Activity: Analyzing Local and Global Behavior of Rational Functions

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Activity: Analyzing local and global behavior of rational functions

A Review of Local Behavior and Introduction to Jump Discontinuity

x  p When we observe the local behavior of a function f (x ) around a specific value , p we look to see if f (x ) approaches a particular value as x approaches from two directions - the left and the right.

Consider the graph of the function f (x ) shown below.

Say we wish to examine the local behavior of the function f (x ) around the value x  1 . We start by examining the behavior of the function as x approaches 1 from the left (i.e.  x  1 ).

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In the above example, we say that "as x approaches 1 from the left, f (x ) approaches 1"  (i.e. if x  1 , then f (x )  1 ). Likewise "as x approaches 1 from the right, f (x )  approaches 3" (i.e. if x  1 , then f (x )  3 ).

Since the function doesn't approach the same value from both sides of x  1 , we say that f (x ) has a jump discontinuity at x  1 . If the ant wished to switch positions from one side of x  1 to the other, we think she would find this terminology quite appropriate!

Rational Functions Activity 1

In this activity, you will use your graphing calculator to explore rational functions - f (x ) functions of the form y  , with f (x ) and g(x ) polynomials. g(x )

The table on the next page contains several rational functions. Complete the missing values of the table by analyzing the local behavior of each function at the location p specified in the table.

Behavior of f (x ) as  p Behavior of f (x ) as x  p Rational Function  x  p (from left) (from right) 1 3 f (x )   f (x )   x  3 1 2 3 x  3   1 3 3 x  3   1 4 3 x  3   1 , n 2n a (x- a ) 1 , n 2n - 1 a (x- a )

Solution Discussion. Below we illustrate how we completed the first row of the table 1 using a graphing calculator. Begin by typing the rational function f (x )  into the x  3 y= editor.

We'll examine local behavior around x  3 using two different methods - with tables and with graphs.

Using a Table. Set your calculator table to "ask mode", then type in values for x that  approach 3 from the left. Examine what happens to values of y as x  3 . Similarly, type in values for x that approach 3 from the right. Examine what happens to values of y  as x  3 .

Tracing a graph. Alternately, graph the function. We chose window settings that focused on values near x  3 .

Next, press 2ND Trace 1: value . Manually enter x values that approach 3 from the left (see below).

Notice the circled y-values on the bottom of each screen. What appears to happen to  these values as x  3 ? Similarly, you can explore local behavior as x approaches 3 from the right by manually entering values of x greater than 3.

Questions for Further Thought

1) Based on the information you compiled in the table, what (if any) generalizations can 1 you make about rational functions of the form k , k . (x- a )

2) Using your knowledge of algebra, make predictions about the local behavior of the x  3 rational function f (x )  near the value x  2 . Use your graphing (x  3)(x  2) calculator to check your predictions.

3) Consider the graph of a rational function of the form 1 f (x )  where a, a , a . Use the graph to predict the (x  a )(x  a )(x  a ) 1 2 3 1 2 3 values of a , a and a . 1 2 3

Check your prediction using graphing features of your calculator.

4) Construct your own rational function f (x )such that f (x ) satisfies the following 4 properties:

(a) As x  3 , y  4  y   (b) As x  0 ,  y   (c) As x  0 , (d) f (6)  1 Project AMP A Quesada Director Project AMP Page 5 Project AMP Dr. Antonio R. Quesada – Director – Project AMP

Introduction to Removable Discontinuity

In the previous activity, you focused much of your attention on rational functions with a 1 constant numerator. Rational functions of the form f( x ) = (x- a1 )( x - a 2 ) ( x - an ) have jump discontinuities at a1, a 2 , , an (i.e. the roots of the denominator).

In this section, we'll examine rational functions with linear factors in both the numerator and denominator. As you will see, rational functions such as these may or may not have jump discontinuities.

An Example (x  2)(x  5) Think about it. Consider the rational function f (x )  . If you asked a (x  5) (x  2)(x  5) student to "simplify" the algebraic expression , what would they likely (x  5) answer? What does this suggest about the graph of f (x )?

