probability

$$P(\text{not A}) = 1 - P(A)$$

Counting Principle

$$\text{total choices} = a \times b \times c \dots$$

Permutations

$\frac{n!}{(n-r)!}$ used when order matters.

Combinations

$\frac{n!}{r!(n-r)!}$ used when order does not matter.

Simple Interest Formula

$I = Prt$ where $P$ is principal, $r$ is rate, and $t$ is time.

Compound Interest Formula

$$A = P(1 + \frac{r}{n})^{nt}$$

Percent Change

$$\frac{\text{change}}{\text{original}}$$

Arithmetic Sequence Formula

$$a_n = a_1 + (n - 1)d$$

Geometric Sequence Formula

$$a_n = a_1(r)^{n-1}$$

Discriminant

$b^2 - 4ac$; results indicate: positive (2 real solutions), zero (1 real solution), negative (no real solutions).

Log Definition

$\log_b(a) = c$ means $b^c = a$