Precalc
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📋 Exam 2 Review — Formula Sheet

All key formulas from 9 sections at a glance.

§3.7

Rational Functions
General Form
f(x)=p(x)q(x)f(x) = \frac{p(x)}{q(x)}

where p(x) and q(x) are polynomials and q(x) ≠ 0

Vertical Asymptotes
Set q(x)=0 (after canceling common factors)\text{Set } q(x) = 0 \text{ (after canceling common factors)}

The x-values where the denominator is zero (and doesn't cancel) give vertical asymptotes

Horizontal Asymptote (degrees equal)
y=anbmy = \frac{a_n}{b_m}

When deg(p) = deg(q), the HA is the ratio of leading coefficients

Horizontal Asymptote (numerator smaller)
y=0y = 0

When deg(p) < deg(q)

Hole Location
If (x−c) cancels from both p(x) and q(x), hole at x=c\text{If } (x - c) \text{ cancels from both } p(x) \text{ and } q(x), \text{ hole at } x = c

§sz.ineq

Polynomial & Rational Inequalities
Sign Chart Method
1.  Move everything to one side2.  Factor completely3.  Find critical points4.  Test intervals\begin{aligned} &1.\;\text{Move everything to one side} \\ &2.\;\text{Factor completely} \\ &3.\;\text{Find critical points} \\ &4.\;\text{Test intervals} \end{aligned}

Critical points = zeros of numerator + zeros of denominator

Polynomial Inequality Setup
p(x)>0   or   p(x)<0   or   p(x)≥0   or   p(x)≤0p(x) > 0 \;\text{ or }\; p(x) < 0 \;\text{ or }\; p(x) \geq 0 \;\text{ or }\; p(x) \leq 0

Always move everything to one side first so you're comparing to 0

Rational Inequality Setup
p(x)q(x)≤0  ⟹  critical points at p(x)=0 and q(x)=0\frac{p(x)}{q(x)} \leq 0 \implies \text{critical points at } p(x)=0 \text{ and } q(x)=0

Never multiply both sides by the denominator — you don't know its sign

Interval Notation Reminders
[a,b] includes endpoints,(a,b) excludes endpoints[a, b] \text{ includes endpoints}, \quad (a, b) \text{ excludes endpoints}

Use ( ) at ±∞ and at values where the function is undefined

Sign Change Rule
Sign changes at single roots; sign stays the same at double (even) roots\text{Sign changes at single roots; sign stays the same at double (even) roots}

A factor like (x - 2)² touches zero but doesn't change sign

§4.1

Exponential Functions
Exponential Function (General)
f(x)=bx,b>0,  b≠1f(x) = b^x, \quad b > 0, \; b \neq 1

b > 1 means growth; 0 < b < 1 means decay

Natural Exponential Function
f(x)=ex,e≈2.71828f(x) = e^x, \quad e \approx 2.71828

The most important base — it makes calculus cleanest

Compound Interest
A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}

P = principal, r = annual rate (decimal), n = compounds per year, t = years

Continuous Compounding
A=PertA = Pe^{rt}

The limit of compound interest as n → ∞

Key Properties of b^x
b0=1,bx>0 for all x,HA: y=0b^0 = 1, \quad b^x > 0 \text{ for all } x, \quad \text{HA: } y = 0

§4.2

Graphs of Exponential Functions
Parent exponential function
f(x)=bxf(x) = b^x

Passes through (0, 1) with horizontal asymptote y = 0

Vertical shift
f(x)=bx+kf(x) = b^x + k

Moves the asymptote to y = k

Horizontal shift
f(x)=bx−hf(x) = b^{x - h}

Shifts the graph h units to the right

Reflection over the x-axis
f(x)=−bxf(x) = -b^x

Flips the graph upside down; range becomes (-∞, k)

General transformed exponential
f(x)=a⋅bx−h+kf(x) = a \cdot b^{x - h} + k

a = vertical stretch/reflect, h = horizontal shift, k = vertical shift

§4.3

Logarithmic Functions
Definition of logarithm
log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = x

b > 0, b ≠ 1, and x > 0

Common logarithm
log⁡(x)=log⁡10(x)\log(x) = \log_{10}(x)
Natural logarithm
ln⁡(x)=log⁡e(x)\ln(x) = \log_e(x)

e ≈ 2.718

Inverse relationships
log⁡b(bx)=xandblog⁡b(x)=x\log_b(b^x) = x \quad \text{and} \quad b^{\log_b(x)} = x

