- WE WANT TO FIND THE EQUATION OF THE VERTICAL ASYMPTOTE
AND THE EQUATION OF THE SLANT ASYMPTOTE
OF THE GIVEN RATIONAL FUNCTION.
BEFORE FINDING THE VERTICAL ASYMPTOTES THOUGH
IT IS IMPORTANT TO FACTOR THE NUMERATOR AND DENOMINATOR
TO SEE IF THERE ARE ANY COMMON FACTORS.
BECAUSE ZEROS OF THE COMMON FACTORS
PRODUCE HOLES IN THE FUNCTION
AND THEY WILL NOT GIVE US VERTICAL ASYMPTOTES.
FOR EXAMPLE, IF WE TAKE A LOOK
AT THESE TWO FUNCTIONS QUICKLY,
NOTICE HOW IN FACTORED FORM WE CAN EASILY SEE
THAT X = -2 AND X - -1.
IN BOTH CASES MAKE THE DENOMINATORS EQUAL TO ZERO.
SO WHILE THESE VALUES WOULD BE EXCLUDED FROM THE DOMAIN
NOTICE HOW IN THIS FIRST FUNCTION
THERE'S A COMMON FACTOR OF X + 1
IN THE NUMERATOR AND DENOMINATOR
AND THEREFORE X = -1.
THERE WOULD NOT BE A VERTICAL ASYMPTOTE,
THERE WOULD BE A HOLE IN THE FUNCTION.
SO THIS RATIONAL FUNCTION ONLY HAS 1 VERTICAL ASYMPTOTE
AT X = -1.
BUT LOOKING AT THE SECOND FUNCTION
NOTICE HOW THERE ARE NO COMMON FACTORS
BETWEEN THE NUMERATOR AND DENOMINATOR
AND THEREFORE, THIS FUNCTION WOULD HAVE
TWO VERTICAL ASYMPTOTES.
ONE AT X = -2 AND ONE AT X = -1.
SO GOING BACK TO OUR EXAMPLE
LET'S BEGIN BY FACTORING THE NUMERATOR.
NOTICE HOW THE DENOMINATOR DOES NOT FACTOR.
SO LET'S SEE IF THIS WILL FACTOR
INTO TWO BINOMIAL FACTORS.
NOTICE HOW THE FIRST TERM IS 8X/SQUARED
SO WE COULD HAVE X AND 8X OR 2X AND 4X.
LET'S TRY 2X AND 4X.
THE SECOND TERMS MUST COME FROM THE FACTORS OF -1
SO THAT THE SUM OF THE INNER AND OUTER PRODUCT
WOULD BE -2X.
SO NOTICE HOW IF WE PUT A -1 HERE OR - 1 AND A + 1 HERE,
THE INNER PRODUCT IS -4X, THE OUTER PRODUCT IS 2X
WHICH DOES SUM TO -2X.
NOW THAT IT'S IN FACTORED FORM
WE CAN SEE THE NUMERATOR AND DENOMINATOR
DO NOT SHARE ANY COMMON FACTORS
AND THEREFORE THIS RATIONAL FUNCTION
WILL NOT HAVE ANY HOLES
AND THEREFORE WE CAN FIND THE EQUATION
OF THE VERTICAL ASYMPTOTE
BY SETTING THE DENOMINATOR EQUAL TO 0 AND SOLVING FOR X.
SO WE WOULD HAVE 4X - 3 = 0.
SO WE WOULD ADD 3 TO BOTH SIDES AND THEN DIVIDE BY 4.
SO WE HAVE X = 3/4 AS THE EQUATION
OF THE VERTICAL ASYMPTOTE.
NOW IT DOESN'T ASK
BUT REMEMBER WE'D ALSO HAVE TO EXCLUDE THIS VALUE
FROM THE DOMAIN OF THE RATIONAL FUNCTION.
SO THE EQUATION OF THE VERTICAL ASYMPTOTE IS X = 3/4.
NOW LET'S TALK ABOUT THE SLANT ASYMPTOTE.
A RATIONAL FUNCTION HAS A SLANT ASYMPTOTE
WHEN THE DEGREE OF THE NUMERATOR IS ONE DEGREE HIGHER
THAN THE DEGREE OF THE DENOMINATOR.
SO NOTICE IN THIS CASE, THE DEGREE OF THE NUMERATOR IS 2,
THE DEGREE OF THE DENOMINATOR IS 1
AND THEREFORE, WE'LL HAVE A SLANT ASYMPTOTE.
AND SO TO FIND THE EQUATION OF THE SLANT ASYMPTOTE,
WE'LL ACTUALLY HAVE TO DIVIDE THE NUMERATOR
BY THE DENOMINATOR
AND BECAUSE THE COEFFICIENT OF THE DENOMINATOR IS 4
WE'RE GOING TO GO AHEAD AND PERFORM LONG DIVISION.
