How to set up a linear, quadratic, or rational function? What is the difference between all of them and when to use each one? what benefit comes as a result of using each one in real life ?

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First consider what a function is. A function has inputs and outputs. It exists where there is one output (y value) for any number of inputs (x values).

  • A linear function is represented by `y=mx+c` where, as you can see, there is only one `x` value for each `y` value....

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First consider what a function is. A function has inputs and outputs. It exists where there is one output (y value) for any number of inputs (x values).

  • A linear function is represented by `y=mx+c` where, as you can see, there is only one `x` value for each `y` value. This is represented by the power of` x` ,( in a linear function, that is `x^1` ).
  • A quadratic function is represented by `y= ax^2 +bx+c` . Note that `x` now has 2 values for every `y` value, as represented by `x^2` (power of 2- 2 `x` values).
  • A rational function has a numerator and a denominator and as the denominator can never be zero, it needs a restriction in order to remain rational. It can be represented by `y=(p(x))/ (q(x))` and has any number of x values for each y value, depending on the question. A simple rational function can be shown in : `y=1/x and x!=0` . Also note that linear and quadratic functions are rational functions, notably when they have fractions, but in any instance because, for example `y= 2x` can be written as `y=(2x)/1`  showing that it has a denominator of 1, even though we are not required to show it.   

Setting up a function requires an understanding of the question. In problem-solving, when you are required to find an unknown, you will form your own equation. This makes functions very useful as creating an equation allows you to solve the problem most readily.

Solving a problem, for example, when a company needs to track its profit in relation to its costs may result in a parabola. Often the costs are higher at the beginning and profit capability reaches a maximum after which time it is no longer quite so profitable. A graph is a visual representation allowing companies to make sense of information quickly and easily and which can be understood by the greatest number of people who otherwise could not interpret the data. 

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A linear function is one of the form f(x) = a*x + b where a and b are constants. The graph of f(x) plotted against x is a straight line. A quadratic function is of the form f(x) = a*x^2 + b*x + c where a, b and c are constants. The graph of f(x) versus x in this case is a parabola. A rational function f(x) is one that can be written as `f(x) = (P(x))/(Q(x))` where P(x) and Q(x) are polynomials of x and Q(x) is not equal to 0. The domain of the function Q(x) has to be set such that it does not equal 0.

All of the functions mentioned are widely used. The conditions that are being modeled decide which of the functions is used. For instance, if the value of f(x) is directly or inversely proportional to that of x, a linear function would have to be used.

An example of a linear function used commonly is the distance traveled by a car in a given duration of time if it is moving at a constant speed. An example of a quadratic function is the distance traveled by a car in a given duration of time if it is accelerating.

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