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Understanding the domain of a function is crucial for graphing it accurately. In simple terms, the domain represents all the possible x-values that can be input into the function without causing issues, like dividing by zero. In our example, the function given is \(y=\frac{6}{x+9}-7\).

To find the domain, we set the condition that the denominator \(x+9\) cannot be zero because division by zero is undefined in mathematics. Solving the equation \(x+9=0\), we find \(x=-9\), which means \(x\) can be any real number as long as it is not \(x=-9\). Thus the domain is represented as the union of the intervals \( (-\infty, -9) \cup (-9, \infty) \) which excludes the single point where the denominator would be zero.

When graphing the function, remember that at \(x=-9\), the graph will not be defined and there will be a break. This is important for overall understanding and sketching the accurate behavior of the function across its entire domain.

The x and y intercepts of a graph are the points at which the graph crosses the x-axis and y-axis respectively. These points are helpful anchors in sketching the rough shape of a graph. In the function \(y=\frac{6}{x+9}-7\), to find the x-intercept, we set \(y=0\) and solve the equation for \(x\).

In our example, solving \(0=\frac{6}{x+9}-7\) leads to \(x=\frac{-69}{7}\), indicating the x-intercept. To find the y-intercept, we set \(x=0\) and solve for \(y\), yielding \(y=-\frac{1}{3}\), indicating the y-intercept.

These intercepts allow us to place exact points on the graph where the function crosses the axes. For graphing rational functions, the presence of intercepts can significantly influence the plot shape especially near the axes.

Asymptotes are lines that a graph approaches but never touches. They are critical to understanding the behavior of rational functions and are divided into two types: vertical and horizontal.

The vertical asymptote of a graph corresponds to the values of \(x\) that make the function undefined (where the denominator is zero). In our function, the vertical asymptote is at \(x=-9\). It dictates that as the function's graph gets close to this line on either side, the y-values will increase or decrease without bound.

On the other hand, a horizontal asymptote is a y-value that the function approaches as \(x\) heads towards positive or negative infinity. It's determined by comparing the degrees of the numerator and the denominator in the function. In this case, the horizontal asymptote is at \(y=-7\), based on the constant terms as the \(x\)-values become very large. When graphing, the function will approach, but not cross, this horizontal line. Knowing the asymptotes helps in visualizing the long-term behavior of the function and is an essential step before sketching the graph.