Week of October 28, 2019

Asymptotes and Graphing Rational Functions

To graph a rational function, find the asymptotes and intercepts, plot a few points on each side of each vertical asymptote and then sketch the graph.

Finding Asymptotes

Vertical asymptotes are "holes" in the graph where the function cannot have a value.  They stand for places where the x-value is not allowed.  Specifically, the denominator of a rational function cannot be equal to zero.  Any value of x that would make the denominator equal to zero is a vertical asymptote. 

It is common practice to draw a dotted line through any vertical asymptote values to denote that the function cannot exist in those places.  The vertical asymptotes inform the domain of the graph.  Specifically, the domain of the graph will be any allowable numbers other than those that create a zero in the denominator — i.e.,  the vertical asymptotes are excluded from the domain.

Horizontal and Slant Asymptotes

A horizontal or slant asymptote shows us which direction the graph will tend toward as its x-values increase.  Unlike the vertical asymptote, it is permissible for the graph to touch or cross a horizontal or slant asymptote.

To find the horizontal or slant asymptote, compare the degrees of the numerator and denominator.


Horizontal Asymptote

If the degree of x in the denominator is larger than the degree of x in the numerator, then the denominator, being "stronger", pulls the fraction down to the x-axis when x gets big. That is, if the polynomial in the denominator has a bigger leading exponent than the polynomial in the numerator, then the graph trails along the x-axis at the far right and the far left of the graph.  The x-axis becomes the horizontal asymptote.

When the degrees of the numerator and the denominator are the same, then the horizontal asymptote is found by dividing the leading terms, so the asymptote is given by:

y = (numerator's leading coefficient) / (denominator's leading coefficient)

The horizontal asymptote may also be approximated by inputting very large positive or negative values of x.

Slant Asymptote

If the numerator is one degree greater than the denominator, the graph has a slant asymptote.  Using polynomial division, divide the numerator by the denominator to determine the line of the slant asymptote.


Finding Intercepts

To find x and y intercepts, set each variable equal to zero and solve in turn.

Plotting Points

Based on information gained so far, select x values and determine y values to create a chart of points to plot.  Select more plots in areas where you think you need information to inform your curve.

Sketching the Graph

Once the points are plotted, remember that rational functions curve toward the asymptotes.  Include additional points to help determine any areas of uncertainty.  In addition, graphing calculators are often used in conjunction with sketches to define the graph.


Example

Graph:

 

1.  Find the vertical asymptotes.
Factor the denominator:  (x + 2)(x - 2) and set equal to zero.
x cannot be 2 or -2, so the vertical asymptotes are x = 2 and x = -2
Sketch these as dotted lines on the graph.

2.  Find the horizontal or slant asymptotes.
Since the degree of the numerator is 1 and the degree of the denominator is 2, y = 0 is the horizontal asymptote.  There is no slant asymptote.  Sketch on the graph.

3.  Find the intercepts
      Set x = 0; y = 0
      Set y = 0; x = 0
      Intercept is (0, 0)

4.  Plot points on either side of the asymptotes.
Select values for x and determine values of y.  Plot.




5.  Sketch the graph.  Consider that the graph must "take off" near the vertical asymptotes and "level off" near the horizontal asymptote.  Sketch with appropriate curves.

Practice

Follow the steps to graph each rational function:

1.  Find the vertical asymptotes.
2.  Find any horizontal/slant asymptotes.
3.  Find intercepts.
4.  Plot points.
5.  Sketch the graph.

1)  y = 1/x












2)  y = x²/(x - 1)












3)        














Answer Key 

1) 

2)  

3)   

Source:https://www.brainfuse.com/jsp/alc/resource.jsp?s=gre&c=36982&cc=108825

Week of October 21, 2019

Asymptotes

In general, you will be given a rational (fractional) function, and you will need to find the domain and any asymptotes. You'll need to find the vertical asymptotes, if any, and then figure out whether you've got a horizontal or slant asymptote, and what it is. To make sure you arrive at the correct (and complete) answer, you will need to know what steps to take and how to recognize the different types of asymptotes.

Let's get some practice:

• Find the domain and all asymptotes of the following function:

$\mathbf{\color{green}{\mathit{y} = \dfrac{\mathit{x}^2 + 3\mathit{x} + 1}{4\mathit{x}^2 - 9}}}$

I'll start with the vertical asymptotes.

They (and any restrictions on the domain) will be generated by the zeroes of the denominator, so I'll set the denominator equal to zero and solve.

$4x^2 - 9 = 0$

$4x^2 = 9$

$x^2 = \dfrac{9}{4}$

$x = \pm \dfrac{3}{2}$

Then the domain is all x-values other than $\pm \frac{3}{2}$, and the two vertical asymptotes are at $x = \pm \frac{3}{2}$.

Next I'll turn to the issue of horizontal or slant asymptotes.

