Download PreCalculus Review Packet

Precalculus Review
Functions to KNOW!
1. Polynomial Functions
Types:
General form
and unique properties
Generic Graph
Constants
Linear
Quadratic
Cubic
Generalizations for Polynomial Functions
- The domain for all polynomial functions is ALWAYS _______________
- The degree of a polynomial is ________________________________
- Graph behavior based on its degree:
End behavior:
Number of ‘turning points’
A function is said to be even if
A function is said to be odd if
2. Rational Functions (a.k.a. Fractional Functions)
- General form:
- Domain:
- Unique Graph attributes:
Vertical asymptotes:
Horizontal or Slanted Asymptotes:
1. degree numerator < degree denominator
2. degree numerator = degree denominator
3. degree numerator > degree denominator
Ex) For the function y =
x +3
determine the following:
x −5
Domain:
Vertical Asymptote
Horizontal or Slanted Asymptote
Also helps to plot the intercepts:
x-intercept
y-intercept
Ex) Determine the asymptotes (vertical, horizontal/slant) for …
x2 − 6x + 7
x
a) f (x) = 2
b) g(x) =
x −9
x −5
3. Inverse Functions
In order for a function f (x) to have an inverse, it must be ______________________
… which means
The inverse of the function f (x) is denoted as ________________
Ex) f (x) = x 3
Ex) h(x) = x 2
The purpose of an inverse function is to ______________________________________
Properties of inverse functions:
- Domain and Range:
- Graph symmetry
Ex) Sketch the graph of the inverse of the function f (x) on the blank axes.
Domain:
Domain:
Range:
Range:
4. Exponential and Logarithmic Functions
Exponential functions of base a and Logarithmic functions of base a are inverses of each other.
General Exp. Function: y = a x
General Log. Function: y = log a ( x)
Domain:
Domain:
Range:
Range:
Intercept:
Intercept:
Asymptote:
Asymptote:
Graph:
Graph:
Most frequently used base is ______
Approximate value:
… whose log inverse is ______________
Properties of Exponentials and Logarithms you’ll need in calculus:
Rewrite between exponential form and logarithmic form:
a x = b can be rewritten as _______________
__________ can be rewritten as ln(b) = x
Cancellation Properties (VERY handy when solving equations)
Solving equations:
Solve the equation 103 x−1 = 45
Solve the equation 6ln(15 − 7x) + 20 = 38
Base Change Formula
The Laws of Logarithms
These are handy when you need to expand or condense logarithmic expressions.
I. ln(UV ) =
II. ln( UV ) =
III. ln(UM ) =
5. Trigonometric Functions
Trigonometric functions were defined in several ways:
-Right Triangle Definitions:
The main three …
sinθ =
cosθ =
tanθ =
REMEMBER:
The roles of ‘OPPOSITE’ and ‘ADJACENT’ depend
on which acute angle you’re calling θ .
- Unit Circle Definitions:
Let t be a radian angle measure and ( x , y) represents
the point on the unit circle paired with the angle t
sin(t ) =
cos(t ) =
tan(t ) =
and their reciprocals
FOR REFERENCE ONLY!!! THIS IS PREREQUISITE MATERIAL!!!
YOU WILL NOT BE ALLOWED TO USE THIS ON THE TEST!!!
You’ll need the unit circle for various reasons this semester:
Ex) Evaluate csc( 74π )
Ex) Solve the equation 3tan2 (x) = 1 on the interval [0,2π ) .
Ex) What interval (or intervals) make 2cos t + 1 ≤ 0 on the interval [0,2π ) ?
- Trigonometric Function Graphs
y = sin(x)
y = cos(x)
Domain of Sine and Cosine: ____________
Range of Sine and Cosine: _____________
Graph is periodic with a cycle repeating every interval of length ______________.
Ex) Sketch two full periods of the graph of y = −8cos(10 x) .
How does the –8 affect the graph?
How does the 10 affect the graph?
The other trigonometric function graphs for reference
(again … prerequisite material won’t be given on the test)
Graph of y = csc(x)
Graph of y = sec(x)
Graph of y = tan( x)
Graph of y = cot(x)
y = tan x
y = cot x
Cosecant and Cotangent have vertical asymptotes at every multiple of π (where Sine has xintercepts)
Secant and Tangent have vertical asymptotes at every ODD multiple of π2 (where Cosine has xintercepts)
Trigonometric Identities we will need in Calculus (KNOW THEM!)
Reciprocal Identities
1
sin u =
csc u
1
csc u =
sin u
1
cos u =
sec u
1
sec u =
cos u
1
tan u =
cot u
1
cot u =
tan u
Quotient Identities
sin u
cos u
cos u
cot u =
sin u
tan u =
Pythagorean Identities
Even / Odd Identities
ODDS
sin 2 u + cos2 u = 1
also … cos2 u = 1 − sin 2 u and sin 2 u = 1 − cos2 u
sin( − u) = − sin u
csc( − u) = − csc u
tan( − u) = − tan u
cot( − u) = − cot u
1 + tan 2 u = sec 2 u
also … tan 2 u = sec2 u − 1 and 1 = sec 2 u − tan 2 u
EVENS
2
2
1 + cot u = csc u
also … cot u = csc 2 u − 1 and 1 = csc 2 u − cot 2 u
2
cos( − u) = cos u
sec( − u) = sec u