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Graphing a Rational Function

Updated on January 5, 2015

Graphing a Rational Function:

A computer or calculator can make a pretty and interesting graph, but if the graph's behavior is understood, then you, obviously, understand the mathematics describing that behavior. Your calculator may not identify aspects of a graph as removable discontinuities, or very inconspicuous local extrema. Understanding the mathematics behind the graph will enable one to discover and identify these things.

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1. Paragraph 10 has the amount of information God stored in a DNA molecule.

CALCULATORS CAN'T REPLACE KNOWLEDGE:

CALCULATORS--FROM SLIDE RULES TO GRAPHING:

2. What do you think I would consider to be higher math? Well, I'll tell ya; it is keeping track of my age. I have seen calculators evolve from slide rules( I still have mine) to the ti-89 graphing calculator. Every time a calculator came along with a new feature, I would buy it. I felt like I just entered heaven when I bought the TI-59( or was it 56?) programmable calculator. When I first encountered the Koch curve( Snowflake Curve) I did not believe it was possible to have a line of infinite length enclosing a finite area. It is described by an infinite series, and I programmed the TI-56 to print out a result every 7 iterations. A friend and I went out for lunch, and when we got back there was a pile of printed- out paper ending with the limit of the calculator--10^99--which represented the length of the line until the calculator said, "I've had it." It is amazing what calculators can do compared to my slide rule. They can even differentiate, integrate, do matrix calculations, and graph numerous functions and equations. We can just sit back and let the calculator do the work.

CALCULATORS CAN GRAPH IT BUT WE STILL HAVE TO INTERPRET IT:

3. Calculators can graph numerous functions and equations, but if you are studying math so you can apply it to your career as, for example, engineering or business, then you must be able to interpret the graphs your calculator produces. Also, your calculator may not reveal all the information contained in a graph. My calculator did not reveal the "removable discontinuity" at the point (x,y) = (1 , --1.3125), which is represented by the black dot in the center curve( line) of E2.

ARE THERE ASYMPTOTES?

4. Technically, an asymptote for a plane curve "is a line which has the property that the distance from a point P on the curve to the line approaches zero as the distance from P to the origin increases without bound . . . ." So the closer the graph gets to an asymptote, the further away it is getting from where the axes cross at (x , y) = (0 , 0). We can suspect this rational function at E1 has a slant( or oblique) asymptote because the polynomial in the numerator is one degree larger than the polynomial in the denominator. After the denominator is factored we can suspect two vertical asymptotes because two of the factors in the denominator approach zero as x approaches 3 and -3. The other factor, (x-1), at L30 also approaches zero at x = 1, but that discontinuity is removable.

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DIVIDE THE FRACTION OUT:

FINDING THE SLANT( OBLIQUE) ASYMPTOTE:

5. The equation in the upper right corner of E2 is the equation of the line representing this function's slant asymptote. It is found by dividing the numerator by the denominator at L19 and L20. This line applies to the curves in the 1st and 3rd quadrants. The first curve in the 3rd quadrant approaches that line as x goes to the left on the number line. The third line in the 1st quadrant approaches that line as x goes to the right on the x axis. Incidentally, the reason this hub starts with L19 rather than L1 is because this was originally a problem I did for a student I was tutoring, and there was more to it than what is included here. Notice in the remainder that the numerator is one degree less than the denominator. As x increases either to the left or right of zero this will cause the remainder to approach zero. All we are left with as x goes away from the origin is the line equation, (1/2)x + (7/2), and this is the slant asymptote. We know this will be a linear line because the degree of the numerator of the original function is one more than the denominator. It is possible to have a quadratic asymptote if the numerator is two degrees larger that the denominator.

