18 RATIONAL FUNCTIONS A RATIONAL FUNCTION is a quotient of polynomials. It will have this form
where g and h are polynomials (h 0). A rational function will have an x-intercept -- y will equal 0 -- only if the numerator g(x) = 0. Consider, for example,
Since the numerator 1 will never be 0, the graph of this function never touches the x-axis.
8 is called a singularity of that function. A singularity of a function is any value of x that would make a denominator 0. Problem 1. In each of the following, which values of x may not be included in the domain? That is, which values are the singularities of the function? To see the answer, pass your mouse over the colored area.
The reciprocal function We are now going to look at the function
This is called the reciprocal function. (It is also the equation of a hyperbola, which, like the parabola, is one of the conic sections.) We are going to see how to draw the graph. We will ask -- and answer -- the following questions. The student should understand that these same questions must be answered when drawing the graph of any rational function. 1. Is the graph symmetrical with respect to the y-axis or to the
We must look at f(−x):
f(−x) = −f(x). Therefore, the graph is symmetrical with respect to the origin. Hence, we need only draw the right-hand side. The left side will follow from the symmetry. 2. Are there any x- or y-intercepts?
No, we cannot, because the numerator 1 is never 0. (Lesson 6 of Algebra.) Therefore there are no x-intercepts. And x may not be 0. Therefore, there are no y-intercepts, either. 3. What happens to the value of y when x is very large?
If x is a very large positive number, then y will be a very small positive number. This means that when we are at the extreme right end of the x-axis, the height of the graph -- the value of y -- will be very close to the x-axis, and on the positive side: 4. Where is the singularity of this function? At x = 0. 5. What happens to the value of y when x is close to a
y becomes extremely large: When x is close to 0 in this case -- when x is a very small positive number -- then y, its reciprocal, will be a very large positive number. Now, we know that there are no intercepts. And x will take every value greater than 0. Therefore, those two ends of the graph must
Asymptotes Briefly, asymptotes are lines that come closer and closer, but never meet. The x-axis is a horizontal asymptote of that graph. The graph and the x-axis come closer and closer, but never touch. In general, "The horizontal line y = b is a horizontal asymptote of the graph of In other words, the distance between the graph and the horizontal asymptote becomes virtually 0. The y-axis is a vertical asymptote of that graph. In general, "The vertical line x = a is a vertical asymptote of the graph of That is, at values of x close to a singularity -- when the denominator is a very small number, almost 0 -- the value of the function becomes an extremely large number. Where, then, do we always find a vertical asymptote of a graph? At a singularity. Problem 2. Write the equation of the vertical asymptote(s) of each of the following.
Problem 3. Each of the following is a translation or transformation of
This is a reflection about the x-axis of graph b).
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