How do you graph #f(x)=(x^2+4x+3)/(-3x-6)# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Julia Adams · Accepted Answer

Below

#f(x)=(x^2+4x+3)/(-3x-6)# #f(x)=-1/3((x^2+4x+3)/(x+2))# #f(x)=-1/3((x+2)(x+2)-1)/(x+2)# #f(x)=-1/3(x+2-1/(x+2))# #f(x)=-1/3(x+2)+1/(3(x+2)# For oblique asymptote, it is #y=-1/3(x+2)# which can be found by letting #x -> oo#. For vertical asymptote, the denominator cannot equal to 0. so #x+2!=0# so at #x=-2#, there is an asymptote. For intercepts, When #x=0#, #y=-1/2# When #y=0#, #x^2+4x+3=0# #(x+3)(x+1)=0# so #x=-1# and #x=-3# Plotting your x and y intercepts as well as your asymptotes, you should have a general idea as to how your graph should appear. The ends of your graph should be approaching your asymptotes but they should NEVER TOUCH the asymptotes. Below is the graph graph{(x^2+4x+3)/(-3x-6) [-10, 10, -5, 5]}

Eli Assouline · Answer

undefined To graph the function f(x) = (x^2 + 4x + 3)/(-3x - 6), we can analyze its holes, vertical and horizontal asymptotes, as well as the x and y intercepts.

1. Holes: To find any holes in the graph, we need to identify values of x that make the denominator zero. In this case, the denominator -3x - 6 equals zero when x = -2. Therefore, there is a hole at x = -2.

2. Vertical Asymptotes: Vertical asymptotes occur when the denominator of a rational function equals zero, but the numerator does not. In this case, the denominator -3x - 6 equals zero when x = -2. Hence, there is a vertical asymptote at x = -2.

3. Horizontal Asymptotes: To determine the horizontal asymptote(s), we compare the degrees of the numerator and denominator. In this case, the degree of the numerator (2) is greater than the degree of the denominator (1). Therefore, there is no horizontal asymptote.

4. X-intercepts: X-intercepts occur when the function's numerator equals zero. To find them, we set the numerator x^2 + 4x + 3 equal to zero and solve for x. Factoring the quadratic equation, we get (x + 1)(x + 3) = 0. Thus, the x-intercepts are x = -1 and x = -3.

5. Y-intercept: The y-intercept occurs when x equals zero. Substituting x = 0 into the function, we get f(0) = (0^2 + 4(0) + 3)/(-3(0) - 6) = 3/-6 = -1/2. Therefore, the y-intercept is y = -1/2.

By considering these aspects, you can plot the graph of f(x) = (x^2 + 4x + 3)/(-3x - 6) accurately.