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FIND COMMON FACTORS:

FACTOR THE NUMERATOR:

6. We want to factor the numerator at L21 to L25.1 to determine if there are any common factors between the numerator and denominator. The best way to factor textbook problems is with a graphing calculator. Where the graph crosses the x-axis is where the factors are. L21 was graphed and the graph crosses the x-axis at -6, and 1. We use synthetic division at L22 and L24 to make sure these are the zeros( roots of the equation).Without a graphing calculator you then can use the Rational Zeros Theorem, Descartes' Rule of Signs, and The Upper and Lower Bounds Theorem to get enough information to apply synthetic division(SD). We use SD at L22 to see if 1 is a root. Since it has a remainder of zero, it is a root. The 3rd line gives us the new polynomial, which is rewritten at L23. Using SD we try -6 on L23. This was done at L24. It has zero remainder and is also a root. The quadratic factor left over has complex roots, and is not involved in finding the asymptotes. The factors for the numerator are listed at L25.

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FACTORING TO FIND THE ASYMPTOTES:

FACTORING THE DENOMINATOR TO FIND VERTICAL ASYMPTOTES:

7. L26 is the polynomial in the denominator, and it is set to equal zero so we can find the roots. We can factor out the 2; this will make our SD a little easier, and will enable us to compare factors of the numerator and denominator. The calculator shows the graph crosses the x-axis at -3, 1, and 3. SD confirms this at L28. The quadratic left over has roots at X = -3, and at x = 3. These factors are listed at L29 and L30. Comparing the factors of the numerator and denominator we have a common factor of ( x - 1); therefore, there is no vertical asymptote at x = 1. I'll explain why soon. The other two factors do represent vertical asymptotes at x = 3 and x = -3.

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REMOVABLE DISCONTINUITY AND WRAPPING UP:

REMOVABLE DISCONTINUITY:

8. The denominator cannot equal zero. When x approaches -3 or 3 then the denominator approaches zero, which causes the graph to go up really fast, or down really fast depending on what side x is coming from. When x approaches 1 the denominator approaches zero also; however, there is no asymptote there because it has a common factor of (x - 1) in the numerator. Under this condition we can call x = 1 a removable discontinuity. There is a dot at x = 1 on the center curve of E2 reminding us that x cannot equal 1. It can approach 1 ever so closely forever but it cannot equal 1.

WRAPPING UP THE GRAPH--PLOTTING POINTS:

9. The asymptotes are the essential features of this graph. You simply plots points to see how the graph behaves. The end behavior on both sides of the graph approach the slant asymptote as x goes to the right or left of the origin. There are two local extrema on the graph whose location can be estimated by plotting points. With calculus you can find exactly where they are. The extrema have zero slope; therefore, we can differentiate the function, and equate the derivative to zero( for zero slope), and then solve for the roots of the derivative. We can get rid of the denominator of the derivative by multiplying both sides of the equation with it( since the right side is equated to zero). This leaves the numerator, which is a 4th degree polynomial and its roots( by the TI-89 calculator) are { -0.523497, 7.3454}. We plug the first value into the original function and get Y = -0.235527, and for the 7.3454 value we get Y = 9.24729. So the two local extrema are at these points: P1 = (-0.523497, -0.235527), and P2 = (7.3454, 9.24729).

INFORMATION IN DNA:

The amount of information God stored in a DNA molecule.

10. If we assign a letter to each nucleotide pair in a single DNA molecule, then we would have 6 billion letters. All those letters are crammed in a volume of 6.86 X 10^(--18) cubic meters( the volume of a single DNA molecule). In comparison, that many letters would require 500,000 sheets of copy paper to print them out on both sides. To get a better idea of how much information God, the Creator, stored in the geometric structure of DNA, take a good look at a dime. Theoretically you could pack enough DNA into that dime to represent 6.644 X 10^22 letters. To print this out would require a stack of copy paper that would be 2.31 X 10^11 miles high or 482,838 round trips to the moon. This stack of papers would weigh 2.746 X 10^13 tons, or the weight of over 75 million Empire State Buildings, but God can store the amount of information that this paper represents in a volume the size of a dime. This is a very low end estimate of the amount of information God packed into a DNA molecule. It is actually much more than is represented by simply assigning a single letter to each nucleotide pair.

11. We pass this way just once, then we die, and then we are judged by the Creator of this universe. Ponder the evidence that God has supplied, and then make a decision to believe in, and obey, Jesus Christ. God never expected anyone to blindly believe in Him. He has inundated us with overwhelming evidence that He is the Creator; therefore, He expects us to make the obvious decision--believe in and obey Jesus Christ.

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