SOLUTION: graph the rational function 2x+3/x+1 Include all asymptotes on your graph as well as x and y-intercpets.

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Question 1155689: graph the rational function 2x+3/x+1
Include all asymptotes on your graph as well as x and y-intercpets.


Found 3 solutions by josgarithmetic, MathLover1, greenestamps:
Answer by josgarithmetic(39621) About Me  (Show Source):
You can put this solution on YOUR website!
You mean, %282x%2B3%29%2F%28x%2B1%29 ?

x-axis intercept at x=-3%2F2, and a asymptote of x=-1.
y-intercept (when x is 0) is y=3 or (0,3).



graph%28400%2C400%2C-6%2C6%2C-6%2C6%2C%282x%2B3%29%2F%28x%2B1%29%29

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
to graph it, make a table, find several points and draw a graph


y=%282x%2B3%29%2F%28x%2B1%29
x-intercepts
0=%282x%2B3%29%2F%28x%2B1%29
0%28x%2B1%29=%282x%2B3%29
0=2x%2B3
2x=-3
x=-3%2F2=> x-intercept is at (-3%2F2,0)

y-intercepts
y=%282%2A0%2B3%29%2F%280%2B1%29
y=3%2F1
y=3=> y-intercept is at (0,3)

asymptotes:
%28x%2B1%29=0
x=-1=> vertical asymptote
y=%282x%2B3%29%2F%28x%2B1%29-> variable is same degree in numerator and denominator, horizontal asymptote is y=2%2F1=2


Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


f%28x%29+=+%282x%2B3%29%2F%28x%2B1%29

(1) y-intercept
This is easy -- set x=0 and evaluate.
y-intercept: (0,3)

(2) zeros (x-intercepts)
The function value is 0 when y is 0. For a rational function, that is where the numerator is 0:
2x%2B3+=+0
2x+=+-3
x=+-3%2F2

x-intercept: (-3/2,0)

(3) vertical asymptotes are where the denominator is 0:
x%2B1+=+0
x+=+-1

vertical asymptote at x=-1

(4) horizontal asymptote
The horizontal asymptote for a rational function is determined by the end behavior of the function -- the y values for very large positive or negative values of x. Since the numerator and denominator are both polynomials of the same degree, the horizontal asymptote is the ratio of the leading coefficients of the polynomials.
horizontal asymptote: y = 2/1 = 2

Note there are many times when working with rational functions that it is helpful to rewrite the function in a different form.

The function in this problem can be rewritten as

%282x%2B3%29%2F%28x%2B1%29+=+%282%28x%2B1%29%2B1%29%2F%28x%2B1%29+=+2+%2B+1%2F%28x%2B1%29

In that form it is easy to see that the horizontal asymptote is y=2.

Although it is not applicable to this problem, note that rewriting a rational function like this can make working a problem if we are using calculus to find derivatives or integrals.

The asymptotes and intercepts we have found are confirmed with a graph:

graph%28400%2C400%2C-5%2C5%2C-10%2C10%2C%282x%2B3%29%2F%28x%2B1%29%2C2%29