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Given the algebraic equation for a quadratic function, one can calculate any point on the function, including critical values like minimum/maximum and x- and y-intercepts.
These calculations can be more tedious than is necessary, however. A graph contains all the above critical points and more, and is essentially a clear and concise representation of a function. If one needs to determine several values on a quadratic function, glancing at a graph is quicker than calculating several points.
Consider the function:
Suppose this models the profit (f(x)) in dollars that a company earns as a function of the number of products (x) of a given type that are sold, and is valid for values of x greater than or equal to 0 and less than or equal to 500.
If one wanted to find the number of sales required to break even, the maximum possible loss (and the number of sales required for this loss), and the maximum profit (and the number of sales required for this profit), one could calculate algebraically or simply reference a graph.
By inspection, we can find that the maximum loss is $750, which is lost at both 0 and 500 sales. Maximum profit is $5500, which is achieved at 250 sales. The break-even points are between 15 and 16 sales, and between 484 and 485 sales.
The above example pertained to business sales and profits, but a similar model can be used for many other relationships in finance, science and otherwise. For example, the reproduction rate of a strand of bacteria can be modeled as a function of differing temperature or pH using a quadratic functionality.
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