Behavior of a Parabola Graph

by

Hieu H Nguyen

For this assignment, we will examine the graphs of a parabola formed by the equation:

,

where a, b, and c are parameters. We begin with the most basic graph for the parabola formed by the equation given below, where a = 1, b = 0 , and c = 0.

Note that the graph of y = x^2 is parabola with minimum point at y = 0 and x-intercepts at 0. The graph is symmetrically increasing because the solutions for y can be the positive or negative of any integer x because the value is squared.

We can then look at the graph below and observe the behavior of the graph as we vary a while keep b = 0, c = 0.

Note that changing the values in front of x^2 can either widen or narrow the parabola. The symmetry stills holds. Increasing variable a to an integer > 1 will narrow the parabola, and decreasing the variable to an integer 1 > a > 0 will widen the parabola. Vice versa, the same characteristics are implied for all sets of integers < 0. The parabola narrows at integers < -1 and widens at integers 0 > a > -1. Basically, the same characteristics apply, but putting a negative symbol in from of the variable will reflect the graph (flip it) across its minimum point.

We now observe the variations of the parabola graph when we vary b and keep a = 1, and c = 0. The set of parabolic equations below are displayed on the graph shown on the right.

Note that the graphs remain symmetric and still increase at the same rate as x^2. The change that can be observed is the locations of x-intercepts. These are the absolute zeros of the equations. For every graph shown, there's one absolute zero which is defined at x = 0 and the other is the negative value of b. For example, if b = 4, then x = -4 and x = 0. If b = -2, then x = 2 and x = 0. Hence, the variable b seems to relocate the graph to a position where the x-intercepts are at 0 and -b. The animated graph below shows illustrates the behavior of the graph x^2 as b is varied between -4 and 4.

Finally, we observe the behavior of graph x^2 as we vary c and keep a = 1 and b = 0. The set of equations below are displayed on the graph to its right.

We've observed that the variable c seems to keep the graph symmetric as well as retain the shape. The observed change is the minimum position of the graph which seems to be located at a y value equivalent to the value of c. Thus, we can conclude that variable c shifts the graph up or down depending on the value of c. Or it can be said that the minimum position of the parabola is located at y = c. The animation below illustrates the behavior of the parabola graph x^2 as c is varied from values of -4 to 4.

In conclusion, we've observed that variable a determines the increased rate of the graph (widens or narrows it.). Variable b determines the absolute zeroes (x-intercepts) while keeping the rate. The minimum point of the graph also changes. Variable c determines the position of the minimum point of the parabola graph, where that point is located at y = c. Variables a, b, and c do not affect the symmetry of the graph.