1
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
TASK 5.3.2: FUNCTIONS 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) f x = (x + 3)(x ! 2) and g x = (x + 2)
()
()
(b) f x = (x + 3) and g x = (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.
2
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
()
()
(c) f x = (x ! 2)(x + 1) and g x = (x + 2)(x ! 1)
()
()
(d) f x = (x ! 3)(x + 2)(x ! 1) and g x = (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.
3
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
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?
•
•
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.
4
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) f x = (x + 3)(x ! 2) and g x = (x ! 2)
(b) f x = (x + 2)(x ! 1) and g x = (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.
5
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
(c)
()
()
f x = (x ! 3)(x + 2)(x ! 1) and g x = (x + 2)(x ! 1)
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.
6
Algebra II: Strand 5. Power, Polynomial, and Rational Functions; Topic 3. Rational
Functions; Task 5.3.2
TASK 5.3.2: FUNCTIONS 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) f x = (x + 3)(x ! 2) and g x = (x + 2)
()
()
(b) f x = (x + 3) and g x = (x + 2)(x ! 1)
()
()
(c) f x = (x ! 2)(x + 1) and g x = (x + 2)(x ! 1)
()
()
(d) f x = (x ! 3)(x + 2)(x ! 1) and g x = (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.
7
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) f x = (x + 3)(x ! 2) and g x = (x ! 2)
(b) f x = (x + 2)(x ! 1) and g x = (x ! 3)(x ! 1)
()
()
(c) f x = (x ! 3)(x + 2)(x ! 1) and g x = (x + 2)(x ! 1)
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.