AP PreCalc

Describe symmetry for even and odd functions

Even- symmetry to the y-axis

Odd- symmetry to the origin

Y-intercepts when domain is all real numbers

Input x=0 and solve

y=f(x-h)+k

-h is horizontal right, k is vertical up

Composite functions

f(g(x)), solve g(x) then f(x)

y-coordinate at x=5

Input x=5 and solve for y

X-intercepts when domain is all real numbers

Input y=0 and solve for x

Domain of a function

Exclude inputs that cause 0 in denominator or negative radicands

BOBO, BOTN, EATS D/C

Defines horizontal asymptotes

BOBO (Big on Bottom) =0

BOTN (Big on Top) =None

EATS D/C (Exponents are the same) =Divide Coefficients

When does a rational function have a hole

Hole if you cancel a factor from numerator and denominator

Relationship between function’s degree and the # of zeroes

Tells the maximum number of zeros (complex/ both real and imaginary)

Relationship between domains & ranges of inverse functions

Domain of f = Range of f^-1

Range of f = Domain of f^-1

Logarithmic to Exponential

log₂x=2

2²=x

Change of base formula for log

log₃12=log12/log3

(If no base is shown, it is base 10)

Unit circle (degrees and radians)

Which trig functions are even and which are odd

Even: cos, sec

Odd: sin, csc, tan, cot

With SOH CAH TOA, sin and tan are 1 and 3, cos is 2 (even), same with inverses

Number of x-ints in sin and cos graph

Sin and cos have infinitely many intercepts

Relationship between sin and cos graphs

cos is a phase-shifted sin graph, left pi/2 radians

Amplitude and period of y=asin(b(x-c))+d

amplitude: absolute value of a

period: 2pi/absolute value of b

Vertical asymptotes of y=tanx, y=cotx, y=secx, and y=cscx

Find where the function is undefined and when the denominator =0

(tanx)(cosx)

(sinx)

Sum and difference formulas for sin and cos

sin(u±v)= sinu cosv ± cosu sinv

cos(u±v)= cosu cosv ∓ sinu sinv

Forms of cos(2u)

1. cos²u-sin²u

2. 2cos²u-1

3. 1-2sin²u

Which shape has interior angles with a sum of 360

Quadrilaterals

How can a triangle be solved (methods)

Law of sines

Law of cosines

Methods to finding triangle area

A=1/2bh

A=1/2absinC

Number of polar descriptions for a point

Infinitely many ways

Number of rectangular descriptions for a point

One way

Values of r and ∅ in polar pairs

r: directed distance from origin

∅: directed angle from polar axis

Polar graphs passing through the pole

Limacons with inner loops

Roses

Polar to rectangular / rectangular to polar form

x=rcos∅

y=rsin∅

r²=x²+y²

tan∅=y/x

Identifying coterminal angles

add or subtract multiples of 360 degrees or 2pi

Techniques for evaluating limits

direct substitution

factoring and canceling

rationalizing the numerator/denominator (conjugating)

Domain of a log function

Set the argument >0 and solve

Marine motto

Semper fidelis

always faithful

Solutions of a linear system

One, none, or infinitely many solutions

three indeterminate forms

0/0

∞/∞

0∞

Solving an absolute value equation

Isolate the express and set the argument to ± the constant

|x-3|=5 turns into x-3= ±5

Two uses of the horizontal line test

1. Determines if function is one-to-one

2. Checks if inverse is a function

Relationships of exponential and log limits

They mirror across line y=x

Vertical asymptotes convert to horizontal asymptotes

3 rules of logs

Product rule: log(uv)=log u + log v

Quotient rule: log (u/v)=log u - log v

Power rule: log (uⁿ)= n log u

Term for successive differences in arithmetic sequences

common difference

like 3,7,11,15 all have a difference of 4

Calculating AROC

(y2-y1)/(x2-x1)

Requirements for continuity at x=b

f(b) is defined

lim as x approaches b of f(x) exists

lim as x approaches b of f(x) = f(b)

Define a limit

the behavior of a function as we approach a certain input value

How are end behaviors determined

Check if the leading coefficient is negative or positive

Check if the highest exponent is odd or even

Limits at infinity and horizontal aymptotes

the lim as x approaches ± ∞ of f(x)=L, and the horizontal asymptote is y=L

Finding horizontal asymptotes of rational functions

Compare degrees of top and bottom

Use the bobo, botn, eats d/c rules

Writing the limit equation

lim as x approaches 3 of x²=9

Write two limit expressions (one left-hand limit and one right-hand limit)

Left: lim as x approaches c- of f(x)

Right: lim as x approached c+ of f(x)

Why do some limits not exist

Left and right limits do not match

The function goes infinitely

The function grows without bound (±∞)