Activity: Graphs of Rational Functions

Objectives:

· Understand definition of a rational function.

· Understand how to find the vertical asymptotes in the set of real numbers of a rational function.

· Understand that the quotient of the division indicated in a rational function is the end behavior asymptotic function.

· Connect how you can tell what the type of polynomial the end behavior asymptotic function will be by the difference in degrees of the numerator and the denominator when the numerator is of higher degree.

· Understand that y = 0 will be the end behavior asymptote when the degree of the numerator is lower than that of the denominator.

• A rational function is a function which is composed of a polynomial divided by another polynomial. In the definition we will assume the denominator polynomial has degree at least 1 when the ratio of the two polynomials is completely reduced.

Examples:

• The numerator polynomial is the trivial monomial, 1. The denominator is the monomial x.

• 
• = (for x not equal to –1) is considered rational with numerator polynomial, 3x-2 and denominator polynomial, x-1. However, (for x not equal to –2/3 ) we will not necessarily consider rational since in reduced form the denominator is trivially equal to 1 and the function is a line with the point at x = -2/3 left out.

• The domain of a rational function will exclude all values which make the denominator equal to zero.
• Composed of division, rational functions have a divisor, a dividend, a quotient, and a remainder.
• = The divisor is 2x-7. The dividend is 4x² + 3x+1. The quotient is 2x² + 7x + 26. The remainder is 183.
• = 0 + . The divisor is x. The dividend is 1. The quotient is 0. The remainder is 1.

• Rational Functions often have asymptotes. An asymptote is a line in which a function approaches arbitrarily closely. You may say that the function levels off around the asymptote. Rational functions may have other curves that they approach arbitrarily closely. We call these other curves, asymptotic curves of the rational function.

o Vertical asymptotes are vertical lines that a rational function approaches arbitrarily closely. The vertical asymptotes occur at the x values in which the denominator equals zero. That is, as values of x get very close to numbers which make the denominator equal to 0, dividing by values very close to zero cause the value of f(x) to soar to positive or negative infinity. Vertical asymptotes are vertical lines and should be expressed as x = appropriate x-value.

• Rational functions also have end behavior asymptotes. The y = the quotient is the function which is the end behavior asymptotic curve for any rational function.
• Example: has y = 2x² + 7x + 26 as an end behavior asymptotic curve. As |x| gets very large, the graph of f(x) levels off around the graph of y = 2x² + 7x + 26.

• If the degree of the denominator is larger than the degree of the numerator, the rational function has y = 0 as a horizontal asymptote since the quotient is 0 as the denominator overpowers the numerator.

• If the degree of the denominator equals the degree of the numerator, the rational function has y = a_n/b_n as a horizontal asymptote. a_n is the leading coefficient of the numerator and b_n is the leading coefficient of the denominator. The numerator and denominator somewhat balance each other out.

• If the degree of the numerator is exactly one higher than the degree of the denominator, the rational function has a slant or oblique asymptote with a defined non-zero slope. This line can be found by completing the long division suggested by the rational function. The result subtract the remainder is the slant asymptote.

• If the degree of the numerator is two higher than the degree of the denominator, then the rational function has an asymptotic parabola. You can extend this idea to an asymptotic cubic ….asymptotic quartic….

More Examples:

Example 1: By the way, if you complete the long division you will obtain quotient x² – 6x –72.5 and remainder 812.5. Thus,

=

• Now look at the graph of f(x)

and zoomed out

• Do you notice a connection between the difference in degrees between the numerator and denominator, the quotient obtained when the division is completed, and the graph ?
• This function has an asymptotic parabola. The graph of f(x) levels off around y = x²- 6x-72.5 as |x| ®µ.
• The vertical asymptote is shown also in the graph. The vertical asymptote is x = -2/5.

Example 2: =

and zoomed out

• Do you notice a connection between the difference in degrees between the numerator and denominator, the quotient obtained when the division is completed, and the graph ?
• This function has an asymptotic line. The graph of f(x) levels off around y = 2x - 27 as |x| ®µ.
• This function does not have any vertical asymptotes. This is because the denominator, x² + 10 never equals 0 in the set of real numbers.

Example 3:

and zoomed out

• Do you notice a connection between the difference in degrees between the numerator and denominator, the quotient obtained when the division is completed, and the graph ?
• This function has an asymptotic cubic function. The graph of f(x) levels off around x³ –6x² + 6x – 7 as |x| ®µ.
• The vertical asymptote is shown also in the graph. The vertical asymptote is

x = -1.

Example 4: =

o This function has the line y = 0 as a horizontal asymptote. As |x| ®µ , f(x) ® 0.

o The vertical asymptote is shown also in the graph. The vertical asymptote is

x = 8.

Example 5: A rational function has x² + x + 1 as an end behavior asymptote and x = 3 and x = -3 as vertical asymptotes. State a formula which could be this rational function.

= and = are two of infinitely many correct formulas. The quotient needs to be x² + x + 1 and the denominator needs to be a(x² – 9).

By the way, = =

Supplementary Exercises:

1) f(x) =

and zoomed out

A) State the divisor, the dividend, the quotient, and the remainder.

B) This function has an asymptotic _______________________. The graph of f(x) levels off around ___________________ as |x| ®µ.

C) The two vertical asymptotes are shown also in the graph. The vertical asymptotes are ______________.

D) The domain of this function is the set __________________________.

2)

A) State the divisor, the dividend, the quotient, and the remainder._______________

____________________________________________________________________

B) This function has a horizontal asymptote. The graph of f(x) levels off around ___________________ as |x| ®µ.

C) The vertical asymptote is shown also in the graph. The vertical asymptote is ______________.

D) The domain of this function is the set __________________________.

3) Given

A) Complete division and state quotient and remainder.

B) Find all asymptotic functions (vertical and end behavior)

C) Sketch the graph of f(x) from what you learned in A) and B).

Label the asymptotic functions and x or y-intercepts.

D) State the domain of f(x).

4) Given

A) Complete division and state quotient and remainder.

B) Find all asymptotic functions (vertical and end behavior)

C) Sketch the graph of f(x) from what you learned in A) and B).

Label the asymptotic functions and x or y-intercepts.

D) State the domain of f(x).

5) Given

A) Complete division and state quotient and remainder.

B) Find all asymptotic functions (vertical and end behavior)

C) Sketch the graph of f(x) from what you learned in A) and B).

Label the asymptotic functions and x or y-intercepts.

D) State the domain of f(x).

6) I am thinking of a rational function with slant asymptote y = -4x+5 and two vertical asymptotes which are x = 12 and x = -4. Name a function formula which fits this description. You may leave the formula in quotient/remainder form.

7) I am thinking of a rational function with an end behavior cubic asymptotic function y =

x³ + 2x² – 4x + 2 and two vertical asymptotes which are x = 0 and x = 1. Name a function formula which fits this description. You may leave the formula in quotient/remainder form.

8) I am thinking of a rational function with an end behavior parabolic asymptotic function y = x² and three vertical asymptotes which are x = 0 , x = -4, and x = 4. Name a function formula which fits this description. Write this formula in standard rational form.