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Effect of Coefficient of x^2 on Parabola Shape

This page and activity authored by James Rutledge, St. Petersburg College. It makes use of an applet created by Larry Green, Lake Tahoe Community College,
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This page first made public: Aug 19, 2010

This material was originally developed through Merlot as part of its collaboration with the SERC Pedagogic Service.


This classroom activity presents College Algebra students with a ConcepTest, a Question of the Day, and a Write-pair-share activity concerning the effect of the coefficient of x^2 on the shape of a parabola where b and c are arbitrarily fixed values in f(x)=ax^2+bx+c.

The results are quite revealing and show that while students may have learned that the coefficient of x^2 affects the shape of a parabola (i.e., when a>0, the parabola opens upward, and when a<0, the parabola opens downward), most do not understand the reason why. The fact that the term with the highest degree is the dominant term in a polynomial is generally unfamiliar to them.

Learning Goals

To enable students to:

Context for Use

This activity can be carried out in either a small College Algebra class or a large lecture setting anytime during or after the topic of graphing quadratic functions has been covered.

The activity is comprised of four segments: ConcepTest, Question of the Day, Write-pair-share exercise, and conclusion involving the use of a demonstration applet. The time required for the entire activity is approximately 40 minutes but fewer segments can be offered as a shorter alternative (see Activity Description below for individual segment times).

Description and Teaching Materials

Activity Description and Teaching Materials

Teaching Notes and Tips

Student understanding of the role of coefficients in polynomial functions is limited at best. Most have not considered the vertex in terms of the values of a, b and c, and many have difficulty in analyzing the effects of changing the values of these coefficients. This activity is designed to help them develop a better grasp of the role of coefficients.


I generally give the students a grade for participation in this activity and sometimes grade their final calculations concerning the work required.

References and Resources

Larry Green's Applet Page-Parameters and Graphs
This applet allows the user to change the values of the coefficients of a quadratic function and immediately see the graphical results.
MERLOT description of this resource

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