QuestionRational Functions, Equations, and Inequalities
NAME Williams 0.
DATE :
181
1212024
K
T
15
c
14
A
112
1) For each function, identify the location of the hole (if applicable), the equation(s) of the vertical asymptote(s) and the equation of the horizontal asymptote.
(a)
(b)
[K-6]
Studdy Solution
STEP 1
What is this asking?
We need to find any holes, vertical asymptotes, and horizontal asymptotes for two given functions.
Watch out!
Don't forget to simplify the functions first!
Holes happen when factors *cancel out* (divide to one), and those canceled factors still affect the graph!
STEP 2
1. Analyze
2. Analyze
STEP 3
Let's **simplify** by **factoring** the denominator.
We've got .
Since we have the same factor of on the top and bottom, we can divide them to one, but *only* when , which means .
So, for .
STEP 4
Since we divided to one, there's a **hole** at .
To find the y-coordinate of the hole, we plug into the simplified function: .
So, the hole is at .
STEP 5
A **vertical asymptote** happens when the denominator of the simplified function is zero.
In our simplified function, , the denominator is zero when .
So, there's a vertical asymptote at .
STEP 6
To find the **horizontal asymptote**, we look at what happens when gets really big (positive or negative).
As gets huge, gets super close to zero.
Therefore, the horizontal asymptote is .
STEP 7
We can **factor** both the numerator and denominator of : .
We can divide to one, as long as .
This gives us for .
STEP 8
Since we divided to one, there's a **hole** at .
Plugging into the simplified function gives us .
So, the hole is at .
STEP 9
The denominator of the simplified function, , is zero when .
So, the **vertical asymptote** is at .
STEP 10
As gets really large, the simplified function approaches , which is just .
So, the **horizontal asymptote** is .
STEP 11
For , there's a hole at , a vertical asymptote at , and a horizontal asymptote at .
For , there's a hole at , a vertical asymptote at , and a horizontal asymptote at .
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