SO WE'LL HAVE THE QUANTITY 8X/SQUARED - 2X - 1
DIVIDED BY THE QUANTITY 4X - 3.
SO WE'RE GOING TO PERFORM THIS DIVISION
THEN DETERMINE WHAT'S HAPPENING TO OUR QUOTIENT
AS X APPROACHES POSITIVE OR NEGATIVE INFINITY.
SO TO BEGIN WE'RE GOING TO DETERMINE
WHAT TIMES 4X IS = 8X/SQUARED.
WELL 2X x 4X WOULD BE 8X/SQUARED.
SO THE FIRST TERM IN THE QUOTIENT IS 2X.
AND IF WE'RE NOT ABLE TO DETERMINE THIS FIRST TERM
WE CAN ALWAYS TAKE 8X/SQUARED AND DIVIDE BY 4X.
NOTICE 8X/SQUARED DIVIDED BY 4X WOULD BE 2X.
AND NOW WE'RE GOING TO MULTIPLY 2X AND 4X - 3.
2X x 4X = 8X/SQUARED AND THEN 2X x -3 = -6X OR - 6X.
NOW WE WANT TO SUBTRACT BUT INSTEAD OF SUBTRACTING
WE'RE GOING TO ADD THE OPPOSITE.
SO WE CAN CHANGE THIS TO ADDITION
IF WE CHANGE THE SIGN OF THIS TERM
AND CHANGE THE SIGN OF THIS TERM.
NOTICE HOW THIS WOULD BE 0.
NOW THE X TERM WOULD BE 4X.
BRING DOWN THE LAST TERM OF -1
AND NOW WE WANT TO KNOW WHAT x 4X WOULD BE EQUAL TO 4X
WHICH WOULD BE 1.
SO THE NEXT TERM WOULD BE + 1
OR AGAIN IF WE NEED TO WE CAN TAKE 4X AND DIVIDE BY 4X
TO GET THE 1.
WELL 1 x 4X - 3 IS JUST 4X - 3.
AND WE'RE GOING TO SUBTRACT THIS BY ADDING THE OPPOSITE.
SO WE'RE GOING TO CHANGE THE SIGN, CHANGE THE SIGN,
AND CHANGE THE SIGN.
SO WE HAVE -1 + 3 THAT'S +2.
WHICH MEANS OUR REMAINDER IS 2.
AND WE CAN WRITE THE QUOTIENT AS 2X + 1 + 2
OVER OUR DIVISOR OF 4X - 3.
AND NOW WE CAN USE THIS QUOTIENT
TO FIND THE EQUATION OF THIS SLANT ASYMPTOTE.
WE WANT TO DETERMINE WHAT'S HAPPENING TO THIS QUOTIENT
AS X APPROACHES POSITIVE OR NEGATIVE INFINITY.
WE WANT TO FOCUS MAINLY ON THIS FRACTION HERE
OR OUR REMAINDER.
NOTICE AS X APPROACHES POSITIVE OR NEGATIVE INFINITY,
THE DENOMINATOR EITHER INCREASES WITHOUT BOUND
OR DECREASES WITHOUT BOUND
AND THEREFORE THIS FRACTION IS GOING TO APPROACH 0
AND THEREFORE, AS X APPROACHES POSITIVE OR NEGATIVE INFINITY
THIS RATIONAL FUNCTION WILL APPROACH THE LINE Y = 2X + 1,
WHICH IS THE EQUATION OF OUR SLANT OR OBLIQUE ASYMPTOTE.
NOTICE HOW THIS QUESTION DOES NOT ASK US
FOR A HORIZONTAL ASYMPTOTE.
IF WE THINK ABOUT IT,
IF THE RATIONAL FUNCTION IS APPROACHING
A SLANTED OR OBLIQUE LINE
THEN THE RATIONAL FUNCTION
WOULD NOT HAVE A HORIZONTAL ASYMPTOTE.
ANOTHER REASON WE KNOW THIS IS
NOTICE HOW THE DEGREE OF THE NUMERATOR IS GREATER
THAN THE DEGREE OF THE DENOMINATOR
AND THEREFORE, AS X APPROACHES INFINITY
NOTICE HOW THE FUNCTION VALUES WOULD ALSO APPROACH INFINITY
VERIFYING THERE IS NO HORIZONTAL ASYMPTOTE.
AND BEFORE WE GO, LET'S VERIFY THESE RESULTS
BY LOOKING AT THE GRAPH OF OUR RATIONAL FUNCTION.
NOTICE HERE WE HAVE OUR VERTICAL ASYMPTOTE
WITH THE EQUATION X = 3/4
AND HERE WE HAVE OUR SLANT ASYMPTOTE
WITH THE EQUATION Y = 2X + 1.
SO THIS GRAPH DOES VERIFY THAT OUR WORK IS CORRECT.
I HOPE YOU FOUND THIS EXPLANATION HELPFUL.