Analyze Rational Function

Status
Not open for further replies.

mathdad

Full Member
Joined
Apr 24, 2015
Messages
737
In the problem below, follow Steps 1 through 8 on to analyze the graph of the function.

STEP 1:

Factor the numerator and denominator of R. Find the domain of the rational function.

STEP 2:

Write R in lowest terms.

STEP 3:

Locate the intercepts of the graph. The x-intercepts are the zeros of the numerator of R that are in the domain of R. Determine the behavior of the graph of R near each x-intercept.

STEP 4:

Determine the vertical asymptotes. Graph each vertical asymptote using a dashed line.

STEP 5:

Determine the horizontal or oblique asymptote, if one exists. Determine points, if any, at which the graph of R intersects this asymptote. Graph the asymptote using a dashed line. Plot any points at which the graph of R intersects the asymptote.

STEP 6:

Use the zeros of the numerator and denominator of R to divide the x-axis into intervals. Determine where the graph of R is above or below the x-axis by choosing a number in each interval and evaluating R there. Plot the points found.

STEP 7:

Analyze the behavior of the graph of R near each asymptote and indicate this behavior on the graph.

STEP 8:

Use the results obtained in Steps 1 through 7 to graph R.

R(x) = (3x + 3)/(2x + 4)

Can you explain, in easy terms, steps 5 through 7?
 
Last edited:
Not a fan.

Omission #1 - There are more kinds of asymptotes than Horizontal, Vertical, and Oblique. Those are just the linear ones.
Omission #2 - The DEGREE of the factor in the denominator that contributes to a vertical asymptote has significance. This consideration removes most of "Step 6".
 
Not a fan.

Omission #1 - There are more kinds of asymptotes than Horizontal, Vertical, and Oblique. Those are just the linear ones.
Omission #2 - The DEGREE of the factor in the denominator that contributes to a vertical asymptote has significance. This consideration removes most of "Step 6".

I just now posted all 8 steps and provided rational function R(x). Please, explain steps 5 to 7 in easy terms. This looks like a fun exercise. More later....
 
Not a fan.

Omission #1 - There are more kinds of asymptotes than Horizontal, Vertical, and Oblique. Those are just the linear ones.
Omission #2 - The DEGREE of the factor in the denominator that contributes to a vertical asymptote has significance. This consideration removes most of "Step 6".

Not a fan of the 8 steps? If so, do you know a better, shorter method for analyzing rational functions?
 
If nothing else, long division exposes non-vertical asymptotes. Give it a go.

Shorter? No. Just track down a the pieces.
 
If nothing else, long division exposes non-vertical asymptotes. Give it a go.

Shorter? No. Just track down a the pieces.

I will try long division. How about using Wolfram to graph R(x)? Can a graph answer the question a bit more easily?
 
I will play with this question on paper and then post my answer STEP BY STEP for all 8 steps, if I can follow safely through. Sullivan has A LOT of questions similar to R(x). I do not have time to answer similar questions going through 8 long steps but I surely will answer 4 on paper and post a full solution reply to R(x) as given here.
 
... STEP 8: Use the results obtained in Steps 1 through 7 to graph R(x) ...
It seems like "the question" in this thread is to graph the given rational function.

... How about using Wolfram to graph R(x)? Can a graph answer the question a bit more easily?
Now you've asked whether it's easier to have a machine do it for you. Are you serious?

?
 
It seems like "the question" in this thread is to graph the given rational function.


Now you've asked whether it's easier to have a machine do it for you. Are you serious?

?

Nevermind. I solved it on paper and will post my work when time allows. The question is not just about analyzing R(x) piece by piece to create a graph.
 
Status
Not open for further replies.
Top