Exercise 32 Page 711

We are asked to write two rational functions with graphs that are identical except that one is shifted vertically with respect to the other. In order to complete this task, we will first write one rational function. Next, we will modify its rule, so that the graph of the new function is shifted vertically with respect to the first function.

First Function

A rational function is a function that can be written as a fraction whose numerator and denominator are polynomials. In particular, some rational functions can be written in the following form. y=a/x- b+ c We will arbitrarily choose the values of a, b, and c. Let's use a= 3, b= - 1, and c= 1. y=3/x-( - 1)+ 1 ⇔ y=3/x+1+1 Functions in this form have a vertical asymptote at x= b and a horizontal asymptote at y= c. Therefore, our function has a vertical asymptote at x= - 1 and a horizontal asymptote at y= 1. Here is its graph.

Second Function

To write the rule of the second function we have to modify the rule of the first function. The graph of the obtained function should be vertically shifted 3 units with respect to the graph of y= 3x+1+1. The graph of the first function can be either moved up 3 units or down 3 units.

A graph of a function is shifted vertically when a constant is added to or subtracted from the function's rule. If we add 3 to the function's rule the graph of the obtained function is moved up with respect to the original function. y=3/x+1+1 + 3 ⇔ y=3/x+1+4 If we subtract 3 from the function's rule the graph of the obtained function is moved down with respect to the original function. y=3/x+1+1 - 3 ⇔ y=3/x+1-2 We have found two equations of rational functions with graphs that are identical expect that one is shifted vertically 3 units with respect to the other. y=3/x+1+1 and y=3/x+1-2 Note that there are infinitely many pairs of such functions. Here are a few examples.