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When working with graph transformations in college algebra, it's important to start with the parent function. In this case, we have the reciprocal function, represented as \( y = \frac{1}{x} \). The reciprocal function is unique because it creates a hyperbola. This means that as the input (x) gets larger or smaller, the output (y) values approach zero but never actually reach it.
One of the key features of the reciprocal function is that it has a vertical asymptote at \( x = 0 \) and a horizontal asymptote at \( y = 0 \). This means the graph approaches these lines but never touches them. Understanding these asymptotes helps you predict the behavior of the function as the values of x become very large or very small.

Graphing the reciprocal function forms two separate curves. One is in the first quadrant (where both x and y are positive), and the other is in the third quadrant (where both x and y are negative). Remember that the function is undefined at x = 0, creating the vertical asymptote.

Horizontal translations are a common graphing technique in algebra, which involves shifting the graph of a function left or right. For the given function \( f(x) = \frac{1}{x-3} \), we perform a horizontal translation.
The term \( x-3 \) indicates that we move the parent function \( y = \frac{1}{x} \) to the right by 3 units. In more general terms, any function \( y = \frac{1}{x-a} \) implies a shift of the graph of \( y = \frac{1}{x} \) by \( a \) units to the right if \( a \) is positive, or to the left if \( a \) is negative.

As we shift the function, the vertical asymptote also shifts. Initially, for \( y = \frac{1}{x} \), the vertical asymptote is at \( x = 0 \). After shifting it right by 3 units, the vertical asymptote will now be at \( x = 3 \). This kind of translation does not affect the shape of the graph, only its position on the x-axis.

Vertical asymptotes are important features in the graphs of rational functions and are lines that a graph approaches but never crosses or touches. For the function \( f(x) = \frac{1}{x-3} \), the vertical asymptote can be found by setting the denominator equal to zero and solving for x.
Since \( x - 3 = 0 \), we find that the vertical asymptote is at \( x = 3 \). This means that as the function values approach \( x = 3 \), the output values (y) increase or decrease without bound.

Vertical asymptotes intersect the x-axis but the function does not actually include the x-value at the asymptote, meaning the function is undefined at \( x = 3 \). Identifying and graphing these asymptotes helps you understand how the function behaves near these critical points.

Graphing techniques are essential for visualizing and understanding functions. Here's a simple approach for graphing transformations:

• Start with the parent function. For this problem, it’s \( y = \frac{1}{x} \).
• Identify the transformations. Here, we have a horizontal translation.
• Apply the transformation. For a horizontal shift, move the entire graph right or left.
• Find and plot the vertical asymptote. In this case, it's at \( x=3 \).

Once these steps are complete, plot key points from the parent function shifted appropriately. This will help you sketch the transformed function. Remember to include the asymptotes as they guide the shape of the graph. The resulting graph will have the same shape as the original but will be positioned according to the transformations applied.