How To Find Horizontal Asymptotes

Definition

A Horizontal Asymptote is a type of asymptote that is a line which a graph approaches but never touches. It is typically represented as a straight line running along the x-axis. This line is a standard way of displaying a graph’s rate of change as it approaches infinity and usually appears as an equation of the form y = mx + b, where m is the slope and b is the y-intercept.

Finding Horizontal Asymptotes

Horizontal Asymptotes can be found by looking for patterns in the coefficients of the equation that represent the graph. Generally speaking, when the degree of the polynomial (the highest power of x in the equation) is even, the x-axis is the horizontal asymptote. When the degree is odd, the asymptote will be some line parallel to the x-axis.

For example, the equation y = 4×2 – 3x + 5 has an even degree, so the x-axis is the horizontal asymptote. The equation y = 5×3 + 6×2 – 8x + 3 has an odd degree, so its horizontal asymptote would be some line parallel to the x-axis.

If the equation can be written in its complete form, then it should have a constant line on the right-hand side of the equation. The horizontal asymptote can then be found by solving for that line.

When the equation is written in its simplified form, then it should be placed in direct form, which has only the highest power of x on the left side (e.g. y = axb + c). Then the horizontal asymptote can be found the same way as the complete form by solving for the constant line on the right-hand side.

Conclusion

The process of finding Horizontal Asymptotes is simple when the equation is written down in an organized way. Pay attention to the equation’s degree, written form, and any constants on the right-hand side to determine the equation’s asymptote. It is important to remember the basic equation of y = mx + b and its relation to horizontal asymptotes. With practice and patience, anyone can easily find the appropriate asymptote for a given equation.