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A vertical asymptote is a vertical line that a graph approaches but never touches or crosses. It occurs where the function is undefined. For the function \( f(x) = \frac{4}{x} \), the vertical asymptote is at \( x = 0 \), because division by zero is undefined. This means that as x approaches zero from the right, f(x) will head towards positive infinity. Similarly, as x approaches zero from the left, f(x) will head towards negative infinity.
To visualize this, imagine you are looking at a graph and you see the curve getting steeper and steeper as it gets closer to the y-axis but never quite touching it.
The vertical asymptote divides the hyperbola into two separate branches, one in the positive x region and one in the negative x region.

A horizontal asymptote is a horizontal line that the graph approaches as x moves towards positive or negative infinity. It indicates the end behavior of the function. For the function \( f(x) = \frac{4}{x}\), the horizontal asymptote is \( y = 0\). This is because as \( x \) becomes very large (positively or negatively), \( f(x) \) approaches zero.
In simple terms, imagine stretching the graph infinitely to the left and right; the curve will get closer and closer to the x-axis but never touch it.
This provides valuable information about the function’s behavior at extreme values and helps in sketching the graph accurately.

When sketching the graph of a hyperbolic function like \( f(x) = \frac{4}{x} \), it's important to consider both the vertical and horizontal asymptotes. Follow these steps:

• Identify the Asymptotes: First, determine the vertical asymptote \( x = 0 \) and the horizontal asymptote \( y = 0 \).
• Calculate Key Points: Find several points to get a sense of the curve. For example, \( f(1) = 4, f(-1) = -4, f(2) = 2, f(-2) = -2, f(0.5) = 8, \) and \( f(-0.5) = -8 \). These points show the behavior of the function in both quadrants.
• Plot the Points: Plot these points on a coordinate plane. Remember the shape of a hyperbola involves a curve that gets closer to the asymptotes.
• Draw the Hyperbola: Connect the points smoothly, ensuring the curve approaches but never touches the asymptotes. The graph will form two distinct branches: one in the first quadrant (positive x and y values), and one in the third quadrant (negative x and y values).

The resulting graph shows how the function behaves for various values of x, providing a complete picture of the hyperbolic curve.