Answer:
To find the horizontal asymptote of the function \(y = \frac{{3x - 6}}{{x + 2}}\), we can analyze the behavior of the function as \(x\) approaches positive and negative infinity.
As \(x\) approaches positive infinity (\(x \to \infty\)), the term with the highest degree in the numerator (which is \(3x\)) becomes dominant. Similarly, in the denominator, the term with the highest degree (which is \(x\)) also becomes dominant. Therefore, we can simplify the function as follows:
\[y = \frac{{3x - 6}}{{x + 2}} \approx \frac{{3x}}{{x}} = 3\]
As \(x\) approaches positive infinity, the function approaches a constant value of \(y = 3\). Hence, the horizontal asymptote of the function is \(y = 3\).
Note that the given x-intercept and y-intercept are not necessary for finding the horizontal asymptote. They are useful for other analysis, such as finding the x- and y-intercepts, but they do not directly affect the horizontal asymptote.