Question

Sketch the graph of the function. $$y=\frac{1}{x-6}-1$$$$y=\frac{1}{x-6}-1$$

Short answer

The graph of the function \( y=\frac{1}{x-6}-1 \) will have a vertical asymptote at \( x=6 \), a horizontal asymptote at \( y=-1 \), and will be above the horizontal asymptote for \( x>7 \) and below it for \( x<7 \).

Step-by-step solution

Identify the vertical asymptote

The vertical asymptote is found by setting the denominator equal to zero and solving for \( x \). In this case, it is \( x-6=0 \), which gives us \( x=6 \), so the vertical asymptote is \( x=6 \). This means the function approaches infinity as \( x \) approaches 6 from either the left or right.

Identify the horizontal asymptote

As \( x \) approaches positive or negative infinity, the \( \frac {1}{x-6} \) becomes increasingly small, so the function will be approaching -1. Hence, the horizontal asymptote is \( y=-1 \). This means as \( x \) moves to the far right or left, the function gets closer and closer to \( y=-1 \).

Identify the overall trend

Because the function \( \frac{1}{x-6} \) is positive for \( x>6 \) and negative for \( x<6 \), the function \( \frac{1}{x-6} - 1 \) will be positive for \( x>7 \) and negative for \( x<7 \). This means the function will be above the horizontal asymptote for \( x>7 \) and below it for \( x<7 \).

Sketch the graph

Taking into account the vertical and horizontal asymptotes identified and the overall trend, you can now sketch the graph. The graph will get close to, but never touch, the vertical asymptote at \( x=6 \) and the horizontal asymptote at \( y=-1 \). The graph will be above the horizontal asymptote for \( x>7 \) and below it for \( x<7 \).