How do you graph rational functions?

Answer 1

See explanation...

Suppose #f(x) = g(x)/(h(x)) = (a_nx^n+a_(n-1)x^(n-1)+..+a_0)/(b_mx^m+b_(m-1)x^(m-1)+...+b_0)#
If #g(x)# and #h(x)# have some common factor #k(x)# then let
#g_1(x) = g(x)/(k(x))# and #h_1(x) = g(x)/(k(x))#.
The graph of #g_1(x)/(h_1(x))# will be the same as the graph of #g(x)/(h(x))#,
except that any #x# where #k(x) = 0# is an excluded value.
Assuming #g(x)# and #h(x)# have no common factor, then there will be vertical asymptotes wherever #h(x) = 0#. If a root is not a repeated root (or is repeated an odd number of times) then the limit on one side of the asymptote will be #oo# and on the other #-oo#. If the root has even multiplicity then the limit on both sides of the asymptote will be the same: #oo# or #-oo#.
If #n < m# then #f(x)->0# as #x->+-oo#
If #n >= m# then divide #g(x) /(h(x))# to get a polynomial quotient and remainder. The polynomial quotient is the oblique asymptote as #x->+-oo#.
For example, if #f(x) = (x^3 + 3)/(x^2 + 2)#, then:
#f(x) = (x^3+3)/(x^2+2) = (x^3+2x-2x+3)/(x^2+2) = x - (2x-3)/(x^2+2)#
So the oblique asymptote of #f(x)# is #y = x#
Intercepts with the #x# axis are where #f(x) = 0#, which mean where #g(x) = 0#.
The intercept with the #y# axis is where #x=0#, so just substitute #x=0# into the equation for #f(x)# to find #f(0) = a_0/b_0#
Apart from all this, just pick some #x# values and calculate #f(x)# to give you coordinates #(x, f(x))# through which the graph must pass.
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Answer 2

To graph rational functions, follow these steps:

  1. Determine the domain of the function by identifying any values that would make the denominator zero. Exclude these values from the domain.

  2. Find the vertical asymptotes by setting the denominator equal to zero and solving for x. These are the vertical lines that the graph approaches but never touches.

  3. Determine the horizontal asymptotes by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater, there is no horizontal asymptote.

  4. Find any holes in the graph by canceling out common factors between the numerator and denominator.

  5. Determine the x-intercepts by setting the numerator equal to zero and solving for x.

  6. Determine the y-intercept by evaluating the function at x = 0.

  7. Plot the vertical asymptotes, horizontal asymptotes, holes, x-intercepts, and y-intercept on the graph.

  8. Use additional points or symmetry to sketch the graph between the asymptotes and intercepts.

Remember to label the axes and provide a title for the graph.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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