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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durbanville | High School Teacher | (Level 2) Educator Emeritus

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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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justaguide | College Teacher | (Level 2) Distinguished Educator

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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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steamgirl | Student, College Junior | (Level 1) Honors

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The difference between any function is the equation. Linear are y=mx+b and make a straight line of a graph. These come in handy visualizing large related quantities of numbers or results such as income revenue, stocks, or gas prices.

Quadratic functions are written as y = mx^2 + b or sometimes y = mx^2+bx+c, though this form can usually be factored into the first form. These function produce a bowl/cup look on a graph. Quadratics, or exponential functions in general, are useful for determining the rate in which populations change, or how quickly bacteria in a petri dish will be produced. Basically, quadratic measures things that either grow or decrease at a more pronounced rate than linear.

Rational functions are fraction functions and are written as  y = f(x) / g(x), only when g(x) doesn't equal zero in any case. These functions can produce strange looking graphs, but the important thing to take note of here is that there is an asymptote involved. These can be either horizontal or vertical and just tells you where this function cannot be. Rational function are useful for force equation in physics, optics, electrical circuits, etc.

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rachellopez | Student, Grade 12 | (Level 1) Valedictorian

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A linear function is represented by the equation y=mx+b and it is simply a straight line. You can use linear functions for finding things like distance, slope, midpoint, etc.

Quadratic functions are represented by the equation ax^2+bx+c=0 and when graphed are called parabolas. You will see a lot of quadratics in algebra 2 and calculus. 

A rational function is represented by the equation  f(x)=P(x)/Q(x) and is just a ratio of polynomials. Q(x) never equals zero or it will be undefined.

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science2014 | (Level 1) Valedictorian

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To set up a mathematical equation, one needs to identify from the description of the problem (in words), what the variables are and how they are related to each other. This will set up an equation that will represent the problem description in mathematical terms.

For example,

The sum of two numbers is 25 and their difference is 10. What are the two numbers?

Let the two numbers be x and y. These are the variables in this problem. And the variables are inter-related as follows:

x + y = 25

x – y = 10

Solving these two equations, one can find the values of x and y.

A quadratic equationis defined as an algebraic equation in which one or more of the terms is squared, but not raised to a power greater than 2.  The general form is y = ax2 + bx + c where a, b and c are constants. It may have one or more variables.

A linear equation is an algebraic equation in which each term is either a constant or a product of constant and single variable. The general form is y = mx + c, where m is the slope, c is the intercept and both are constants. It can have one or more variables.

A rational equation is an algebraic equation in which one or more of the terms is/are fractional. While solving a rational equation by cross-multiplication or least common denominator (LCD) method, it can take the form of a linear equation or quadratic equation.

Difference in Shape of Plot: The difference between linear and quadratic equations is that when a linear equation is plotted it gives a straight line (e.g. x - y = 8), whereas when a quadratic equation is plotted it gives a parabola when one variable is squared (e.g. y = -x2 + 2x + 4) and it gives a circle when two variables are squared (e.g. x2 + y2 = 25)

Difference in the value of Exponent: The difference between linear equation and any other equation is that linear equation is of first degree, meaning that the exponents of the variables are always one. On the other hand, equations of any other degree (e.g. quadratic equation) are non-linear.

Difference in Slope: The slope of a linear equation is constant, whereas the slope of a quadratic equation varies at each points.

Use of equations in real life:

Linear equations can be applied to any steady progression application, for example, the cost of cell phone plans for family members. Suppose for each individual, the cell phone plan costs $35. So the linear equation here is: y = 35x, where y = total cost and x = number of family members.

Quadratic equation is often applied to the motion of an object. We state, s = ut + ½ at^2 where s = distance traveled, u = initial velocity, a = acceleration, and t = time.

Examples of exponential equation is the evaluation of rate of growth of bacteria.

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