Rational+Functions

Rational Functions
Y=1/x When we input 1/x in a table the variable x can't be 0 because you cant divide by 0 but you can divide by any other number thus the range becomes x ≠ 0. This is also the asymptote of 'x' which is why the hyperbolas in the graph never meet 0. This is why when the x variable is replaced by numbers closer to 0 the hyperbolas will become in a graph because the results of 1/x will be larger numbers. This also explains why when x becomes a larger number the results of 1/x will be numbers closer to 0 thus the hyperbolas 'shrink'. The Range of 1/x would be y≠0 since 1/y is just the inverse function of 1/x and the same properties that are applied to x are also applied to y. Thus the asymptote of y is also 0 and this is why 1/x is displayed as an equilateral hyperbola in a graph.
 * Basic Equation:**
 * Domain:**
 * Range:**

Rational functions can be easily defined with the equation y=1/x, or f(x)=1/x, but this equation has variations, therefore changing the basic rational function equation to F(x)= a / b(x-h) + k
 * Equations, domain & range, graphs and transformations:**

Each of the variables in the equation affects the function, therefore changing the graph. The graph of the function 1/x (if put into the basic equation: 1/1(x-0) + 0) is the following: (Domain: all real numbers except 0, range:all real numbers except 0)



Now, when the variable H changes in the basic equation, the vertical asymptote of the graph changes, hence the graph moves either to the left or to the right. When H is a negative number, then the asymptote is the same value but positive, and vice versa, so if the value is negative, the graph moves to the right and if it's positive it moves to the left. The only restriction of the value of H is that it cannot equal X due to the fact that the whole equation would be undefinable.The common domain for the function is all real numbers except X, and the range is all real numbers except 0 (if K is NOT changed, k=0), so it's actually all real numbers except K. The graph of the function 1/1(x-1) + 0 is the following: When the variable K changes, the horizontal asymptote of the graph changes, hence the graph moves up or down. The value and position of the asymptote in the graph is the same as the value of K, so if it's 3, the graph moves up by three units and vice versa. There are no restrictions for the value of K. The domain is equal to all real numbers except X and the range is all real numbers except K. The following grph shows a change in tha horizontal asymptote due to the equation 1/1(x-1) + 3: Finally, when the variable B changes, the stretch of the lines change meaning that they come closer and closer to the asymptotes defined. When B is a negative value, the lines change of quadrants from 1 and 3 to 2 and 4, but this is only when the equation is 1/-1(x-0) + 0. If the values for both H and K were different, then all the quadrants of the graph are included.

The changes of variable A are very similar to those of variable B, but with the only difference that when A changes, the stretch of the lines gets bigger and bigger as A increases, which is the opposite of what happens when B changes.

Like previously mentioned, the simplest way to express a rational function is with the equation F(x)=1/x. The inverse of this function would be F-1(x)=1/y. This function is clearly represented in the previous graphs; it's the line that is in quadrant 3 (first graph). As you can see, in the graph the lines of each functions are reflections of one another, and therefore, the values of each one are the sames as the other ones but with a different sign. The specific name for this is an equilateral or rectangular hyperbola due to the fact that the asymptotes are perpendicular to one another. The inverse of the basic rational function (f(x)=a/b(x-h) + k) is much more complicated to find but it finally turns out to be f(x)=a+k/by-bh, but the concept is still the same. Now, the restrictions for the equation are basically that the sum of a and k cannot equal 0, and the subtraction by and bh cannot equal 0 as well.
 * Inverse:**

Key Features: Like mentioned before, rational functions contain asymptotes that might vary with the values of the variables K and H. With the simple rational equation f(x)=1/x, the asymptotes are implied to be equal to 0 nothing is being added or subtracted from the equation. When the variable H changes, the position of the lines move right or left, hence changing the vertical asymptote. If H is negative, the lines move to the right and if H is positive, the lines move to the left. The value of the number of spaces the lines move is equal to the value of H, but with an opposite sign. When K changes, the horizontal asymptote changes, so the lines move either up or down, depending on the value of K. If K is negative, the asymptote moves down and when K is positive, the asymptote moves up, and the umber of spaces the lines move is equal to the value of K.

In the real world, rational functions are used to finds an amount or measurement of something in proportion to something else. Example: -Young's rule. This is a rule to measure a childs dosage of medicines in proportion to the dosage of an adult. It is worked out firstly by adding 12 to the childs age and then dividing the result by the childs age, you then divide the adult dose by the figure obtained. The equation of this function would be A/((x+12)/x) where A is adult dose and A is age.
 * Applications to the real world:**

- Focus lenses in cameras.

- Satelite dishes.

- Rocket nozzles.

- Cooling towers of nuclear power stations.

- Tracking particles in particle physics.

- Gas properties.

-Math of rainbows.

- Analyzing caplliary forces.

- Orbits of some space crafts.

Bibliography: Mathdoc. "Answers.com - Real Life Application of Hyperbola." //WikiAnswers - The Q&A Wiki//. Web. 29 Mar. 2011. [].

Donaghue, Doug. "How Are Hyperbolas Applied to the Real World? - Yahoo! Answers." //Yahoo! Answers - Home//. Yahoo!, 2007. Web. 29 Mar. 2011. [].

John. "What Are Real Life Examples of Parabolas, Hyperbolas, and Ellipses? - Yahoo! Answers." //Yahoo! Answers - Home//. Yahoo!, 2008. Web. 29 Mar. 2011. [].

"Pre-Calculus Advanced Polynomial and Rational Functions Rational Function Graphs." //Winnipeg School Division//. Web. 29 Mar. 2011. [advanced/poly and rational functions/Rational Function Graphs/ratfuncgraphs.htm]. [|Weisstein, Eric W.] "Rectangular Hyperbola." From [|//MathWorld//]--A Wolfram Web Resource. [] Back to Wiki Assignment