Identify all atributes of the rational function below. Write "none" if there is none for any category.

Question

Hole(s):  

Vertical Asymptote(s): x=  

End Behavior Asymptote: y= 

x-intercept(s):  

y-intercept:  

Domain: x​=  

Answer 1

Given: y = (x^2 - 5x - 6) / (x^2 + 5x + 4)

Find: VA, End behavior, intercepts, domain

Solution:

y = (x^2 - 5x - 6) / (x^2 + 5x + 4)

y = [(x + 1)(x - 6)] / [(x + 1)(x + 4)]

y = (x - 6) / (x + 4). HOLE at x = -1

VA: x = -4

HA: y = 1

When: x = -1

y = (-1 - 6) / (-1 + 4)

y = (-7) / (3)

Hole at point: (-1, -7/3)

End Behavior is determined by HA

As x approaches +/- infinity, y approaches 1

X-intercept ==> y = 0

y = (x - 6) / (x + 4)

0 = (x - 6) / (x + 4)

0 = x - 6 Numerator must be zero

6 = x

X-intercept at point (6,0)

y-intercept ==> x = 0

y = (x - 6) / (x + 4)

y = (0 - 6) / (0 + 4)

y = (-6)/(4) = -3/2

Y-intercept at point (0, -3/2)

Domain:

Skip over values of x that make the denominator = 0

or where the function is not defined (e.g. the hole)

The domain is valid values of x.

Note: x is NOT equal to -1 or 4

Domain: (-infinity, -1) U (-1, 4) U (4, infinity)