Coefficients of a Parabola

Kim Seay

EMAT 6680

The standard form of a quadratic equation is given above (where a, b, and c are constants). The graph will be a parabola. I would like to explore what happens to the graph of the parabola as the coefficients a, b and c vary.

The coefficient "a"

By definition of a quadratic equation, "a" cannot equal zero. Let's begin by looking at the simple case where a=1 , b=0, and c=0.

What happens to the graph as "a" changes? I will now graph several variations of "a" on the same axis to compare.

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The parabola is keeping it's same vertex, but seems to be getting more narrow as "a" increases. Let's look at some negative "a" values.

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The graphs of the negative "a" values are a reflection of their positive graphs.

Click here to see a Quick Time movie of the graph as a changes from -100 to 100. Note that when a=0, the equation becomes y=0, and the graph is a horizontal line through the x-axis.

The coefficient "b"

In order to see the changes in the coefficient b, I will keep a = 1 and c = 0. The first graph where b = 0 has been graphed above. I will now graph b=0, b=1, b=-1, b=5, and b=-5 on the same graph.

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In each of the above equations, the parabola intersects the x-axis at "-b".

To investigate this further, click here for a Quicktime movie to see what happens as "b" changes from -20 to 20. The parabolas continue to cross the x-axis at "-b". The graph passes through quadrants 1 and 4 when "b" is negative, and quadrants 2 and 3 when "b" takes on positive values.

The coefficient "c"

Again, we will keep the other two coefficients constant while we observe changes in the graph as "c" varies. When a=1 and b=0, let's graph the functions where c=0, c=1, c=-1. c=5, and c=-5 on the same graph.

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As "c" changes, the parabola moves up or down the y -axis. In fact, it appears that "c" is where the graph intercepts the y-axis. Click here to see changes if this remains true as "c" varies from -50 to 50.

Conclusion:

I think this exploration would be very beneficial to students. Every Algebra 2 student is required to know how the coefficients of a parabola affect its' graph, but I'm sure very few are given the opportunity to discover this on their own. Most are probably given rules to copy and memorize where "a", "b", and "c" have no real meaning. Students may not be able to do this type of visualizing in their head. I chose this as a write-up, because it is such a common mathematical problem that all college bound students should be exposed to in this method.

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