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

Question

Savannah Anderson · Accepted Answer

You may rewrite to #f(x)=-1/2 xx(x+4)/(x+3)# The vertical asymptote will then be clear: #x=-3# as the demoninator may not be #=0# If #x# gets larger and larger (both netaive or positive), the #+4# and #-3# matter less and the function will converge to: #-1/2 xx x/x=-1/2# So #y=-1/2# is the horizontal asymptote . graph{(x+4)/(-2x-6) [-11.25, 11.25, -5.625, 5.625]}

Jace Anderson · Answer

undefined To graph the function f(x) = (x+4)/(-2x-6), we can start by determining the vertical and horizontal asymptotes, as well as the x and y intercepts.

Vertical asymptote:
To find the vertical asymptote, we set the denominator (-2x-6) equal to zero and solve for x:
-2x-6 = 0
-2x = 6
x = -3
Therefore, the vertical asymptote is x = -3.

Horizontal asymptote:
To find the horizontal asymptote, we compare the degrees of the numerator and denominator. In this case, both have a degree of 1. Since the degrees are the same, we divide the leading coefficients:
1 / -2 = -1/2
Therefore, the horizontal asymptote is y = -1/2.

X-intercept:
To find the x-intercept, we set the numerator (x+4) equal to zero and solve for x:
x+4 = 0
x = -4
Therefore, the x-intercept is (-4, 0).

Y-intercept:
To find the y-intercept, we substitute x = 0 into the function:
f(0) = (0+4)/(-2(0)-6)
f(0) = 4/-6
f(0) = -2/3
Therefore, the y-intercept is (0, -2/3).

Holes:
To determine if there are any holes in the graph, we check if any factors in the numerator and denominator cancel out. In this case, there are no common factors, so there are no holes in the graph.

Now, we can plot these points and draw the graph accordingly.

Savannah Anderson · Answer

undefined To graph $ f(x) = \frac{x + 4}{-2x - 6} $:

1. Determine the vertical asymptote by setting the denominator equal to zero and solving for $ x $.
   $$ -2x - 6 = 0 $$
   $$ x = -3 $$

2. Determine the horizontal asymptote. Since the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote.
   $$ \lim_{x \to \pm \infty} f(x) = \frac{1}{-2} = -\frac{1}{2} $$

3. Find the $ x $-intercept by setting $ f(x) = 0 $ and solving for $ x $.
   $$ \frac{x + 4}{-2x - 6} = 0 $$
   $$ x + 4 = 0 $$
   $$ x = -4 $$

4. Find the $ y $-intercept by setting $ x = 0 $ and evaluating $ f(x) $.
   $$ f(0) = \frac{0 + 4}{-2(0) - 6} = -\frac{2}{3} $$

5. Plot the vertical asymptote at $ x = -3 $, the horizontal asymptote at $ y = -\frac{1}{2} $, the $ x $-intercept at $ (-4, 0) $, and the $ y $-intercept at $ (0, -\frac{2}{3}) $.

6. To account for the hole in the graph, simplify the function by factoring out any common factors from the numerator and the denominator.
   $$ f(x) = \frac{x + 4}{-2(x + 3)} $$

7. Since there's a common factor of $ x + 3 $, the graph will have a hole at $ x = -3 $.

8. Plot the hole at $ (-3, h) $, where $ h $ is the value of the simplified function at $ x = -3 $.