To solve this problem, we'll first need to understand how translating a function affects its equation, and then identify the asymptotes of the translated function.
### Step 1: Understanding the reciprocal parent function
The reciprocal parent function is given by:
[tex]\[ f(x) = \frac{1}{x} \][/tex]
### Step 2: Translation of the function
#### Translation 8 units left:
When a function is translated horizontally, we modify the [tex]\( x \)[/tex]-variable inside the function. Specifically, translating a function 8 units to the left means replacing [tex]\( x \)[/tex] with [tex]\( x + 8 \)[/tex] in the function. Therefore, the function becomes:
[tex]\[ f(x) = \frac{1}{x + 8} \][/tex]
#### Translation 3 units up:
When a function is translated vertically, we add or subtract a constant from the entire function. Translating a function 3 units up means adding 3 to the function. So the function now is:
[tex]\[ f(x) = \frac{1}{x + 8} + 3 \][/tex]
So, the translated function is:
[tex]\[ f(x) = \frac{1}{x + 8} + 3 \][/tex]
### Step 3: Identifying the asymptotes
#### Vertical asymptote:
A vertical asymptote occurs where the denominator of the function is zero because the function approaches infinity at this point. For the function:
[tex]\[ f(x) = \frac{1}{x + 8} + 3 \][/tex]
The denominator [tex]\( x + 8 \)[/tex] becomes zero when:
[tex]\[ x + 8 = 0 \][/tex]
[tex]\[ x = -8 \][/tex]
So, the vertical asymptote is at:
[tex]\[ x = -8 \][/tex]
#### Horizontal asymptote:
A horizontal asymptote is the value that the function approaches as [tex]\( x \)[/tex] tends to infinity. In our translated function:
[tex]\[ f(x) = \frac{1}{x + 8} + 3 \][/tex]
As [tex]\( x \)[/tex] goes to infinity, the term [tex]\( \frac{1}{x + 8} \)[/tex] approaches 0. Therefore, the function approaches the constant term left:
[tex]\[ y = 3 \][/tex]
So, the horizontal asymptote is at:
[tex]\[ y = 3 \][/tex]
### Summary
- The equation representing the translated reciprocal function is:
[tex]\[ f(x) = \frac{1}{x + 8} + 3 \][/tex]
- The vertical asymptote is at:
[tex]\[ x = -8 \][/tex]
- The horizontal asymptote is at:
[tex]\[ y = 3 \][/tex]
This is a detailed, step-by-step solution to translating the reciprocal parent function and identifying its asymptotes.
