Graph the rational function
Fofx=2x+2/x+4
To graph the rational function f(x) = (2x + 2)/(x + 4), we need to follow these steps:
1. Determine the vertical asymptote(s): Set the denominator equal to zero and solve for x.
x + 4 = 0
x = -4
Therefore, the vertical asymptote is x = -4.
2. Determine the horizontal asymptote: If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote. In this case, the degrees are the same (both 1), so there is a horizontal asymptote at y = 2/1 = 2.
3. Determine the x-intercept(s): Set the numerator equal to zero and solve for x.
2x + 2 = 0
x = -1
Therefore, the x-intercept is (-1, 0).
4. Determine the y-intercept: Plug in x = 0 into the function.
f(0) = (2(0) + 2)/(0 + 4) = 2/4 = 1/2
Therefore, the y-intercept is (0, 1/2).
Now, let's plot these points and identify the behavior near the asymptotes.
As x approaches -4 from the left side (x < -4), f(x) becomes infinitely large (positive infinity).
As x approaches -4 from the right side (x > -4), f(x) becomes infinitely large (negative infinity).
Here is the graph of the rational function f(x) = (2x + 2)/(x + 4):
Graph:
|
| /
| /
|/
------------------|-----------------
| / |
| / |
| / |
| / |
| / |
| / |
------------------|-----------------
|
-4 -1 0
(Note: The graph does not include the vertical asymptote at x = -4 because the program cannot display a vertical line.)
To graph the rational function f(x) = (2x + 2)/(x + 4), we can follow these steps:
Step 1: Find the vertical asymptote(s)
The vertical asymptote(s) occur where the denominator of the rational function equals zero. So, set x + 4 = 0 and solve for x:
x + 4 = 0
x = -4
So, the vertical asymptote is x = -4.
Step 2: Find the horizontal asymptote
To find the horizontal asymptote, determine the degrees of the numerator and denominator polynomials. In this case, both the numerator and denominator are linear functions, so we'll have a slant asymptote.
To find the slant asymptote, perform long division of the numerator by the denominator:
2x + 2
__________
x + 4 | 2x + 0
-(2x + 8)
__________
-8
The quotient is 2, and there is a remainder of -8. So, the slant asymptote is y = 2.
Step 3: Find the x-intercept(s)
To find the x-intercept, set the numerator equal to zero and solve for x:
2x + 2 = 0
2x = -2
x = -1
So, the x-intercept is x = -1.
Step 4: Find the y-intercept(s)
To find the y-intercept, substitute x = 0 into the equation:
f(0) = (2(0) + 2)/(0 + 4)
f(0) = 2/4
f(0) = 1/2
So, the y-intercept is y = 1/2.
Step 5: Plot the points and draw the graph
Using the information from steps 1-4, we can plot the points (-4, not defined), (-1, 0), and (0, 1/2) on a coordinate plane. We can also draw the vertical asymptote x = -4 and the slant asymptote y = 2.
Finally, we can sketch the graph by connecting the points and following the asymptotes.