How do I find the asymptote?

Question

For question 18 on the algebra 2 common core test for June 2016 it says it's false that the graph of the equation c(x)=logBASE6x has an asymptote of y=0. I'm still confused how I would know this.

Answer 1

c(x) = log ₆ X

 

Here is the General equation for the log function:

 

           Y = log _b (X - h) + k

 

The graph of:  y = log _b x will always pass threw the point (1,0) ,

no matter what the base is (this is called the locator point);  

the asymptote will be one unit to the left of the locator pt.:  y-axis

 

The h value moves the entire graph : right if h is positive, left if h is negative

The k value moves the entire graph : up if k is positive, down if k is negative

 

The asymptote is always one unit to the left of the locator point

 

Given your equation:  c(x) = log ₆  x;  h=0 and k=0  NO MOVEMENT OF

LOCATOR POINT  (1,0)

 

Your asymptote will be one unit to the left of (1,0) ;  therefor it is the y-axis

 

 

 

Answer 2

For a>0 (a≠1), the graph of y=log_ax has the y-axis as vertical asymptote, but has no horizontal asymptote.  

 

An equation of the y-axis is x = 0 (not y = 0).   

Answer 3

Examine the given function:

 

y(x) = log₆x

 

The domain in x is (0,∞).  Note that as x → 0, y → -∞.  As x → ∞, y → ∞.  The function does not "blow up" anywhere between the two extremes.  It is also well defined at y = 0:

 

y = log₆x ⇒ 6^y = x

 

For y = 0, 6^y = 6⁰ = 1 = x.  Then you have the point (1,0).  The function is well defined here, so there is no asymptote at y = 0.