Question

Write a rational expression with at least one in the denominator and then analyze it using the following questions:a. Set your expression equal to 0. Does this expression have any solutions? Why, or why not?b. Set your expression equal to and graph the equation on a graphing utility or on your calculator. Are there any values that cannot be? Explain.c. How do the numbers and variables in your equation relate to the features of the graph?d. Identify the denominator of your expression; add 6 to the denominator only. How does the graph change?

Answer 1

A rational expression is an expression of the form a/b. If we want at least one x in the denominator, we can write the following

`(x-1)/(x-5)`

Part a.

If we make the expression equal to 0 and we solve, we get:

`\begin{gathered} (x-1)/(x-5)=0 \\ \\ x-1=0(x-5) \\ x-1=0 \\ x=1 \end{gathered}`

So, it has a solution because the numerator is equal to 0 when x = 1 and x = 1 doesn't make the denominator equal to 0.

Part b.

If we make the expression equal to y, we get:

`y=(x-1)/(x-5)`

Then, the graph of the expression is

So, the expression doesn't have a value for x = 5 and it doesn't have a value of x that makes y = 1.

Part c.

The expression doesn't have a value for x = 5 because the denominator is equal to 0 at x = 5 and it doesn't have a value of x that makes y = 1 because there is no solution to the equation

`(x-1)/(x-5)=1`

Part d.

If we add 6 to the denominator, we get the following expression

`y=(x-1)/(x-5+6)=(x-1)/(x+1)`

Then, the graph is

Therefore, we can see that the vertical asysmptote change from x = 5 to x = -1 because the denominator change from x = -5 to x = 1.