Unit G Summary Question # 10 Range of a Rational Function

While the domain of a rational function depends on DIVAH, what do you think the range of a rational function depends on? Give an example.

• The range of a rational function depends on the horizontal asymptote and its holes. (RIHAH). Just as the vertical asymptotes we the "boundaries" of the domain, the horizontal asymptotes will be the boundary of the range. Holes in the graph are still considered the bad values.

Unit G Summary Question #9 X-intecepts of Rational Functions

 

Describe how to find the x-intercepts of a rational function. Include both the long way and the shortcut way, explaining why the shortcut makes mathematical sense.

• To find the x-intercept of a rational function, we must first have the complete factored of the equation done, then set the whole equation to zero. When we do so, we must get rid of the denominator of the equation, so, we multiply the denominator on both sides of the equation. so, in the end, we are simply setting then numerator to zero. 
• Do not forget to put it in correct notation! 
• Ex) (4,0)

Unit G Summary Question #7 Vercical Asmptote Limit Notation

 


Describe how to write limit notation for vertical asymptotes and what the notation means.

• To right the limit notation for a vertical asymptote, we must first find the equation for the vertical asymptote. To do that, we set the denominator of the function equal to zero. When we find the asymptotes, we can do the limit notation. Let's say x=1. This is ow the notation will look like:
• as x->3 +, f(x)->___
• as x->3 -, f(x)->___
• This notation means as the graph is approaching  towards to the vertical asymptote x=3, the f(x) will either be inf or -inf. And as the graph is going away from the vertical asymptote, the graph will either be inf. or -inf.

Unit G Summary Question # 2 Horizontal Limit Notation

Describe what the limit notation for the horizontal asymptotes actually means.

• The limit notation is describing the graphs boundaries. As the graph goes further to the left or further to the right,it can not go pass the asymptote. So if the horizontal asymptote is y=1/2, then the graph will not go pass the horizontal 2. the limit notation will look like this: 
• as x->inf., then f(x)-> 1/2
• as x-> -inf., then f(x)->1/2

Unit G Summary Question # 1 Horzaontal Asymptotes

How do we know if a graph has a horizontal asymptote? What are the three options?

• A graph has a horizontal asymptote only when the degrees of the numerator and the denominator are being compared. When the degree in the denominator is bigger, the asymptote will be y=0. When the degree of the the numerator and the denominator are the same, then the asymptote will be the ratio of the coefficients. Although, if the degree is bigger in the numerator, there is no horizontal asymptote.

Unit G Summary Question #8 Y-intercept of Rational Function

How do you find the y-intercept of a rational function? Does this need to be done in the original or simplified equation?

• To find the y-intercept of a rational function, first we must take the simplified factored equation of the function. this includes any factors that may have been canceled out when finding holes. Simply, plug in zero in the x-values of the simplified equation on all the factors. the y-intercept may be a fraction or not, but the answer will give you the y-intercept.

Unit G Summary Question #6 Plotting Holes

How do we find the appropriate place to plot a hole if the y-value is undefined when plugged into the original equation?

• When the y-value is undefined when plugged into the original equation, there is no hole to plot. When we do have a hole to plot, we take the canceled factors of the vertical asymptotes equation. we equal those to zero. We will get the x-value from there. Next, as we plug the value back to the original equation , thus getting the y-value. Finally, simply plot the hole on the graph. Make sure it is a HOLE.

Unit G Summary Question #4 Vertical Asymptotes and Holes

4. What is the difference between a graph having a vertical asymptote and a graph having a hole?

• A graph having a vertical asymptote are simply the borderline where the graph will not pass through or even touch at all. A graph will a hole is similar, the point that is a hole does not exists on the graph. The point that represent the holes derive from the the same factorization of finding the vertical asymptote. When factors cancel when finding a vertical asymptote, those factors are then equaled to zero. That will give you the x-value. In order to find the y-value, we must plug in the point separately  to the simplified factored equation. So the hole on the graph are pints the graph does not touch.

Unit G Summary Qestion #3 Slant Asmptotes

When does a graph have a slant asymptote? How do you find the slope of the slant asymptote?

• A graph has a slant asymptote only if the rational function's numerator has a degree that is one bigger than the denominator. In order to find the equation of the slant asymptote, long division must be used. the numerator will obviously be divided by the denominator. the final answer will be the new equation, excluding the remainder that may be present. 

Unit G Summary Question # 5 Graph Crossing Asymptotes

      

Describe the conditions in which a graph can cross through an asymptote? 

• The graph majority of the time, graphs will not touch asymptotes. However, for a horizontal asymptotes, the graph will sometimes pass through the middle of the graph. Same goes for a slant asymptote, sometimes through the middle. However, a graph can never touch a vertical asymptote.

STUDENT VIDEO #1: Unit F Concept 10

This video covers a problem from Unit F concept 10 which is finding all zeros of polynomials of real or irrational imaginary. The problem we chose was number 7 from the extra problems on the SSS packet. What does the viewer need to pay special attention to in order to understand the concept? The viewer must understand the rules of finding the PQs to find all the possible positive and negative zeros in the given polynomial. Also, the viewers must know by using the synthesis, the we are able to find the zero when getting a zero at the last number.