Logs and exponentials cancel each other

Change-of-base formula
log⁡b(x)=ln⁡(x)ln⁡(b)=log⁡(x)log⁡(b)\log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log(x)}{\log(b)}

Use this to evaluate any log on a calculator

§4.4

Graphs of Logarithmic Functions
Parent logarithmic function
f(x)=log⁡b(x)f(x) = \log_b(x)

Passes through (1, 0); vertical asymptote at x = 0

Horizontal shift
f(x)=log⁡b(x−h)f(x) = \log_b(x - h)

Asymptote moves to x = h; domain is (h, ∞)

Vertical shift
f(x)=log⁡b(x)+kf(x) = \log_b(x) + k

Shifts graph up/down; asymptote and domain unchanged

General transformed log
f(x)=a⋅log⁡b(x−h)+kf(x) = a \cdot \log_b(x - h) + k

a = vertical stretch/reflect, h = horizontal shift, k = vertical shift

Exponential-log reflection relationship
y=bx  ⟷  y=log⁡b(x) reflected over y=xy = b^x \;\longleftrightarrow\; y = \log_b(x) \text{ reflected over } y = x

§4.5

Logarithmic Properties
Product Rule
log⁡b(MN)=log⁡b(M)+log⁡b(N)\log_b(MN) = \log_b(M) + \log_b(N)

Multiplication inside becomes addition outside

Quotient Rule
log⁡b ⁣(MN)=log⁡b(M)−log⁡b(N)\log_b\!\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)

Division inside becomes subtraction outside

Power Rule
log⁡b(Mn)=n⋅log⁡b(M)\log_b(M^n) = n \cdot \log_b(M)

Exponent inside becomes coefficient outside

Change of Base Formula
log⁡b(M)=ln⁡Mln⁡b=log⁡Mlog⁡b\log_b(M) = \frac{\ln M}{\ln b} = \frac{\log M}{\log b}
Log of 1 and Log of the Base
log⁡b(1)=0andlog⁡b(b)=1\log_b(1) = 0 \quad\text{and}\quad \log_b(b) = 1

Useful for simplifying terms quickly

§4.6

Exponential & Logarithmic Equations
Same-Base Strategy
bf(x)=bg(x)  ⟹  f(x)=g(x)b^{f(x)} = b^{g(x)} \implies f(x) = g(x)

Only works when both sides share the same base

Take-Log-of-Both-Sides Strategy
ax=c  ⟹  x=ln⁡cln⁡aa^x = c \implies x = \frac{\ln c}{\ln a}

Use when bases cannot be matched

Log-to-Exponential Conversion
log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = x

The key move for solving logarithmic equations

One-to-One Property of Logarithms
log⁡b(M)=log⁡b(N)  ⟹  M=N\log_b(M) = \log_b(N) \implies M = N

If two logs with the same base are equal, their arguments are equal

Extraneous Solution Check
Domain requirement: arguments of all logs must be >0\text{Domain requirement: arguments of all logs must be } > 0

Always verify solutions in the original equation

§4.7

Exponential & Logarithmic Models
Exponential Growth/Decay Model
N(t)=N0 ektN(t) = N_0 \, e^{kt}

k > 0 for growth, k < 0 for decay

Doubling Time
tdouble=ln⁡2kt_{\text{double}} = \frac{\ln 2}{k}

Time for a quantity to double (assumes k > 0)

Half-Life
t1/2=ln⁡2∣k∣t_{1/2} = \frac{\ln 2}{|k|}

Time for a quantity to halve (assumes k < 0)

Newton's Law of Cooling
T(t)=Ts+(T0−Ts) ektT(t) = T_s + (T_0 - T_s)\,e^{kt}

T_s = surrounding temp, T_0 = initial temp, k < 0

Logistic Growth Model
P(t)=c1+a e−btP(t) = \frac{c}{1 + a\,e^{-bt}}

c = carrying capacity; growth slows as P approaches c