Since the degrees of the numerator and the denominator are the same (each being 2), then this rational has a non-zero (that is, a non-x-axis) horizontal asymptote, and does not have a slant asymptote. The horizontal asymptote is found by dividing the leading terms:

$y = \dfrac{x^2}{4x^2} = \dfrac{1}{4}$

Then the full answer is:

domain: $\mathbf{\color{purple}{ \mathit{x} \neq \pm \frac{3}{2} }}$

vertical asymptotes: $\mathbf{\color{purple}{ \mathit{x} = \pm \frac{3}{2} }}$

horizontal asymptote: $\mathbf{\color{purple}{ \mathit{y} = \frac{1}{4} }}$

slant asymptote: none

---

A given rational function may or may not have a vertical asymptote (depending upon whether the denominator ever equals zero), but (at this level of study) it will always have either a horizontal or else a slant asymptote.

Note, however, that the function will only have one of these two; you will have either a horizontal asymptote or else a slant asymptote, but not both. As soon as you see that you have one of them, don't bother looking for the other one.

• Find the domain and all asymptotes of the following function:

$\mathbf{\color{green}{\mathit{y} = \dfrac{\mathit{x} + 3}{\mathit{x}^2 + 9}}}$

The vertical asymptotes come from the zeroes of the denominator, so I'll set the denominator equal to zero and solve.

x2 + 9 = 0

x2 = –9

Oops! This has no solution. (Duh! The denominator is a sum of squares, not a difference. So of course it doesn't factor and it can't have real zeroes. I should remember to look out for this, and save myself some time in the future.)

Since the denominator has no zeroes, then there are no vertical asymptotes and the domain is "all x".

Since the degree is greater in the denominator than in the numerator, the y-values will be dragged down to the x-axis and the horizontal asymptote is therefore "y = 0". Since I have found a horizontal asymptote, I don't have to look for a slant asymptote.

My full answer is:

domain: all x

vertical asymptotes: none

horizontal asymptote: y = 0 (the x-axis)

slant asymptote: none

The Special Case with the "Hole"

We've dealt with various sorts of rational functions. When you were first introduced to rational expressions, you likely learned how to simplify them. You'd factor the polynomials top and bottom, if you could, and then you'd see if anything cancelled off.

What if you've found the zeroes of the denominator of a rational function (so you've found the spots disallowed in the domain), but one or another of the factors cancels off? Let's look at an example of exactly that situation:

• Find the domain and all asymptotes of the following function:

$\mathbf{\color{green}{\mathit{y} = \dfrac{\mathit{x}^2 - \mathit{x} - 2}{\mathit{x} - 2}}}$

It so happens that this function can be simplified as:

$y = \dfrac{x^2 - x - 2}{x - 2}$

$= \dfrac{(x - 2)(x + 1)}{x - 2}$

$= \dfrac{\color{blue}{(x - 2)}(x + 1)}{\color{blue}{(x - 2)}}$

$= x + 1$

So the entire rational function simplifies to a linear function. Clearly, the original rational function is at least nearly equal to y = x + 1 — though I need to keep in mind that, in the original function, x couldn't take on the value of 2. But what about the vertical asymptote? Is there one at x = 2, or isn't there?

If there is a vertical asymptote, then the graph must climb up or down it when I use x-values close to the restricted value of x = 2. I'll try a few x-values to see if that's what's going on.

x = 1.5, y = 2.5

x = 1.9, y = 2.9

x = 1.95, y = 2.95

x = 1.99, y = 2.99

Week of October 14, 2019

Safety First!
Keeping Students with Disabilities Safe at Home and in the Community
Parent-Only Event Hosted in Partnership with Cobb County School District
Safety Is No Accident! Parents, come to the Safety Village to get specialized instruction from Cobb Fire and Police Educators. Learn strategies to keep your children with disabilities safe at home and in the community. You will tour Sparky’s House to discuss fire safety practices and learn to be safe in and around water. We will also have a session on car seat safety for students with special needs. And much, much more.
Sign up at: signup.com/go/aDrXNkp
Cobb County Safety Village
1220 Al Bishop Dr. Marietta, GA 30008
CCSD’s Special Education Parent Mentors have partnered with the Cobb County Safety Village for this event, one of our Parent University programs for families of students with disabilities.
Questions?
Call 770-529-0046 to speak with a mentor.
Monday October 21st
6.30p.m - 8.30 pm
Registration begins at 6:00 pm

Cobb Special Education
Parent Mentors
2019-20 Programs
Date Time Parent University - Location
08/26/19 6:30 – 8:30pm School Year Success – Campbell MS
09/16/19 6:30 – 8:30pm Understanding the IEP Process – Pine Mountain MS
10/21/19 6:30 – 8:30pm Safety First – Cobb Safety Village
11/18/19 6:30 – 8:30pm Instructional Strategies – Smitha MS
01/27/20 6:30 – 8:30pm Community Connections Resource Fair – Wheeler HS
02/10/20 6:30 – 8:30pm Fostering Independence – Durham MS
03/17/20 6:30 – 8:30pm Evening of Inspiration – McEachern HS
04/20/20 6:30 – 8:30pm Looking Toward School Year 2020-21 – East Cobb MS
Date Time VIP Parent Workshops – Hawthorne Center
08/21/19 9:30 – 11:00am School Year Success
10/09/19 9:30 – 11:00am Understanding the IEP Process
11/13/19 9:30 – 11:00am Q & A with Special Education Leadership
01/15/20 9:30 – 11:00am Transitioning from ES to MS to HS and Beyond
02/12/20 9:30 – 11:00am Emotional Regulation
03/12/20 9:30 – 11:00am Fostering Communication & Social Regulation
04/15/20 9:30 – 11:00am Looking Toward School Year 2020-21
Cobb Special Education Parent Mentors
Stacy Greene: Stacy.Greene@cobbk12.org Antoinette Nichols: Antoinette.Nichols@cobbk12.org
Phone: 770-529-0046 Website: www.cobbk12.org/specialeducationparentmentors/
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The Spanish versions are below:


