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.
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.
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