CW 2.6 Name: ________________________________ Rational Functions Hour: _____ Precalculus Rumppe Rational functions are just functions that are fractions – basically, there’s at least an “x” in the bottom. Let’s make sure we’re OK with classic fractions first. 1. !

!+ !

!        = 2. !

!− !

!"        = 3. !

!∙ !!        = 4. !

!÷ !

!        =

All the same rules still apply:

• When adding or subtracting, the denominators need to be the same. Multiply by clever forms of one to make this happen.

• When dividing, multiply by the reciprocal of the divisor (flip the 2nd fraction).

• Thou shalt not dividest by zero.

• Multiplying is still the easiest operation.

So now try some problems involving rational functions. If you have trouble with factoring quadratics, you’re in big trouble. Try Khan Academy or the Math Center if you need to brush up. Ex. !!!

!!  !!!!"− !

!!! 1. !

!!  !!!!+ !

!!! 2. !

!(!!  !!)− !

!!!

= !!!

(!!!)(!!!)− !

!!!∙ !!!!!!

; (x ≠ {-5, 6} = !!!

(!!!)(!!!)− !!!!"

(!!!)(!!!) ; (x ≠ {-5, 6}

= !!!!"

(!!!)(!!!) ; (x ≠ {-5, 6}

3. 5− !!!!

4. !!!!!!

− !!!!

− !"!!!!!!!

5. !!!!!  !!!!!!"

∙ !!!!!!!

!!!!!!!

6. !!!!!

!!!!÷ !"!!!!

!! 7. !

!!!!!"!!!

÷ !!!!!!"#!!"!!!"

÷ !!!!"!!!!!!

The Rational Function Family As with all function families, there’s a parent function. The parent function for rational functions is 𝑦 =   !

!.

7. Create a table and plot the points on the grid.

Notice the asymptotes! The function never crosses the lines x = 0 (the vertical asymptote) and y = 0 (the horizontal asymptote).

In a graphing calculator, explore what the following functions look like, and get to know them. Sketch each function in the grid provided, plot the requested points, and write the equations of the asymptotes. 8. 𝑦 = !!

! 9. 𝑦 = !

!!

Be sure to plot (-1, 1) and (1, -1) Be sure to plot (-1, 1) and (1, 1) Vertical asymptotes: Vertical asymptotes: Horizontal asymptotes: Horizontal asymptotes:

10. 𝑦 = !

!!! 11. 𝑦 =   !!

!!!

Be sure to plot (-4, -1) and (-2, 1) Be sure to plot (-1, 1), (1, 3), (3, -3), (5, -1) Vertical asymptotes: Vertical asymptotes: Horizontal asymptotes: Horizontal asymptotes:

Before we get into those asymptotes, let’s get some general ideas down. In the equations below, the parmeter a stands for any value. For this exercise, imagine a is positive. 12. A function of the form 13. A function of the form 𝑦 =   !

! looks something like this: 𝑦 =   !

!! looks something like this:

14. A function of the form 15. A function of the form 𝑦 =  !!

! looks something like this: 𝑦 =  !!

!! looks something like this:

Asymptotes – these are lines that aren’t actually a part of the graph of the function, but are barriers that define the end behavior of a function. As the value of x gets further from zero, the function never reaches the asymptote but comes very close to it. For instance, in a graphing calculator, graph the function 𝑦 =   !

!!!. It has a vertical asymptote at

x = 2. The value x = 2 is “not defined” for the function (otherwise you’d be dividing by zero) and therefor the domain is D:{(-∞, 2), (2, ∞)}. (Those are all x-values when we write the domain in “interval notation” by the way.) But as we investigate points very close to x = 2 (like say, x = 2.0001) we get values that approach infinity. For this function, we get values that approach negative infinity by looking at values on the other side of x = 2 (like x = 1.9999). We can determine the vertical asymptotes by looking at the denominator and seeing what values for x make the denominator zero. Horizontal asymptotes occur when we see what the value of a rational function approaches at the far left and right of the graph. Typically you can enter very large (+ and - ) values to find the horizontal asymptote, but there are some short cuts we’ll look at later. For the function 𝑦 =   !

!!!

the horizontal asymptote is y = 0. If we enter x = -1,000,000, the value of the function is y = -0.000001…, or basically zero. 16. Graph the function 𝑦 =   !!!!"

!!! in a graphing calculator. The vertical asymptote is easy to

find. It’s got to be x = 2, since this value makes the denominator zero. Look far to the left and right at the graph to find the horizontal asymptote. What is it? Tricks to find horizontal asymptotes:

• If a rational function has a higher degree on the denominator, the horizontal asymptote is y = 0.

• If a rational function has the same degrees for both the numerator and denominator, the horizontal asymptote is the ratio of the leading coefficients.

Ex. Find the horizontal asymptote of the function 𝑦 =   !!!!!

!!!!!!!!.

Since the degrees on the top and bottom are the same, the horizontal asymptote is 𝑦 =   !

!.

Find the horizontal asymptotes of the following functions: 17. 𝑦 =   !!

!!!!!!!!!!!!!!

18. 𝑦 =   !!!!!!!!!!!!!

19. 𝑦 =   (!!!)(!!!!)(!!!!)(!!!!)

x-intercepts – These are values of the function that make y = 0. For a rational function to equal zero, the numerator must be zero (for instance, !

! ). Factoring functions allow these to be easy to

find. Ex. What are the x-intercepts of the function = !!!!!!!"

!!!!!!!" ?

Factor: 𝑦 =   (!!!)(!!!)

(!!!)(!!!)

x-intercepts are easy to see, at (5, 0) and (-8, 0) since x = 5 and x = -8 make the numerator zero. Just for fun, the vertical asymptotes are the zeros of the denominator, x = 6 and x = -2. The horizontal asymptote is y = 1, since the degrees on the top and bottom match, and both leading coefficients are 1. Holes – these are open values of the graph, simply a “hole” in the curve, and occur when the same zero occurs in both the numerator and denominator. These values are never x-intercepts or vertical asymptotes. To find the location of the holes, cancel out the “like” factors and enter the x-value of the hole into the remaining terms. Ex. Find the hole of the function =   !!

!!!"!!!"!!!!!!!"

. Factor: 𝑦 =   !(!!!)(!!!)

(!!!)(!!!) The hole is when x = -3

Cancel: 𝑦 =   !(!!!)

(!!!)

Enter x = -3: 𝑦 =   !(!!!!)

(!!!!)=   !(!!)

!!=   !"

!= !

!

The hole is at (-3,!

! )

20. Find the following for the function 𝑦 =   !!

!!!!!!"!!!!!!!!"

• Equations of horizontal asymptotes

• Equations of vertical asymptotes

• Coordinates of x-intercepts

• Location of holes.