Pre-Calculus A 2nd Hour, Fall 2012

Friday, November 16, 2012

Chapter 2 Final Exam Review- Rational Fucntions

RATIONAL FUNCTIONS:

A rational function can be written in the form:

f(x)=N(x) / D(x)

- where N(x) and D(x) are polynomials

-From rational functions you can find vertical asymptotes, horizontal asymptotes, and the intercepts. You can then graph the function using this information.

Finding vertical asymptotes:

The line x=0 is a vertical asymptote.
Vertical asymptotes occur at the zeros of D(x).

Example:

f(x)= 2x/ x-3

VA: Set the denominator equal to 0 to find the zeros at D(x).

x-3=0
x=3
Vertical asymptote: x=3 (dont forget to include the "x=" portion!)

Finding horizontal asymptotes:

The line y=0 is a horizontal asymptote.
Horizontal asymptotes can be found by observing the coefficents of the leading term of N(x) and D(x).

H.As exist in these conditions:

1- If the degree of the numerator is equal to the degree of the denominator
**Take ratio of leading coefficents to find the H.A.

2- If the degree of the denominator is greater than the degree of the numerator
***y=0 is the H.A.

If the degree of the numerator is greater than the denominator there is no H.A.

Finding x and y intercepts:

Find x intercepts by setting y to zero

Find y intercepts by setting x to zero

To graph:

Sketch the Vertical and Horizontal asymptotes and the x and y intercepts and sketch a corresponding graph. The graph should intersect at points of intersection and head off in the direction of positive or negative infinity in relation to the asymptotes.

(see below videos for examples)

Here are some helpful videos on rational functions and their graphs:

Rational functions in general-

Graphing rational functions:

Finding x and y intercepts:

Good luck studying!!

Chapter P Overview

Chapter P overview

P.1 Graphing Representation of Data

( x ,y )
x is the direct distance from y axis
y is the direct distance trom x axis

The Distance Formula

The Midpoint Formula

Standar form of the Equation of a circle

P.2 Graphs of Equation

then plug in points

P.3 Line in the Plane

if the dinaminator is 0 the slope is undifine
if the numerator is zero the slope is 0

point slope form:

P.4 Solving Equation Algebraically and Graphically

solve for:

solution:
The LCD is:

Multiply by the LCD and simplyfy

Point (a,0) is the x-intercept

Polynomial Equations

Degree: Name:

1st Linear

2nd Quadratic

3rd Cubic

4th Quartic

5th Quintic

Solving Inequality

Linear:

Polynomial:

Thursday, November 15, 2012

Graphs of Secant, Cosecant, Tangent, and Cotangent

Tangent Function: y = tan x

One cycle occurs between -p/2 and p/2
There are vertical asymptotes at each end of the cycle

The asymptote that occurs at p/2 repeats every p units

Period = p
Amplitude = none, graphs go on forever in vertical directions

Cotangent Function: y = cot x

One cycle occurs between 0 and p
There are vertical asymptotes at each end of the cycle

The asymptote that occurs at p repeats every p units

Period = p
Amplitude = none, graphs go on forever in vertical directions

Cosecant Function: y = csc x

There are vertical asymptotes. The asymptote that occurs at p repeats every p units.
Period = 2p
Amplitude: none, graphs go on forever in vertical directions.
The maximum values of y = sin x are minimum values of the positive sections of y = csc x. The minimum values of y = sin x are the maximum values of the negative sections of y = csc x.
The x-intercepts of y = sin x are the asymptotes for y = csc x.

Secant Function: y = sec x

There are vertical asymptotes. The asymptote that occurs at p repeats every p units.
Period = 2p
Amplitude: none, graphs go on forever in vertical directions.
The maximum values of y = cos x are minimum values of the positive sections of y = sec x. The minimum values of
y = cos x are the maximum values of the negative sections of
y = sec x.
The x-intercepts of y = cos x are the asymptotes for
y = sec x.

Here are some helpful videos you might want to watch:

Graphs Of Sine and Cosine Functions

Graphs of Sine and Cosine Functions

Y=cos x

Y=sin x

The following two equations:

Y=d+a sin(bx-c)

And

Y=d+a cos(bx-c)

can experience various transformations as the functions change.

Amplitude

The amplitude of y= a sin x and y= a cos x respresents half of the distance between the max and min values and the equation below can be used:

Amplitude= IaI , where the absolute value of a is taken.

Increasing or decreasing the value of a will either vertically shrink or stretch the graph.

Example #1:

Consider the values:

Y=sin x

Y=2sin x

Y=1/2sin x

Y=cos x

Y=2cos x

Y=1/2cos x

Period

The period of a function of y=a sin bx and y= a cos bx can be found by the equations:

Period= 2 /b

Examples:

Shifting of Sine and Cosine graphs

Below you can see both the original graph of y =sin(x) and the graph of the translation

y = sin(x) + 1

y = (1/2)Cos 3x

Identify each before you graph:

Amplitude = 1/2
Period = 2p/ 3
Maximums are at the beginning point (0, 1/2) and
End point (2p/ 3, 1/2)
minimum point at ( p/3, -1/2)
Zeros at ( p/ 6, 0) and ( p/ 2, 0)

y = -2 Sin (p/2)x

Amplitude = | -2 | = 2

Period = 2p/ (p/ 2) = 4
Note that this graph is a reflection about the x-axis. This interchanges the maximum and minimum values.
                        zeros : (0, 0), ( 2, 0), ( 4, 0)
                        minimum :( 1, -2)
                        maximum : ( 3, 2)