Graphic Behavior of Rational Equations 

By Pierce Ducsay

-The function "y=k/x" has the 'x' in the denominator, which creates a graph as shown, with the 'y' reaching closer to infinity (plus or minus) as x reaches zero. As such, the domain of this function is all reals besides zero: "[x:x≠0]"

On the flip side of the same coin, 'x' reaches closer and closer to infinity (plus or minus) as 'y' reaches zero. As such, the range of this function is: "[y:y≠0]"

As for the x and y intercepts--in which both 'x' and 'y' must equal zero, respectively--this function has no solution, as making either variable equal to zero would end up making infinity, so no solution is the answer.


This function also makes figuring the horizontal and vertical asymptotes a cake walk as well. Without any shifts, each end stretches out to infinity; the asymptotes will be the values that the domain and range cannot equal; in the case of this non-shifted graph, both the horizontal and vertical asymptotes are equal to zero.


Any vertical shift on the graph would require an addition or subtraction outside of 'k/x'. An addition will shift the function closer to Quadrants I and II (move it 'up'), while a subtraction will shift the function closer to Quadrants III and IV (move it 'down').
Any horizontal shift on the graph would require an addition or subtraction directly to 'x' in the denominator of 'k/x'. An addition will shift the function closer to Quadrants II and III (move it 'left'), while a subtraction will shift the function closer to Quadrants I and IV (move it 'right').


For example, "y=(1)/(x - 3) + 5" would result in this graph: (the orange lines are the asymptotes)



Subtracting three from 'x' has shifted the function horizontally three units to the right. Adding five to '1/x' has shifted the function vertically five units up.
-The function "y=(ax^m)/(bx^m)" has the 'x' in the denominator. Again, because of this 'x' cannot equal zero without resulting in infinity. As such, the domain of this function is all reals besides zero: "[x:x≠0]"


As for variable 'y', the 'x' variable existing in both the numerator and denominator--since their exponents must be the same 'm'--will cancel each other out and result in (a/b)(1). So, 'y' will be a horizontal line dictated by the value of: "[y:y=a/b]"


A rather boring-looking graph, if I do say so myself. (While this graph is assuming both 'a' and 'b' to equal one, a straight line will always be the result as long as both 'a' and 'b' are real numbers. 'X' cannot equal zero, and neither can 'b'. Also, while you can't see it in the graph shown, this horizontal line is non sequitur; at 'x=0', 'y' becomes undefined.)

...However, I found there were some cases in which 'm', the exponent, can be a number that results in this horizontal line staying only within the first or fourth (positive x) Quadrants. The example of such shown below was using the exponent '3.52' for 'm'; it appears multiple decimal numbers will get this result, and honestly, I don't have an explanation as to why.

Anyways,


With the x and y intercepts, the math becomes tricky again as before. Making 'x' equal to zero is, again, impossible, so the y-intercept is no solution. As for the x-intercept, 'y' can only equal zero is 'a' is equal to zero on the numerator; assuming this is the case, 'x' has the potential to equal any and all real numbers except for zero. [x:x≠0]



Since this function must always result in a horizontal line, there is no vertical asymptote, so it is no solution. With the horizontal asymptote, it ends up being both parallel and identical to the line given (or only extends in the positive direction, apparently) so the horizontal asymptote equals 'a/b'.


The vertical shift of the function works the same as the previous function: adding a number outside the fraction shifts up, subtracting shifts down. What is interesting is the horizontal shift of the functions, which has unique-looking curves based upon which part of 'x'--numerator or denominator--is being added or subtracted. A graph is provided below along with a plus one or minus one on both numerator and denominator; 'a', 'b', and 'm' all equal one.

The results are all reminiscent of our previous function of "y=k/x". As you can see, though, the only two relevant lines are the orange and green--both of which have the the adder or subtractor on the denominator, shifting the function right one or left one, respectively.