Graph the rational function

Question

X/x^2-49 graph a rational function

Answer 1

x / (x² - 49)

 

 

To graph, you need several important points.  You need to find the domain, range, vertical and horizontal asymptotes, and intercepts.

 

First, find the domain and vertical asymptotes by setting the denominator equal to zero.

 

x² - 49 = 0

 

(x - 7)(x + 7) = 0

 

x = 7     and      x = -7

 

 

The domain is in the interval    (-∞-7)∪(-7, 7)∪(7, ∞).

 

Vertical asymptotes are      x=-7        and          x = 7

 

 

Find the x-intercept.  Set the numerator equal to zero.

 

x = 0

 

Your x-intercept is (0, 0).

 

 

Find the y-intercept.  Evaluate the function when x=0.  Your y-intercept is (0, 0).

 

To find the horizontal asymptote, we look at the degrees of the numerator and denominator part.  Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is       y=0.  This is the x-axis.

 

 

Finally, we have to see how the endpoints behave as the x values increase.  Evaluate the function using test points.

 

f(-11) = -11 / ((-11)² - 49)) = -0.1528

 

f(-9) = -9 / ((-9)² - 49)) = -0.2813

 

f(-8) = -8 / ((-8)² - 49)) = -0.5333

 

--------------------------------------------------------- vertical asymptote point at x=-7

 

f(-6) = -6 / ((-6)² - 49)) = 0.461

 

f(0) = 0

 

f(6) = 6 / (6² - 49) = -0.461

 

----------------------------------------------------------vertical asymptote point at x=-7

 

f(8) = 8 / (8² - 49) = 0.5333

 

f(9) = 9 / (9² - 49) = 0.2813

 

f(11) = 0.1528

 

 

In the interval (-∞, -7) the function decreases.

In the interval (-7, 7), the function decreases.

In the interval (7, ∞), the function decreases.

 

 

You now have all the information you need.  Use it to sketch a graph of the function.