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When we talk about horizontal shifts, we're looking at how we can slide a graph left or right along the x-axis. It's a simple transformation, but it can significantly alter the appearance of a function. For a function defined as \(f(x)\), shifting it to the right by \(k\) units means we replace every \(x\) in the function with \(x - k\).
This may seem counterintuitive at first, but remember, we're essentially telling the function, "Now, respond as though the x-values have been increased by \(k\)." For example, if your original function is \(f(x) = \frac{1}{x}\), and you want to shift it 3 units to the right, the new function becomes \(f(x) = \frac{1}{x - 3}\).
The idea is simple yet profound, affecting the domain and the location of vertical asymptotes of rational functions. So, the horizontal shift doesn't change the y-values directly but instead affects where the x-values "start."
This kind of manipulation is powerful for graphing functions exactly where you need them.

• Shift right: replace \(x\) with \(x - k\).
• Shift left: replace \(x\) with \(x + k\).

In contrast to horizontal shifts, a vertical shift moves a graph up or down along the y-axis. This transformation changes the output of the function directly. If you have a function \(f(x)\), shifting it vertically involves adding or subtracting a constant from the function.
When we shift down, we subtract a number from the function to lower all its y-values by that amount. For instance, with \(f(x) = \frac{1}{x}\) and a vertical shift down by 4 units, your new function becomes \(f(x) = \frac{1}{x} - 4\).
This transformation maintains the same shape of the graph but moves every point on it vertically downward, affecting features like horizontal asymptotes and intercepts directly.
Unlike horizontal shifts, the variable inside the function remains unchanged.

• Shift up: add a number to the function \(f(x) + c\).
• Shift down: subtract a number from the function \(f(x) - c\).

Rational functions are a unique category in mathematics, characterized by variables in the denominator. A classic example is \(f(x) = \frac{1}{x}\). These functions can become undefined if the denominator equals zero, creating vertical asymptotes where the function can "shoot off" to infinity.
Importantly, rational functions can also have horizontal asymptotes, which represent a line that the function approaches but never quite reaches as \(x\) becomes very large or very small.
Transformations such as horizontal and vertical shifts can change the location of these asymptotes but not their general behavior. For instance, in the equation \(f(x) = \frac{1}{x - 3} - 4\), we have shifted the original function both horizontally and vertically.

• The denominator \(x - 3\) shows a horizontal shift right, moving the vertical asymptote to \(x = 3\).
• The subtraction \(- 4\) indicates a vertical shift downward, moving the horizontal asymptote to \(y = -4\).

Understanding rational functions and their transformations helps us not only graph them effectively but also predict behaviors across their domains.