18 ## RATIONAL FUNCTIONSA RATIONAL FUNCTION is a quotient of polynomials. It will look like this:
where A rational function will have an Consider
Since the numerator 1 will never be 0, the graph of that function never touches the Now a denominator may not be 0. The symbol has no meaning. Therefore, in the rational function
, A singularity of a function is any value of the variable that would make a denominator 0.
Problem 1. In each of the following, which values of To see the answer, pass your mouse over the colored area.
The reciprocal function In Topic 8 we saw the graph of the reciprocal function,
That is also the equation of a hyperbola, which, like the parabola, is one of the conic sections. But say that we did not know how to draw the graph. Then we would ask -- and answer -- the following questions. The student should understand that these same questions must be answered when drawing the graph of 1. Is the graph symmetrical with respect to the Answer. We must look at
Since 2. Are there any
No, we cannot, because the numerator is not 0. (Lesson 5 of Algebra.) Therefore there are no And 3. What happens to the value of
If the height of the graph -- the value of 4. Where is the singularity of this function? At 5. What happens to the value of
When Now, we know that there are no intercepts. And
Briefly, an asymptote is a straight line that a graph comes closer and closer to but never touches. The More precisely: "The horizontal line In other words, the distance between the graph and the horizontal asymptote becomes The It is true that the distance between the graph and the vertical axis becomes almost 0. But what it is important is to see that, as a graph approaches a vertical asymptote, its absolute value becomes extremely large. "The vertical line Thus at values of Problem 2. Where do we always find a vertical asymptote of a graph? At a singularity. Problem 3. a) What does the equation of a vertical asymptote look like?
b) What does the equation of a horizontal asymptote look like?
See Lesson 33 of Algebra.
Problem 4. Write the equation of the horizontal asymptote of
Problem 5. Write the equation of the vertical asymptote(s) of each of the following.
Problem 6. Each of the following is a translation or transformation of
This is a reflection about the
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