2.7 Graphs of Rational Functions

Guidelines for Graphing Rational Functions

Let , where N(x) and D(x) are polynomials that have no common factors. To graph this function,

1. Find the y-intercept by solving for x = 0

2. Solve for N(x) = 0. The real solutions are the x-intercepts.

3. Solve for D(x) = 0. The real solutions are the vertical asymptotes.

4. Find the horizontal asymptote

5. Plot each of these on the graph, representing the intercepts as points, and the asymptotes as dashed lines.

6. Plot a point between and a point beyond each x-intercept and vertical asymptote, if necessary. Connect the points with smooth curves between and beyond the vertical asymptotes

Example

1. Y-intercept(s): f(0) gives no y-intercepts

2. X-intercept(s): N(x)=0 gives no x-intercepts

3. Vertical asymptotes: x=0

4. Horizontal asymptotes: y=0

5 and 6. Graph. Both asymptotes are along the axes in this example, so they are not easily seen.

A basic rule of thumb for graphs of this type is that around the vertical asymptotes, the graph often does opposite things on either side (for example, the graph above falls on the left of the vertical asymptote, and rises on the right). This is not always true, but we will learn about the predictable exceptions to this rule later.

Hole in a graph

Sometimes, holes can be found in graphs. This occurs when the numerator and denominator of the rational function have exactly the same factor.

For example, the function can be written as , in which the factor (x-1) can be found in the numerator and the denominator. If this is cancelled out in the numerator and the denominator, we are left with the function , just like in the graph above. But what is different about these two graphs?

The answer is that there is a hole in the graph of at x = 1, the location of the eliminated factor (x-1). When looking at this graph in a graphing calculator, it is useful to view the zoom decimal window, in order to best see this hole. When graphing this type of function by hand, the graph can be drawn with an open circle in the line at x = 1.

It is easy to tell that there will be a hole in a graph, if, when graphing the function, there is an x-intercept and a vertical asymptote at the same place. At this point in the graph, the intercept and asymptote should be removed, and at this location, there should be a hole.

2-6 Rational Functions and Asymptotes

Rational Function is a function that can be written as

F(x)= N(x)/D(x)

Where N(x) and D(x) are polynomials.

Asymptotes of Rational Functions

1) The graph of F has the vertical asymptotes at the zeros of D(x).

2) The graph of F has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x).

a) If n < m, the line y=0(x axis) is a horizontal asymptote at the zeros of D(x)

b) If n=m, the line y=a/b is a horizontal asymptote.

c) If n>m the graph of f has no horizontal asymptotes.

As or the distance between the horizontal asymptotes and the points on the graph must approach zero.

Example 1

Graph of 1/x In the graph 1/x the Horizontal Asymptote is the X axis and the Vertical Asymptote is the Y axis. Because the denominator is zero when x=o, the domain is of F is all real numbers except x=0.

Example 2:

Graph of 1/(x-3)

Finding the Vertical asymptote:

D(x)= x-3

0=x-3

x=3

3 is a zero of the function, so the vertical asymptote on the graph

Example 3

Graph of 1/(x-3) +2

Vertical Asymptote

D(x)= x-3

X=3

Vertical Asymptote= 3

Horizontal Asymptote

Horizontal Asymptote= 2

If the exponent is greater in the denominator than in the numerator, the horizontal asymptote will be x=o as the denominator approaches infinity. Therefore in this example, as 1/(x-3) approaches zero, the horizontal asymptote is two, f(x) approaches 2.

 

2.5 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra:

If has a polynomial degree of , the number of zeros is also

Conjugate Pairs:

and

Finding a Polynomial with Given Zeros:

Example:

The Imaginary Unit, i

The equation to the left has no real solution because there is no real number x that be squared to equal -3. For that reason, there is an imaginary unit, i.

You can create a set of complex numbers by adding real numbers to real multiples of i, which
can be written in standard form.

Standard form: a+bi

i equals:

There is a pattern to this, i to the fifth degree equals i, i to the sixth degree equals -1, and so on.

This pattern will help you solve equations like i^170. You can divide 170 by four (because i to the fourth equals one), and the remainder is two. i to the second power equals -1, which is your answer.

Examples Problems

Put i in standard form:

1.

** the -2i^2 turns into a +2 because you times it by (-1)

2.

Polynomial Functions of Higher Degree

Graphs of Polynomial Functions

The graph of a polynomial function is continuous, which means that it has no breaks, holes, or gaps. Also, they have smooth rounded graphs, without any sharp turns.

D=0 D=1

D=2 D=3

D=4 D=5

Zeros of Polynomial Functions

For a polynomial function of degree n, the following are true:

• The graph of has at most n real zeros
• The function has at most n-1 relative extrema (relative minimums or maximums)

Minima and Maxima are the plural form of minimum and maximum.

End Behavior

End behavior is what happens at the left and the right of the graph. Instead of simply using words, notation is used.

For the following graph:

Translation:

As x approaches infinity, approaches infinity

As x approaches negative infinity, approaches negative infinity

For a polynomial function of degree n, if n is...

• Even, the end behaviors are the same
• Odd, the end behaviors are different
• Even and the leading coefficient is positive, the graph rises to the left and right
• Even and the leading coefficient is negative, the graph falls to the left and right
• Odd and the leading coefficient is positive, the graph falls to the left and rises to the right
• Odd and the leading coefficient is negative, the graph rises to the left and falls to the right

ex.

Both ends of the graph would rise or

because n is even, and the leading coefficient is positive.

The graph would rise to the left and fall to the right or

because n is odd and the leading coefficient is negative.