Question

Sketch the graph of the function. $$ y=3^{x-1} $$

Short answer

After conducting the above procedure correctly, the graph of the function \( y=3^{x-1} \) should demonstrate an exponential growth, crossing the x-axis at x = 1, and having a horizontal asymptote at y = 0.

Step-by-step solution

Understanding the Function

First, it is essential to identify the type of function. The function \( y=3^{x-1} \) is an exponential function, which is generally characterized by rapid growth or decay.

Identify Key Features of the Function

Two features are of particular importance when sketching the graph: its intercepts and asymptotes. Here, the x-intercept is x=1 because if we set \( y = 0 \), we get \( x = 1\). There is no y-intercept because the x doesn't equal to 0. As for the horizontal asymptote, it exists at \( y = 0 \) because as x approaches negative infinity, y will approach 0.

Draw the Graph

Start off by marking the x-intercept on the x-axis. Draw a horizontal asymptote at \( y = 0 \). Sketch a curve that starts from the horizontal asymptote in the third quadrant, goes through the intercept, and continues to grow to the upper right of the second quadrant.

Key concepts

Exponential Function Growth

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.

Graph X-intercepts

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.

Horizontal Asymptotes

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.