Effect of Coefficient of x^2 on Parabola Vertex (a < 0)
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This page first made public: Aug 19, 2010
This material is replicated on a number of sites as part of the SERC Pedagogic Service Project
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 vertex of a parabola where a<0, b>0 and 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, most do not realize that it also affects the position of the vertex when b is not equal to 0. Algebraically analyzing the effects requires a good bit of effort on the students' part, but the results are rewarding.
To enable students to:
- develop their understanding of the effects of coefficients in quadratic functions
- exercise their mathematical intuition and verify it via appropriate calculations
- exercise the skills of algebraic and mathematical analysis
Context for Use
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
- Instructor presents a ConcepTest (Rich Text File 75kB Aug18 10) in the form of a straw poll (either show-of-hands or written) concerning the effect of increasing the value of the coefficient of x^2 on the position of the vertex. Each student is asked to make a conjecture and the instructor records the results for the class to see. (~5 minutes)
- After the straw poll, the question becomes the Question of the Day (Rich Text File 72kB Aug18 10) and students work in pairs to share and explain their reasoning in written form. (~10 minutes)
- Lastly, students are asked to carry out the appropriate calculations in a write-pair-share activity (Rich Text File 136kB Aug18 10) to verify their conjectures. (~15 minutes)
- In conclusion, the instructor presents a classroom demo involving a Java applet that quickly and easily demonstrates the effects of changing the value of the coefficient of x^2 in a quadratic function. http://www.ltcconline.net/greenl/java/Other/MovingGraph/MovingGraph.html
In addition, the instructor ensures that students understand the algebraic and mathematical analysis involving the expanded form of the formula for the vertex by clearly explaining how the coefficient affects each of the x- and y-coordinates of the vertex. (~10 minutes)
Teaching Notes and Tips
The concept of a limiting value for the vertex, (0,c), as a approaches negative infinity may also be addressed.
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