RATIONAL FUNCTION GRAPHIC ORGANIZER

Introducing Properties of Rational Functions:

• the form of h(x)=f(x)/g(x), where f(x) & g(x) are polynomial functions and g(x)≠0
• The zeros of a rational function h(x)= f(x)/g(x) are the zeros of f(x) if h(x) is expressed in simplified form.

ex: f(x)= x(x-2)/(x+2)(x-1) will have zeros of 0 and 2

From the graph of f(x)=x(x-2)/(x+2)(x-1), it demonstrates that zeros occur when f(x) equals 0, which also means that zeros will occur if the numerator of a rational function equals 0.

Introducing Properties of Rational Functions:

• the y-intercept of a rational function h(x) is the value of the simplified form of the function when x=0.

ex: h(x)= (x+2)(x-5)(x-3)/(x+3)(x-2)(x-6), y-intercept will be: 5/6

From the graph of h(x)= (x+2)(x-5)(x-3)/(x+3)(x-2)(x-6), it demonstrates that y-intercept will occur when x equals 0

What is an Asymptote?

An asymptote is a line or curve that approaches a given curve arbitrarily (infinitely) close.

Like this:

Yes, there can also be a removable discontinuity instead of a infinite discontinuity, removable discontinuity is also known as holes.

Removable discontinuity usually occurs when the function can be simplified by dividing the denominator and numerator by a common factor that includes a variable.

ex:

Therefore, at rational function's restriction values, a rational function can either have a removable/ infinite discontinuity or both.

To sum up...

The connections between a vertical asymptote and a hole are that vertical asymptote and hole both came from restrictions which means they are also undefined; yet the major difference between a vertical asymptote and a hole is that vertical asymptote is an infinite discontinuity while hole is a removable discontinuity.

SIMILARITIES & DIFFERENCES between horizontal and vertical asymptote

Obtain horizontal asymptote through division; vertical asymptote through restrictions

We use division to obtain the H.A (E.B.A) of a rational function:

• Degree of numerator < Degree of denominator

Horizontal occurs at f(x)=0

ex: f(x)= x/x^2

• Degree of numerator = degree of denominator

H.A will be the ratio of the leading coefficient

ex: f(x)= 2(x-1)/x; H.A occurs when f(x)=2

• Degree of numerator is exactly 1 degree greater than the denominator: Oblique(slant) asymptote

ex: f(x)= x^2/(x+1)

H.A occurs when f(x)= x-1

• Degree of numerator is exactly 2 degrees greater than denominator: Parabolic asymptote

ex: x^3/(x+1)

H.A occurs when f(x)= x^2+1

Different behavior near asymptotes

f(x)= 1/x

Determining V.A and the behavior near V.A

Therefore, As x--> 0-, f(x) --> negative infinity

As x--> 0+, f(x) --> infinity

Determining E.B.A and describe its behavior

As x --> infinity, f(x) --> 0 (approaching from above)

As x --> negative infinity, f(x) --> 0 (from below)

WHY DO RECIPROCAL FUNCTIONS GIVE CURVED LINES WHEN THE ORIGINAL FUNCTION HAS A STRAIGHT LINE?

Let’s look at this question with the function of y=x and the reciprocal function of y = 1/x.

1. When x equals 0 as a denominator, the value of y is undefined, therefore, the reciprocal function does not goes through the point (0,0) like the function ‘y = x’ does. Therefore, the domain of the reciprocal transformation will be: {XER|X ≠ 0} and the range will be: {YER| Y ≠ 0}. The domain and the range of the reciprocal transformation demonstrates that as |x| get closer to the vertical asymptote (which is 0 in this case), the absolute value of y becomes greater. However, as |x| move further away from the vertical asymptote, the absolute value of y becomes smaller (also never touches the asymptote). Therefore, the reciprocal transformation of y= 1/x will never go across its asymptote to form a straight line like the function y = x does.

2. There can only be two points where the values of ‘x’ and ‘y’ are both whole numbers in a reciprocal transformation, which is when the absolute value of the denominator equals the value of the numerator (ex: (1,1) (-1,-1) in the case of y=1/x). As a result, curved lines must occur in a reciprocal transformation, because if the value of ‘x’ is a whole number, the value of ‘y’ will never be a whole number (unless the value of ‘x’ equals the value of the numerator); yet, when the value of ‘y’ is a whole number, the value of ‘x’ cannot be a whole number, meaning that a straight line will not occur.

Moreover, due to the fact that a straight line have a constant slope and a reciprocal function may have an increasing slope or decreasing slope; thus, reciprocal transformation give curved lines when the original function has a straight line.

A function can go across its horizontal asymptote while it cannot go across its vertical asymptote

If rational function can cross its E.B.A, what will be the point of intersection?

This question will lead to solving EQUALITIES...

for example, solve:

, point of intersection?

Solving inequalities is similar as determining behavior near V.A, since both of them requires an interval table. For example,

Electricity that we use everyday:

As a biologist:

In medical field:

• The topic rational function becomes more and more significant in our society as doctors are required to calculate the concentration of a drug injected in a patient in order to perform the right medical procedures.
• Rational function is also important in the medical field as it portrays how far the drug has affected the patient before starting a treatment, since the outcome can be influenced by the concentration of the drug.

For example, surgeries and operations can only begin after a concentration of a drug reaches a desired level. Therefore, rational function is helpful for doctors to determine after how many hours they should start surgery based on the concentration level they prefer.

Therefore, Real-life first, mathematics second.

In other words, RATIONAL FUNCTIONS= REAL-LIFE

References:

• Ms. Mendaglio
• http://myclass.peelschools.org/sec/12/63175/default.aspx
• http://www.mccc.edu/~greenbay/documents/Sec.2.6RationalFunctions.pdf
• http://prezi.com/yortopyaqutx/copy-of-application-of-functions-in-the-real-world/

Background Music:

• Brahms - Waltz in A flat Op. 39, No.15

After learning rational functions, as a student, we may wonder, what is the connection between rational functions and real-life? When will we ever use rational functions?

Well, here are some real-life examples:

RATIONAL

FUNCTION

GRAPHIC

ORGANIZER

By Candy Su

Domain is the most important thing about a function. The importance of domain becomes even more obvious in rational functions as the restrictions of the function will lead to the formation of vertical asymptote or hole.

If there is

would there also be finite(removable)?

a hole

a vertical asymptote

both

The domain of a rational function consists of all the real numbers except for the zeros of the denominator.

Domain of f(x):

Domain of h(x):

{XER|X≠-3 and x≠2 and x≠6}

{XER|X≠ 1 and X≠ -2}

Through division, we can sum up that:

Connection between restrictions and asymptotes/holes:

Process of obtaining a H.A

Point of Intersections

INEqualities

A function can have more than 1 vertical asymptote and also more than 1 horizontal asymptote

H.A answers the question "what does f(x) do x gets infinitely large in either direction?"

Horizontal asymptote mimics the behavior of the function while vertical asymptote came from the restrictions

SIMILARITIES & DIFFERENCES between horizontal and vertical asymptote

Serve as an End behavior asymptote

Imaginary line

Vertical asymptote is always straight, yet horizontal asymptote can be slanted or even in a parabolic shape

An asymptote is a line or curve that approaches a given curve arbitrarily close

Restrictions lead to asymptotes/holes:

Extensive Question:

!

WARNING

__________

Thank you.

Brain at work