¡SEGURIDAD PRIMERO!
Manteniendo Estudiantes con Discapacidades a Salvo
Evento de Padre organizado en cooperación con El Distrito Escolar del Condado de Cobb
¡La Seguridad NO Es Accidente ! Padres, les exhortamos a que vengan al Pueblo de Seguridad donde obtendra n instruccio n especializada por los educadores del departamento de Bomberos y de Policí a de Cobb. Aprendera n estrategias para mantener seguros a los nin os con discapacidades en la casa y en la comunidad. Se recorrera la casa de Sparky para hablar de las pra cticas de seguridad de fuego y aprender a mantenerse fuera de peligro en, o cerca del agua. Tambie n habra una sesio n sobre la seguridad del asiento para autos relacionado con los estudiantes con necesidades especiales. Y mucho, mucho ma s.
Registro: signup.com/go/aDrXNkp
Cobb County Safety Village
1220 Al Bishop Dr. Marietta, GA 30008
Los Mentores del departamento de Educación Especial de CCSD han colaborado con El Pueblo de Seguridad del Condado de Cobb en este evento de Seguridad Primero, lo cual es uno de nuestros Programas Universitarios de Padres de estudiantes con discapacidades. ¿Preguntas? Contacte al 770-529-0046 para hablar con un mentor.
Registración
comienza a las
6:00 pm



Cobb Special Education
Parent Mentors
2019-20 Programs – SPANISH
Programas de Padres Mentores de Educación Especial de Cobb 2019-2020
Fecha Hora Universidad para padres (Parent University)
08/26/19 6:30 – 8:30pm Exitoso Año Escolar - Campbell MS
09/16/19 6:30 – 8:30pm Entendiendo el proceso de IEP (Plan de educación individualizada) –
Pine Mountain MS
10/21/19 6:30 – 8:30pm Seguridad Primero - Cobb County Safety Village
11/18/19 6:30 – 8:30pm Estrategias de enseñanza - Smitha MS
01/27/19 6:30 – 8:30pm Feria de recursos y conexiones en la comunidad - Wheeler HS
02/10/19 6:30 – 8:30pm Fomentando independencia - Durham MS
03/23/19 6:30 – 8:30pm Tardes de inspiración - McEachern HS
04/20/19 6:30 – 8:30pm Mirando hacia el año escolar 2020-21 - East Cobb MS
Fecha Hora Talleres VIP para padres - Hawthorne Center
08/21/19 9:30 – 11:00am Exitoso Año Escolar
10/09/19 9:30 – 11:00am Entendiendo el proceso de IEP (Plan de educación individualizada)
11/13/19 9:30 – 11:00am Preguntas y Respuestas (Q&A) del liderazgo de Educación Especial
01/15/20 9:30 – 11:00am Transición de Primaria (ES k-5º) a Intermedia (MS 6º-8º) a Secundaria/Prepa (HS 9º-12º) y más allá
02/12/20 9:30 – 11:00am Regulación emocional
03/12/20 9:30 – 11:00am Fomentando la comunicación y la regulación social
04/15/20 9:30 – 11:00am Mirando hacia el año escolar 2020-21
Padres Mentores de la Educación Especial de Cobb
Stacy Greene: Stacy.Greene@cobbk12.org Antoinette Nichols: Antoinette.Nichols@cobbk12.org
Teléfono: 770-529-0046 Página de internet: www.cobbk12.org/specialeducationparentmentors/
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Los Padres Mentores de Educación Especial de Cobb brindan orientación, recursos y programas para padres de estudiantes con discapacidades durante todo el año. ¡Esté atento a los correos electrónicos que enviaremos y síganos en Facebook!

Week of October 7, 2019

Simplify rational expression

An algebraic expression where both the numerator and the denominator are polynomials e.g.

x+3x$$\frac{x+3}{x}$$

is called a rational expression. Since the denominator can't be zero there are values of x which are excluded from the rational expression. The expression above has an excluded value of zero.

To simplify a rational expression you have to eliminate all factors that are common of the numerator and the denominator. To accomplish this use the greatest common factor (GCF) of the factors e.g.

4xy6y2=2y⋅x2y⋅3y=/2y/2y⋅x3y=x3y

Source:https://www.mathplanet.com/education/algebra-1/rational-expressions/simplify-rational-expression