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

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

Finally, we observe the behavior of graph x^2 as we vary

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

In conclusion, we've observed that variable

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