Solving rational equation and inequalities.

Hi, my beloved friend. Seriously now, I am really not in the mood of writing. So just enjoy this video. For real.

All this video cover the whole chapter 3.4. I think, there's no need to explain in words anymore. Plus I am really not in the mood of writing. So I hope you enjoy this video. I just want to say that there are 3 method we can use to solve rational equation and inequalities. 

1. Using normal method to solve algebra expression, but this is only for rational equation not inequalities.

2. Using number line and test value.

3. Using the graph, we have to draw the graph and observe the solution in the graph.

Thumbs up everybody!! We are at the end of this chapter. And I can assure that learning this chapter is more interesting with the three video above. Have fun reading my it!! See you next time, Bye!!!

Rational Function

Hi there!!! We meet again. I am not in the mood to write something. So just enjoy this video about rational function.

Upps... sorry I cannot find any video that are related to our syllabus, this is because rational functions is so wide thus the video i found talks about reciprocal, not what we learn in this chapter. But this video up there is a portion of what we learn in class. So I have no choice and just have to write it.


The general equation for rational functions is f(x)=ax+b/cx+d {a,b,c and d E R}.So i will demonstrate to you the key features of the function using the a,b,c and d term.

• Domain - {x E R, x cannot equal to -d/c}
• Range - { y E R, y cannot equal to a/c}
• x-intercept - x equal to -b/a.
• y-intercept - y equal to b/d.
• vertical asymptote - x equal to -d/c.
• horizontal aymptote - y equal to a/c.
• restriction - x cannot equal to -d/c.

How to find horizontal and vertical asymptotes manually?

1. Horizontal asymptote

• We have to substitute x negative infinity and positive infinity into the equation.
• and see the y values approaches what value, that value is the horizontal asymptote.
• or even easier just divide all the term in the equation with the x term with the highest degree

2. Vertical asymptote.

• To find the vertical asymptote we have to substitute the closest value of the restriction from the left and from the right. 
• The result must approaches negative and positive infinity. If not the restriction is not the horizontal asymptote.

HUH!!! Is that how to find the vertical and horizontal asymptote?? YES!! That's the way. In case if there are a question about how to find the VA and HA and required to show the solution, just use the method above. I think that all for now. See you next time!! bye!!

Reciprocals of a Quadratic Function.

HA....(yawning). Look at the time it is 5.00 in the morning!! Last night I was preparing for my presentation for International Business, then I thought of writing my blog after that. Unfortunately, I was so sleepy. I took my pillow and sleep on the floor, hoping that I will woke up at_____maybe_____around three a.m. However, I did not manage to do so even though I slept on the floor. And now, here I am telling you all my sad story........


Anyways, let's get to the real stuff. Today, we are still in chapter 3, so previous post we learn about the reciprocal of a linear. We discover how its looks like and also we describe the key features of the function. For today's lessons we will doing the same thing, but using quadratic function.


How does quadratic function looks like? Basically quadratic function has this form of general equation, (F(x)=ax^2+bx+c). For example, f(x)=5x^2+10x+1. The shape of the graph is like this :

This graph may not shown the actual measurement for the example of equation I gave you, but this how quadratic function looks like. Now lets see how the reciprocal of quadratic looks like...

Again, this picture may not show the real exact measurement of the equation given but still this is how reciprocal of quadratic looks like. Now lets discover and describe the key feature of this reciprocal ;

• The restriction on the domain - the denominator cannot be zero, so equate the denominator to zero and calculate the x-value. 
• Domain and range - { x E R, x cannot equal to the restriction}, { y E R, y cannot equal to the horizontal asymptote}
• x-intercept and y-intercept - To find the x-intercept we have to substitute y=0, y-intercept we have to substitute x=0.
• The end behaviour 
• as x approaches vertical asymptote from the left, y approaches......and as x approaches vertical asymptote from the right, y approaches.........
• as x approaches negative infinity, y approaches........and as x approaches positive infinity, y approaches........
• The horizontal asymptote is equal to the restriction of domain, the vertical asymptote equal to the restriction of range.
• We can also use the increasing and decreasing interval or the increasing and decreasing of slope to describe the key features of a reciprocal of quadratic.

Here are videos about reciprocal of quadratic function in details.

These three videos explain in details about reciprocal of a quadratic function. It shows how to find the key features of reciprocal and also how to draw the graph without using graphic calculator. Enjoy!! See you next time with chapter 3.3....Bye!!