How do you identify all asymptotes or holes for #h(x)=(2x^2+9x+2)/(2x+3)#?

Question

Elias Ackerman · Accepted Answer

#"vertical asymptote at "x=-3/2#
#"oblique asymptote "y=x+3# The denominator of h(x) cannot be zero as tis would make h(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote. #"solve "2x+3=0rArrx=-3/2" is the asymptote"# Horizontal asymptotes occur when the degree of the numerator #<=# degree of the denominator. This is not the case here hence there are no horizontal asymptotes. Oblique (slant ) asymptotes occur when the degree of the numerator #># degree of the denominator. This is the case here hence there is an oblique asymptote. #"dividing out"# #x(2x+3)+3(2x+3)-7# #rArrh(x)=(2x^2+9x+2)/(2x+3)=x+3-7/(2x+3)# #"as "xto+-oo,h(x)tox+3# #rArry=x+3" is the asymptote"# Holes occur when there is a cancellation of a common factor on the numerator/denominator. This is not the case here hence there are no holes. graph{(2x^2+9x+2)/(2x+3) [-10, 10, -5, 5]}

Theodore Amidon · Answer

undefined To identify the asymptotes or holes of $ h(x) = \frac{{2x^2 + 9x + 2}}{{2x + 3}} $:

1. Check for vertical asymptotes: Set the denominator equal to zero and solve for $ x $. Any $ x $ values that make the denominator zero are vertical asymptotes.

2. Determine if there are any holes: Simplify the function and see if there are any common factors in the numerator and denominator that can be canceled out. If there are, then there might be a hole at that point.

3. Horizontal and slant asymptotes: For rational functions like this one, there might be horizontal or slant asymptotes. To find these, compare the degrees of the numerator and denominator.

4. Determine any vertical asymptotes by setting the denominator equal to zero and solving for $ x $.

5. Simplify the function to see if there are any common factors that can be canceled out.

6. Compare the degrees of the numerator and denominator to determine if there are horizontal or slant asymptotes. If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at $ y = 0 $. If the degree of the numerator is exactly one more than the degree of the denominator, there is a slant asymptote, which can be found using polynomial division.

7. Plot any vertical asymptotes, holes, and asymptotes on a graph to visualize the behavior of the function.