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Understanding exponential function growth is essential when dealing with problems like sketching the graph of the function y=3^{x-1}. An exponential function is described by an equation where the variable appears as the exponent. In this case, the base is 3, a positive number greater than 1, which indicates that as x increases, the value of y grows exponentially.

Exponential functions are unique because they increase at a rate proportional to their current value, leading to a rapid growth as the input gets larger. This concept is captured visually in the steepness of the graph as you move to the right along the x-axis. To master these functions, remember that the bigger the base (greater than 1), the steeper the curve and the more quickly the function will grow.

Graphing x-intercepts is a critical step when sketching the graph of a function like y=3^{x-1}. The x-intercept(s) are the point(s) where the graph crosses the x-axis. This occurs when the output of the function, y, is zero.

To find the x-intercept of an exponential function, you set the output - in our example, y - equal to zero and solve for x. For the function y=3^{x-1}, we calculate that the x-intercept is at x=1. This signals that the graph will touch the x-axis at the point (1, 0). Notably, exponential functions will have at most one x-intercept because they are always increasing or decreasing and thus can only cross the x-axis once.

A horizontal asymptote is a horizontal line that the graph of a function approaches as x moves towards positive or negative infinity. In the context of the function y=3^{x-1}, the graph has a horizontal asymptote represented by the line y=0.

This line is a boundary that the function will never cross as x trends towards negative infinity. It effectively represents the 'limit' as to how low the function's graph will go without ever actually touching the asymptote. Even as the value of x gets increasingly negative, the value of y will approach, but never reach, zero. This concept helps to understand the behavior of exponential functions at the extremes and is important for correctly sketching their graphs.