Question

Sketch the graph of the function. $$y=\frac{2}{x-6}+9$$

Short answer

The graph of the function \(y=\frac{2}{x-6}+9\) has a vertical asymptote at \(x=6\) and a horizontal asymptote at \(y=9\). The function is defined for all values except \(x=6\).

Step-by-step solution

Identify the Vertical Asymptote

The vertical asymptote corresponds to the value that makes the denominator of the fraction equal to zero. Thus, set \(x-6=0\) to find \(x=6\). This means the graph will approach but never touch the line \(x=6\).

Identify the Horizontal Asymptote

The horizontal asymptote is the value that the graph approaches as \(x\) tends to positive or negative infinity. For this type of function, the horizontal asymptote is the constant term, which is 9 in this case. So, the graph will approach but never touch the line \(y=9\).

Plot Relevant Points

Choose some points for \(x\) that does not make the denominator zero and compute corresponding \(y\). You can choose points from both sides of the vertical asymptote. For example, choose \(x=5\) and \(x=7\) to get the points (5,11) and (7,11), respectively. These points can guide you while sketching the graph.

Sketch the Graph

Now, sketch the graph using the identified asymptotes and the points. On either side of the vertical asymptote (\(x=6\)), the graph should approach the horizontal asymptote (\(y=9\)).