Unlike the vertical asymptote, it is permissible for the graph to touch or cross a horizontal or slant asymptote. To find the horizontal or slant asymptote , compare the degrees of the numerator and denominator. Horizontal Asymptote If the degree of x in the denominator is larger than the degree of x in the numerator, then the denominator, being "stronger", pulls the fraction down to the x-axis when x gets big. That is, if the polynomial in the denominator has a bigger leading exponent than the polynomial in the numerator, then the graph trails along the x -axis at the far right and the far left of the graph.
The x -axis becomes the horizontal asymptote. When the degrees of the numerator and the denominator are the same, then the horizontal asymptote is found by dividing the leading terms, so the asymptote is given by:. The horizontal asymptote may also be approximated by inputting very large positive or negative values of x. Slant Asymptote If the numerator is one degree greater than the denominator, the graph has a slant asymptote. Using polynomial division, divide the numerator by the denominator to determine the line of the slant asymptote.
Finding Intercepts To find x - or y -intercepts, set the other variable equal to zero and solve in turn. Plotting Points Based on information gained at this point, select x -values and determine y -values to create a chart of points to plot. Select more plots in areas where you think you need information to inform your curve. Sketching the Graph Once the points are plotted, remember that rational functions curve toward the asymptotes. Include additional points to help determine any areas of uncertainty.
In addition, graphing calculators are often used in conjunction with sketches to define the graph. Example Graph:. Find the vertical asymptotes. Sketch these as dotted lines on the graph. Find the horizontal or slant asymptotes. There is no slant asymptote. Sketch this on the graph. Also, although the graph of a rational function may have many vertical asymptotes, the graph will have at most one horizontal or slant asymptote.
It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction. For instance, if we had the function. The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator.
Find the horizontal asymptote and interpret it in context of the problem. Both the numerator and denominator are linear degree 1. Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. In the numerator, the leading term is t , with coefficient 1. In the denominator, the leading term is 10 t , with coefficient The horizontal asymptote will be at the ratio of these values:. First, note that this function has no common factors, so there are no potential removable discontinuities.
The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The numerator has degree 2, while the denominator has degree 3. A rational function will have a y -intercept when the input is zero, if the function is defined at zero. A rational function will not have a y -intercept if the function is not defined at zero. Likewise, a rational function will have x -intercepts at the inputs that cause the output to be zero.
Since a fraction is only equal to zero when the numerator is zero, x -intercepts can only occur when the numerator of the rational function is equal to zero. We can find the y -intercept by evaluating the function at zero.
The x -intercepts will occur when the function is equal to zero:. Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a rational function. Then, find the x — and y -intercepts and the horizontal and vertical asymptotes. Skip to main content. Rational Functions. Search for:. Identify vertical and horizontal asymptotes By looking at the graph of a rational function, we can investigate its local behavior and easily see whether there are asymptotes.
Vertical Asymptotes The vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not common to the factors in the numerator. How To: Given a rational function, identify any vertical asymptotes of its graph. Factor the numerator and denominator. Note any restrictions in the domain of the function. Reduce the expression by canceling common factors in the numerator and the denominator. Note any values that cause the denominator to be zero in this simplified version.
These are where the vertical asymptotes occur. Note any restrictions in the domain where asymptotes do not occur. These are removable discontinuities.
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