precalc
Unit 1: Functions
What is a function?
a rule describing the dependence of one quantity, A, on another, B
A function f is a rule that assigns to each element x in a set A exactly one element called f(x) in set B
f(input) = output
Each input has one and only one output
Set of all inputs = domain
Set of all outputs = range
Interval notation: denotes start and end numbers of interval
“Hard” [ ] include end values
“Soft” ( ) exclude end values
Set Builder notation: describes a set by saying what property its members have
{ x| x >0 } “The set of all x such that x is greater than 0”
Finding the domain of a function: unless explicitly give, algebraically find the set of all real numbers that defines the expression as a real number, use two types of domain restriction
Denominator restriction: denominator cannot equal zero
Radicand restriction: radicand must be greater than or equal to zero
Finding the range: rearrange f(x) to g(y) and then look for set of y
Solve for x- find the inverse
Find domain of inverse relation → this is the range of the original f(x)
Check: does this make sense?
Greatest Integer Function: f(x) = [[x]] → “greatest integer less than or equal to x” “step function”
Notating increasing/decreasing intervals: use interval notations, read graph from left to right, use inclusive brackets for endpoints
Average rate of change: ARC between x=a and x=b is ARC=f(b)-f(a)b-a
Even function: symmetry over y-axis f(-x) = f(x)
Odd function: symmetry about origin f(-x)= -f(x)
Transformations
y= af(b(xc))d
Translation +c left -c right
Translation + d up -d down
Vertical |a|>1 stretch 0<|a|<1 shrink
Horizontal 0<|b|<1 stretch |b|>1 shrink
Reflection 0>a over x- axis 0>b over y-axis
One-to-one: only one x-value for every y-value . must pass vertical and horizontal line tests
Original D → new R, Original R → new D
Cancellation property: if f-1(x) is the inverse of f(x), then:
f(f-1(x))= x AND f-1(f(x)) = x
NEED TO CHECK BOTH
Involution: function equal to its inverse
Composition of Functions
Algebra of Functions:
Sum: Df+g= DfDg
Difference Df-g= DfDg
Product: Dfg= DfDg
Quotient: Df/g= DfDg where g(x) 0
Composition: the new function has a domain of D and a range of R
Units 2-4: Trigonometry
sin cos tan | |||
Even-Odd Properties
sin(-t) = -sin(t) odd
csc(-t) = -csc(t) odd
cos(-t) = cos(t) even
sec(-t) = sec(t) even
tan(-t) = tan(t) odd
cot(-t) = cot(t) odd
Reciprocal Identities
Pythagorean Identities
Proving Trig Identities:
Goal: transform one side into the other
Start with one side
Use algebra and known identities
Rewrite in terms of sin and cos if stuck
tips: each step must be reversible! Conjugate method! Can work with both sides separately
Area of a Non-Right Triangle: A = 12absin
Law of Sines
In any triangle the lengths of the sides are proportional to the sines of corresponding angles
sinAa= sinBb=sinCc
Ambiguous cases → test for both triangles
Law of Cosines
Works for SAS and SSS cases
a2=b2+c2-2bccosA
Cofunction Identities
Evaluating Functions
Simplify using even/odd properties
Write the angle as 90° n
If n is even, keep same trig fx
If n is odd, take “opposite” trig fx
Sign is decided by original angle in the quadrant
Angle Addition and Subtraction Formulas
Pay attention to signs!
Double Angle formulas
Can be derived from addition formulas if forget
Half-Angle Formulas
Sign depends on quadrant
Bro shut up
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Product to Sum and Sum to Product
Sine/Cosine Graphs
y= asin(bx) + d
Midline → d
Period → 2/b horizontal |b|<1 stretch |b|>1 shrink
Amplitude → |a| vertical |a|>1 stretch |a|>1 shrink
Graphing tan + cot functions
Tan + cot function shave output values that repeat after every period of
y=atanb(x-c)
[-2,2]
As x → 2, sinx → 1 cosx → 0 tanx → +
As x → -2, sinx → -1 cosx→ 0 tanx→ -
As x→ 0, sinx→ 0 cosx → 1 tanx → 0
y=acotb(x-c)
[0,]
As x → 0, sinx → 0 cosx → 1 cotx→ +
As x → 2, sinx → 1 cosx→ 0 cotx →0
As x→ , sinx→ 0 cosx → -1 cotx→ -
/b= period
a= vertical stretch ( but does not change asymptotes)
c= phase shift
Reciprocal Functions
Have period of 2
y=acscb(x-c) and y=asecb(x-c)
a= vertical
2/b = horizontal stretch/shrink
c= phase shift
Easier to graph sin/cos first
Simple Harmonic Motion
Modeling real life movement
Amplitude = |a| → max displacement
Period = 2/ → time required for one cycle
Frequency = /2 → number of cycles per unit of time
Damped Harmonic Motion
In real life, things slow down. Amplitude decreases exponentially, period doesn’t change
y=ke-ctsin(t)
k= initial amplitude
2/= period
c=dampening constant
As k increases, initial amp increases
As c increases, function dies down faster
Trig Inverses
Must limit domain and range for it to work
Composition of Trig Inverses
Combining sine + cosine functions
asinx+bcosx=a2+b2sin(x+) where satisfies cos= aa2+b2and sin=ba2+b2
Unit 5: Polar Coordinates & Complex Numbers
Rectangular → polar
r2=x2+y2 tan=yx
Polar → rectangular
x=rcos y=rsin
Polar Equations (from rectangular)
What does it look like on rectangular?
Express in polar coordinates
What does it look like on polar?
**tip: express the polar equation(manipulate) so that it has rcos or rsin
r=a circle of radius |a| centered at origin
=b (in radians) line with a slope of tanb
r=2asin circle with radius a centered at (a,2) or (a,0)
Cardioids and Limacons
Hi r u mad.
Roses
r=asin(n) and r=acos(n) when n=odd, n petals, when n=even, 2n petals
Lemniscates
r2=a2cos and r2=a2sin
Tests of Symmetry
Graphing sets of complex numbers
a+bi = complex number
moduli= absolute value of complex number z, |z|= a2+b2= distance away from the radius
Converting complex numbers to polar form
z=a+bi has the polar form z=r(cos+isin) where r=|z| and tan=ba
Demoivre’s Theorem:
If z=r(cos+isin), then zn=rn(cosn+isinn)
Multiply and Divide Complex Numbers
z1=r1(cos1+isin1) and z2=r2(cos2+isin2)
z1z2=r1r2(cos(1+2)+isin(1+2))
z1/z2=r1/r2(cos(1-2)+isin(1-2))
Nth root of a complex number
All solutions have same modulus r → circle and nth roots are equally spaced apart points on the circle = UNITY OF ROOTS