📋 Exam 4 — Formula Sheet

All key formulas from 10 sections at a glance.

§sys.eq

Substitution Method

\begin{aligned} &1.\;\text{Solve one equation for } y \\ &2.\;\text{Substitute into the other equation} \\ &3.\;\text{Solve for } x \\ &4.\;\text{Back-substitute to find } y \end{aligned}

Best when one equation is already solved for a variable, or a coefficient is $1$ or $-1$

Elimination Method

\text{Multiply to match coefficients} \;\Longrightarrow\; \text{add/subtract equations} \;\Longrightarrow\; \text{solve} \;\Longrightarrow\; \text{back-substitute}

Best for linear systems when coefficients line up or nearly line up

Number of Solutions

\text{Line–Line: } 0, 1, \text{ or } \infty \qquad \text{Line–Conic: } 0, 1, \text{ or } 2 \qquad \text{Conic–Conic: } 0, 1, 2, 3, \text{ or } 4

A line can be tangent to a conic (1 solution) or miss it entirely (0 solutions)

§circles

Circles

Standard form of a circle

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

center $(h, k)$ , radius $r$

General form to standard form

x^2 + y^2 + Dx + Ey + F = 0 \;\Longrightarrow\; \left(x + \tfrac{D}{2}\right)^2 + \left(y + \tfrac{E}{2}\right)^2 = \tfrac{D^2 + E^2 - 4F}{4}

center $\left(-\tfrac{D}{2},\, -\tfrac{E}{2}\right)$ , radius $\tfrac{\sqrt{D^2 + E^2 - 4F}}{2}$

Distance formula (for finding radius)

r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2}

where $(x_1, y_1)$ is any point on the circle

§10.1

The Ellipse

Standard Form (Horizontal Major Axis)

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, \quad a > b

Center (h, k), vertices at (h ± a, k), co-vertices at (h, k ± b)

Standard Form (Vertical Major Axis)

\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1, \quad a > b

Center (h, k), vertices at (h, k ± a), co-vertices at (h ± b, k)

Foci Distance

c^2 = a^2 - b^2

c = distance from center to each focus; foci lie on the major axis

Eccentricity

e = \frac{c}{a}, \quad 0 < e < 1

e near 0 → nearly circular; e near 1 → elongated

§10.2

The Hyperbola

Standard Form (Horizontal Transverse Axis)

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

Center (h, k), vertices at (h ± a, k), branches open left and right

Standard Form (Vertical Transverse Axis)

\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1

Center (h, k), vertices at (h, k ± a), branches open up and down

Asymptotes

\text{Horizontal: } y - k = \pm\frac{b}{a}(x - h) \\[4pt] \text{Vertical: } y - k = \pm\frac{a}{b}(x - h)

Slopes depend on which axis is the transverse axis — don't mix them up!

Foci Distance

c^2 = a^2 + b^2

PLUS, not minus — opposite of ellipses. Foci lie on the transverse axis, beyond the vertices.

Eccentricity

e = \frac{c}{a}, \quad e > 1

e near 1 → narrow branches; larger e → wider, more open branches

§11.1

Sequences & Notation

Explicit Formula

a_n = f(n)

Gives the nth term directly as a function of n — no previous terms needed

Recursive Formula

a_n = f(a_{n-1}), \quad a_1 = \text{(given)}

Each term depends on the previous term; you must know the starting value

Factorial

n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1, \qquad 0! = 1

Grows extremely fast; 0! = 1 by definition

§11.2

Arithmetic Sequences

Explicit (nth term) formula

a_n = a_1 + (n - 1)d

Jumps directly to the $n$ th term without computing every term before it.

Recursive formula

a_n = a_{n-1} + d

Defines each term from the previous one; you also need the first term $a_1$ .

Sum of an arithmetic series (version 1)

S_n = \frac{n}{2}(a_1 + a_n)

Use when you know both the first and last term.

Sum of an arithmetic series (version 2)

S_n = \frac{n}{2}\bigl(2a_1 + (n - 1)d\bigr)

Use when you know $a_1$ , $d$ , and $n$ but not $a_n$ .

§11.3

Geometric Sequences

Explicit (general) term

a_n = a_1 \cdot r^{n-1}

jumps directly to the $n$ th term; $a_1$ is the first term, $r$ is the common ratio

Recursive definition

a_n = r \cdot a_{n-1}, \quad a_1 \text{ given}

each term equals the previous term times $r$

Finite geometric series

S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1

sum of the first $n$ terms

Infinite geometric series

S = \frac{a_1}{1 - r}, \quad |r| < 1

only converges when $|r| < 1$ ; otherwise the series diverges

§9.3

Summation Notation

Sigma Notation

\sum_{i=m}^{n} a_i = a_m + a_{m+1} + \cdots + a_n

i is the index, m is the start, n is the end, aᵢ is the expression

Sum of a Constant

\sum_{i=1}^{n} c = c \cdot n

Adding the same number n times

Sum of First n Positive Integers

\sum_{i=1}^{n} i = \frac{n(n+1)}{2}

The classic Gauss formula: 1 + 2 + 3 + ⋯ + n

Sum of First n Squares

\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}

Linearity of Summation

\sum_{i=1}^{n}(a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i, \qquad \sum_{i=1}^{n} c \cdot a_i = c \cdot \sum_{i=1}^{n} a_i

Split sums apart and pull constants out — just like distributing

§11.4

Series & Their Notations

Arithmetic series (partial sum)

S_n = \frac{n(a_1 + a_n)}{2}

Average of first and last term, times the number of terms.

Geometric series (partial sum)

S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1

Multiply by $r$ , subtract, and almost everything cancels.

Infinite geometric series

S_\infty = \frac{a_1}{1 - r}, \quad |r| < 1

Only converges when $|r| < 1$ ; diverges otherwise.

Sigma notation

\sum_{k=m}^{n} a_k = a_m + a_{m+1} + \cdots + a_n

§11.6

Binomial Theorem

Binomial Coefficient

\binom{n}{k} = \frac{n!}{k!\,(n-k)!}

Also written $C(n,k)$ or ${}_nC_k$ . Counts the ways to choose $k$ items from $n$ .

Binomial Theorem

(a + b)^n = \sum_{k=0}^{n} \binom{n}{k}\, a^{n-k}\, b^{k}

Specific Term Formula

\text{The } (k+1)\text{th term} = \binom{n}{k}\, a^{n-k}\, b^{k}

Use $k = 0$ for the 1st term, $k = 1$ for the 2nd, etc.