How do you determine the vertical and horizontal asymptotes of the graph of each function #f(x) = (3x)/(x+4)#?

Question

Liam Brennan · Accepted Answer

There is a vertical asymptote at #color(red)(x=-4)# and a horizontal asymptote at #color(red)(y=3)#.

#f(x)=(3x)/(x+4)#

Step 1. Find the vertical asymptotes.

Set the denominator equal to zero and solve for #x#.

#x+4=0#

#x=-4#

There is a vertical asymptote at #x=-4#.

Step 3. Find the horizontal asymptotes.

Since the numerator and denominator are the same degree, we must divide the coefficients of the highest terms.

#3/1=3#

The horizontal asymptote is at #y=3#.

Anna Allen · Answer

undefined To determine the vertical and horizontal asymptotes of the graph of a rational function like $f(x) = \frac{3x}{x + 4}$:

1. **Vertical Asymptotes**:
   Vertical asymptotes occur where the denominator of the rational function is equal to zero, but the numerator is not zero. To find vertical asymptotes, set the denominator $x + 4$ equal to zero and solve for $x$. The solutions are the vertical asymptotes.

$x + 4 = 0$  
   $x = -4$

So, the vertical asymptote is $x = -4$.

2. **Horizontal Asymptotes**:
   Horizontal asymptotes can be determined by examining the behavior of the function as $x$ approaches positive or negative infinity.

If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is the line $y = 0$.

If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

In this case, the degree of the numerator (1) is less than the degree of the denominator (1). So, the horizontal asymptote is $y = 0$.

Therefore, the vertical asymptote is $x = -4$ and the horizontal asymptote is $y = 0$.