Encourage students to work in pairs on the activity sheets. The
discussion generated by questions in the activity is beneficial. Each
student needs the activity sheets, three different colored pencils, and
a strip of paper or a ruler. The activity is appropriate in either
first- or second-year algebra, as soon as students have a good
foundation in linear functions, some knowledge of quadratic functions,
and an understanding of what is meant by a polynomial function.
Building Polynomial Functions
Building Polynomial Functions Activity Sheet
Students start by identifying a linear function and putting the equation in slope/x-intercept form, y = m(x - c), where c is the x-intercept. This form serves as a connector with other classes of polynomial functions and forces students to focus on the x-intercept of the graph. They then choose another function in the form y = m(x - c)
and graph this function on the same axes. Students predict how a new
function, formed by taking the product of the two linear expressions,
would appear graphically. After making their prediction, they graph the
resulting quadratic function and compare the actual function with their
prediction. Students can use a graphing utility to check the function
formed by taking the product of the linear factors, but only after
making the prediction.
Activity questions that compare the linear functions with the resulting
quadratic function focus the students' attention on the parts of the
graphs to be emphasized. Students learn that the quadratic function has
the same x-intercepts as the linear functions, which can be quite a revelation, and that the y-intercept of the quadratic function is the product of the
y-intercepts of the linear functions. In fact, the y-coordinate of the parabola for a given x-value is always the product of the y-coordinates of the lines for that x-value. Seeing this relationship is easier when x equals 0 and the y-coordinates are lined up on the y-axis.
Students then use a strip of paper or a ruler to cover parts of the graph. This part of the activity shows that the sign of the y-coordinate for any point on the parabola can be determined by observing whether the y-coordinates
of the lines for that section of the graph are positive or negative.
For example, if both lines in a section of the graph are above the x-axis, then the parabola will be above the x-axis, that is, (+) • (+) = (+). If one line in a section of the graph is above the x-axis and the other is below the x-axis, then the parabola is below the x-axis,
that is, (+) • (–) = (–). This result corresponds to the sign table
that students have traditionally used as an aid to graph functions and
inequalities.
Working Backwards
Working Backwards Activity Sheet
The second part of the activity requires students to work in the
opposite direction, that is, take a graph of a quadratic function and
break it into its linear components. Classroom experiences often focus
on factoring quadratic expressions into linear factors. The graphical
counterpart to this process is to break the graph of a quadratic
function into its components, the lines. This illuminates factoring in
a visual way. You should caution students that a graph that appears to
be a parabola may actually be the graph of a fourth-degree or higher
even-degree polynomial function. Beyond this activity, students should
not assume that a graph that has two x-intercepts is the graph of a second-degree polynomial function.
When students first attempt this activity, they usually focus on having their lines go through the parabola's x-intercepts, but they may not consider the y-intercepts. They should also check sections of the graphs before and after the x-intercepts to make sure that the product of the y-coordinates of the linear factors gives the sign and values of
the y-coordinates on the parabola in that section.
When asked whether the choice of these lines is unique (it is not),
students may have some conflict. They have been told that quadratic
expressions factor uniquely into linear expressions; however, they may
not have been told that this outcome is unique only up to unit factors.
Unit factors can therefore be split over linear expressions to give an
infinite number of combinations of lines; for example, (2x - 3) • (x + 4) is equivalent to (1/2)(2x - 3) • 2(x + 4) = (x - (3/2)(2x + 8). The only limitations for possible lines are that the product of the y-intercept for the parabola and the x-intercepts must remain the same.
The last graph in this section has no x-intercepts, but
students are asked to try to sketch lines that could be its components.
This example should prompt some good discussion. Students should
eventually conclude that an absence of x-intercepts implies
that no real roots exist - that is, lines cannot be drawn because the
quadratic equation cannot be factored into linear expressions over the
real numbers. This visual display fosters insight into why quadratic
equations sometimes cannot be factored in the real-number system.
Higher Degree Polynomials
Higher Degree Polynomials Activity Sheet
When students have experience building quadratic functions from
linear expressions and working backward to find linear components,
extending these ideas to polynomials of degree three is a natural
progression. The teacher should warn students that graphs of
fifth-degree or higher odd-degree polynomials can closely resemble the
graph of a cubic polynomial. Therefore, beyond this activity, they
should not assume that a graph is a cubic because it has three x-intercepts.
Summarizing the Lesson
After students have completed this activity, the teacher should have
the class discuss the material. Students should share their insights
and ask questions.
This new way of looking at polynomial functions enriches and broadens
students' conceptions about polynomial functions. Visualizing the
algebraic representation through a graph gives more meaning to the
symbols. These activities should make the connections between the x-intercept and the factors of the polynomial more salient, as well as highlight for students "the
glue" that holds the classes of polynomial functions together.
Activity Sheet Answer Keys
References
- Buck, Judy Curran, October 2000 edition of Mathematics Teacher Journal.
- Curran, Judy. "An Investigation into Studentsç Conceptual Understanding of the Graphical Representation of Polynomial Functions." Ph.S. diss., University of New Hampshire, 1995.
- National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000.
- Schwartz, Judah L., Michal Yerushalmy, and Educational Development Center. The Function Supposer: Explorations in Algebra. Pleasantville, N.Y.: Sunburst Communications, 1988. Software.