Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
T ASK 5.3.2: F UNCTIONS AND THEIR QUOTIENTS
Solutions
1. Graph the following functions and their quotient. (Hint: Put Function 1 in Y1=,
Function 2 in Y2=, then make Y3= Y1/Y2. Change the graph style for Y3 to show the path (“football” icon). This will make it easier to view the graph of the quotient). Use the window shown in Problem 7 of task 5.3.1.
(a) ( )
=
( x
+
3)( x
!
2) and ( )
=
( x
+
2)
1
(b) ( )
=
( x
+
3) and ( )
=
( x
+
2)( x
!
1)
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.

Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
2
(c) f x ( )
=
( x
!
2)( x
+
1) and ( )
=
( x
+
2)( x
!
1)
(d) ( )
= ( x !
3)( x + 2)( x !
1) and ( )
= ( x !
2)
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.

Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
3
Compare the graph of the quotient with the graphs of the numerator and denominator. Compare the graphs with the function values found in the function table. You will need to cursor to the right to view the function values for Y3. What observations can you make?
•
The graph of the quotient crosses the x-axis at the zeros of the numerator. The xintercepts of the rational function are determined by the zeros of the numerator.
This is shown graphically from the plot and in tabular form by comparing Y1 and
Y3 in the table.
• The rational function does not exist at the zeros of the denominator. The zeros of the denominator determine the location of vertical asymptotes. This is shown graphically from the plot and in tabular form by comparing Y2 and Y3 in the table.
Extension question:
Ask, “Will the graph of the rational function ever cross a vertical asymptote? Why, or why not?”
The graph will never cross a vertical asymptote because the function value is undefined at the value of x at which the asymptote occurs. The function may approach a vertical asymptote, but it can never cross it. To cross it requires that the function exist at the value of x that produces the zero in the denominator.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.

Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
2. Graph the following functions and their quotient.
(a) ( )
=
( x
+
3)( x
!
2) and ( )
=
( x
!
2)
4
(b) ( )
=
( x
+
2)( x
!
1) and ( )
=
( x
!
3)( x
!
1)
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.

Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
(c) ( )
=
( x
!
3)( x
+
2)( x
!
1) and ( )
=
( x
+
2)( x
!
1)
5
Compare the graph of the quotient with the graphs of the numerator and denominator.
Compare the graphs with the function values found in the function table. You will need to cursor to the right to view the function values for Y3. What observations can you make?
When the numerator and denominator have a common factor, the function value does not tend to ± ∞ near the common zero; therefore, a vertical asymptote does not occur. Instead, the function vanishes at this point, as its value is undefined.
The function is discontinuous at this point. This discontinuity is represented on the graph as a hole. It is called a removable discontinuity. This discontinuity may, or may not, appear in the graph, depending upon the viewing WINDOW . It will always be evident in the table if the TblSet is set to show the zeros of the numerator and denominator. The discontinuities may be seen by comparing Y1, Y2, and Y3 in the table. The discontinuities appear as ERROR in Y3, across from zeros in both Y1 and Y2.
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.

Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
6
T ASK 5.3.2: F UNCTIONS AND THEIR QUOTIENTS
1. Graph the following functions and their quotient. (Hint: Put Function 1 in Y1=,
Function 2 in Y2=, then make Y3= Y1/Y2. Change the graph style for Y3 to show the path (“football” icon -0). This will make it easier to view the graph of the quotient). Use the window shown in Problem 7 of task 5.3.1.
(a) ( )
=
( x
+
3)( x
!
2) and ( )
=
( x
+
2)
(b) ( )
= ( x + 3) and ( )
= ( x + 2)( x !
1)
(c) ( )
= ( x !
2)( x + 1) and ( )
= ( x + 2)( x !
1)
(d) ( )
= ( x !
3)( x + 2)( x !
1) and ( )
= ( x !
2)
Compare the graph of the quotient with the graphs of the numerator and denominator. Compare the graphs with the function values found in the function table. You will need to cursor to the right to view the function values for Y3. What observations can you make?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.

Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
2. Graph the following functions and their quotient.
(a) ( )
=
( x
+
3)( x
!
2) and ( )
=
( x
!
2)
(b) ( )
=
( x
+
2)( x
!
1) and ( )
=
( x
!
3)( x
!
1)
(c) ( )
= ( x !
3)( x + 2)( x !
1) and ( )
= ( x + 2)( x !
1)
7
Compare the graph of the quotient with the graphs of the numerator and denominator. Compare the graphs with the function values found in the function table. You will need to cursor to the right to view the function values for Y3. What observations can you make?
December 16, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas at Austin for the Texas Higher Education Coordinating Board.