Week 7

This week we looked again in Chapter 3, Polynomial and Rational Functions.

On Day 1, instead of starting another lesson, we looked again on what we looked at last week which included Polynomial Functions, Division of Polynomial Functions and Synthetic Division, How to Find Zeros of Polynomial Functions and Finding the Real Zeros of Polynomial Functions.

We reviewed on problems to see how they work and if we are doing the problems correctly.

On Day 2, We looked at Approximating Zeros. Some Zeros of Graphs are not Solid Numbers like 1, 2 or 3 but are instead decimals or fractions like -0.481923 or 3/47. In this lesson, we were trying to figure out how to find the zeros to the most approximate number as possible.

The steps to find them is:

1) Divide the interval [a, b] in half by finding its midpoint, 

m=a+b

     2

2) Compute f(m)

3) If f(a) & f(m) have opposite signs, then f has a zero in the interval [a, m]

If f(m) and f(b) have opposite signs then f has a zero in the interval [m,b] 

If f(m)=0 then m is a zero of f and the problem is done.

When solving the problem at the end, you would want the 

error< 0.0005

You find if you are done and find the error with the equation 

1/2 (b-a)

Below is a picture of how you would solve it. 

The problems may take a long time and you would do it until you find a number with 0.0005.

Sometimes you would find it with 1 dec which is < 0.05

2 dec < 0.005

3 dec < 0.0005

4 dec < 0.00005

However the most common one is 3 dec.

That is what we did on day 2.

On day 3, we looked at 3.6 which is Rational Functions. We looked at this in Algebra 2 and it is when you use Vertical and Horizontal Asymptotes. 

Usually the equations would look like 

f(x)=P(x)/Q(x)

To find the Vertical Asymptotes of a problem you would set the denominator=0 and the answer would be x= the answer.

To find Horizontal Asymptotes of a problem, you look at the degree of the first equation:

It would look like

f(x)=anX^n/bmX^m.

n= degree of numerator

m= degree of denominator

an= L.C. of numerator

bm= L.C. of denominator

It would be like n/m

To find the horizontal asymptote though, there are three possibilites

When the degrees are both equal. n=m you would use the coefficient of both answers as the answer. If its 2/2 then it would be y=1

Then n<m the answer would be y=0

n>m then the answer DNE.

However, something we have not seen before is Slant Asymptotes which is only possible if the horizontal asymptote is n>m.

f(x)=m(x)/d(x)

The answer from the vertical asymptote is used to divide numerator of the equation. For example x=2 is the VA. You would divide it by its numerator, for example x^2+4. y= answer.

For all equations, there will be x-intercepts and y-intercepts. You just plug in zero for the opposite to find it. If looking for the x-intercept, make y=0 and solve and vice versa.

To find the holes in a problem. You factor the equation and see if anything can be canceled out. Whatever cancels out is the answer and it would usually look like (x,y). If nothing cancels out, there is no hole. 

This is how Rational Functions work.

On day 4, we took a quiz on 3.4-3.6 to test if we knew the information. We then reviewed quickly on some of the material covered last week and the week before. 

That is what we did on Day Four. Next week we will have a test on Friday and will cover one more lesson before that. 

See You Next Week!