Dig Deeper. With your graphing calculator, construct a graph of f (x ).

Your graph should resemble that of the linear function f (x )  x  2 , with one small difference. Complete the following statements to learn more about the rational function (x  2)(x  5) f (x )  . (x  5)

 y  _____ (a) As x  5 , (fill in the blank)  y  _____ (b) As x  5 , (fill in the blank)

(c) Does f (x ) have a jump discontinuity at x  5 ? Why or why not?

(d) f (5)  _____ (fill in the blank)

As you probably discovered, the graph of the rational function doesn't "jump" at x  5 . As the following screenshots suggest, as x  5 , f (x )  3 . Since the same y-value is approached from both sides, the function doesn't "jump".

However, since evaluating the function at x  5 results in division by zero the function itself is undefined at x  5 .

Because the graph breaks at x  5 , we still say that the function is discontinuous at x  5 . The graph highlights an example of a removable discontinuity (some find it helpful to think of "removable" discontinuities as values of x where "holes" occur in the graph of the function).

In this activity, you will use your graphing calculator to explore rational functions with linear factors in the numerator and denominator. Some of the functions have removable discontinuities. Some have jump discontinuities. Some have both types.

The table on the next page lists several rational functions. Complete the missing values of the table by analyzing the local behavior of each function at the locations specified in the table. Then determine whether the function has a jump or removable discontinuity at the specified location.

Behavior of f (x ) Type of Behavior of f (x ) as p  discontinuity at Rational Function  as x  p (from x  p (from left) x  p right) x  3 3 f (x )   f (x )   Removable x  3 (x  3) 2 3 (x  3) (x  3)2 3 (x  3) (x  3) 3 3 (x  3) (x  2) 5 2 (x  2) (x  2)5 2 (x  2) (x  2)5 5 2 (x  2)

5) Based on the information you compiled in the table, what (if any) generalizations can you make about rational functions of the form

(x  a )m , m  n  0 and m  n an even integer. (a) n x  a  

(x  a )m , m  n  0. (b) n x  a  

(x  a )m , m  n  0 and m  n an odd integer. (c) n x  a   Project AMP A Quesada Director Project AMP Page 8 Project AMP Dr. Antonio R. Quesada – Director – Project AMP

6) Based on the information you compiled in the table, what (if any) generalizations can you make about rational functions of the form

(x  a )m , m  n  0 and m  n an even integer. (a) n x  a  

(x  a )m , m  n  0 and m  n an odd integer. (b) n x  a  

7) Using your knowledge of algebra, make predictions about the local behavior of the (x2 - 6 x + 9) rational function f( x ) = near the value x  3 . Use your graphing (x - 3) calculator to check your predictions.

(x2 - 2 x - 8) 8) Consider the rational function f( x ) = . (x2 + 5 x + 6)

(a) Factor the numerator and denominator of the function. At what values of x will the function have a removable discontinuity?

(b) Using factored form of numerator and denominator, simplify the function by "cancelling out" like factors. The resulting function should be in the (x  a ) form with a  b . (x  b)

Think about it. You can use the "long division algorithm" to write the simplified form of the function in a way that clearly illustrates locations of jump x  1 discontinuities. Here's an example using the rational function g(x )  . x  2

1 x  2 x  1 x  1 1 x  2 so  1  there's a jump at x=2. x  2 x  2 ------1

1 Note that graph of g(x ) is actually the graph of translated to the x right 2 units and translated up 1 unit.

(b) Complete similar steps to find the jump discontinuities of the function you wrote in part (b).

9) Consider the graph below. An equation for the function that generates the graph may (x  a )(x  b)(x  c) be written in the form f (x )  where a,b,c,d ,e ¢ (with (x  d )(x  e) values not necessarily distinct).

Find values of a,b,c,d , and e that generate the graph. Check your prediction using graphing features of your calculator.

