Sketch the asymptotes and graph the function y equals start fraction 6 over left parenthesis x minus 2 right parenthesis end fraction plus 4.
To sketch the asymptotes of the function \(y = \frac{6}{{(x-2)}} + 4\), we need to identify the vertical and horizontal asymptotes.
Vertical asymptote:
The vertical asymptote occurs when the denominator of the fraction equals zero. In this case, the vertical asymptote occurs at \(x = 2\) because when \(x = 2\), the denominator \((x-2)\) becomes zero.
Horizontal asymptote:
To find the horizontal asymptote, we need to look at the leading terms of the numerator and the denominator as \(x\) approaches positive or negative infinity. In this case, the leading term of the numerator is 6, and the leading term of the denominator is \(x\). Therefore, the horizontal asymptote is \(y = 0\).
Now, let's plot the asymptotes and sketch the graph of the function.
- Vertical asymptote at \(x = 2\)
- Horizontal asymptote at \(y = 4\)
By plotting these asymptotes and considering the behavior of the function near the asymptotes, we can sketch the graph of the function.
The graph of the function \(y = \frac{6}{{(x-2)}} + 4\) will approach the vertical asymptote at \(x = 2\) and the horizontal asymptote at \(y = 4\) as \(x\) approaches positive or negative infinity.