10) Construct your own rational function f (x )such that f (x ) satisfies the following 3 properties:

 y   (a) As x  3 ,  y   (b) As x  3 , (c) A removable discontinuity exists at x  5

For those who like a challenge, modify your rational function to satisfy the above three properties and the additional property listed below:

(d) f (2)  4

Sketching Rational Functions with Multiple Factors in Numerator and Denominator Project AMP A Quesada Director Project AMP Page 11 Project AMP Dr. Antonio R. Quesada – Director – Project AMP

Recall that the global behavior of a function f (x ) describes values that function approaches as x   (i.e. the end behavior of f (x ) on the left) and x   (i.e. the end behavior of f (x ) on the right).

If we know the end behavior of a function and we know information about all points of discontinuity, we can quickly draw a "rough sketch" of the function.

An Example

Suppose a rational function f (x ) possesses the following characteristics: o Global (end) Behavior: f (x )  0 as x   , f (x )  0 as x   A single discontinuity at :  ,  o x = 3 f (x )   as x  3 f (x )   as x  3

With this information, you should be able to sketch the "general idea" of the graph of the function. Examine the above characteristics, then make a quick sketch predicting the shape of the function's graph (don't use your calculator).

Discussion: Many students find it helpful to draw a dotted line to indicate the location of any jump discontinuities. The graph will "jump" at x  3 , so we sketch the dotted vertical line x  3 . Project AMP A Quesada Director Project AMP Page 12 Project AMP Dr. Antonio R. Quesada – Director – Project AMP

Next, we draw small "arcs" to indicate the local behavior of the function "near" x  3 :

Next, we draw additional "arcs" to indicate the global behavior of the function - i.e. the left and right end behavior.

Lastly, we connect the arcs to complete our sketch.

From the sketch, you should be able to construct an equation that "roughly" generates the 1 graph. The sketch resembles the function f (x )  flipped over the x-axis and x 1 translated 3 units to the right (i.e. ). (x  3)

We can graph the function on a graphing calculator to test the reasonableness of our prediction.

The graph looks quite similar to our sketch.

In this activity, you will use your your knowledge of local and global behavior to sketch graphs of rational functions that satisfy particular characteristics. The functions may have linear factors in the numerator, denominator, or both. Some of the functions have removable discontinuities. Some have jump discontinuities. Some have both types of discontinuties.

In the following three exercises, we list characteristics of several rational functions. Perform the following for each exercise: (a) Draw a rough sketch of each function based on the characteristics; (b) Construct a function that you believe will generate a curve similar to your sketch; and (c) Use your graphing calculator to check your work. If your function doesn't match your sketch, revise it and test again.

Exercise 1 o Global (end) Behavior: f (x )  1 as x   , f (x )  1 as x   A single discontinuity at :  ,  o x = -2 f (x )   as x  2 f (x )   as x  2

(a) Sketch:

(b) Function that generates sketch:

Exercise 2 o Global (end) Behavior: f (x )  0 as x   , f (x )  0 as x   o Two discontinuities - one at x = -3 , one at x = -2 :   f (x )  1 as x  3 , f (x )  1 as x  3   f (x )   as x  2 , f (x )   as x  2

(a) Sketch:

Exercise 3 o Global (end) Behavior: f (x )   as x   , f (x )   as x   A single discontinuity at :  ,  o x = 1 f (x )  2 as x  1 f (x )  2 as x  1

(a) Sketch:

In problems 11-15, you are provided with a rational function. For each, perform the following tasks: (a) identify location of all jump discontinuities; (b) identify location of all removable discontinuities; (c) sketch a graph of the function without your calculator; (d) check your work by graphing with your calculator (if possible).

You may wish to factor or use the long division algorithm on one or more of the items to help you construct an informed sketch.

1 f (x )  11) 2 (x  2) (x  3)

x 2  5x  6 f (x )  12) 2 (x  2) (x  3)

x 2  x  6 13) f (x )  x  2

x 2  6x  9 14) f (x )  2 x  5x  6

(x  a )(x  b)(x  c) 15) f (x )  (x  a )(x  c)

(x- a )2 ( x - b ) 16) Graph the function y=, where a < b < 0 < c < d . (x- c )( x - d )2

17) Find a rational function whose graph is given

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