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- WELCOME TO A LESSON ON DETERMINING SLANT ASYMPTOTES
OF A RATIONAL FUNCTION.
THIS VIDEO DOES ASSUME THAT YOU'VE ALREADY WATCHED THE VIDEO
ON VERTICAL AND HORIZONTAL ASYMPTOTES
OF RATIONAL FUNCTIONS.
THE GOALS OF THIS VIDEO ARE TO DETERMINE THE SLANT ASYMPTOTES
OF A RATIONAL FUNCTION
AND THEN GRAPH A RATIONAL FUNCTION.
HERE WE SEE A GRAPH OF A RATIONAL FUNCTION
THAT HAS A SLANT ASYMPTOTE AS WE SEE HERE.
A SLANT ASYMPTOTE IS JUST LIKE A VERTICAL
OR HORIZONTAL ASYMPTOTE
WHERE IT'S THE LINE THAT THE GRAPH APPROACHES
BUT INSTEAD OF BEING A VERTICAL OR HORIZONTAL LINE
IT'S A SLANTED LINE.
SO LET'S TAKE A LOOK AT HOW WE CAN DETERMINE
WHEN A RATIONAL FUNCTION HAS A SLANT ASYMPTOTE
AND THEN HOW TO DETERMINE IT.
SLANT ASYMPTOTES SOMETIMES CALLED OBLIQUE ASYMPTOTES
ARE LINES THE GRAPH APPROACHES.
A GRAPH HAS A SLANT ASYMPTOTE
IF THE DEGREE OF THE NUMERATOR IS ONE DEGREE HIGHER
THAN THE DEGREE OF THE DENOMINATOR.
SO IF IT SATISFIES THIS CONDITION--
TO FIND THE SLANT ASYMPTOTE
WE'LL DIVIDE THE NUMERATOR BY THE DENOMINATOR
AND THEN DETERMINE WHAT LINE THE GRAPH APPROACHES
AS X APPROACHES POSITIVE OR NEGATIVE INFINITY.
AND WHAT WE'LL FIND IS THE EQUATION OF THE LINE
WILL BE FROM ONLY THE QUOTIENT PART OF OUR ANSWER,
MEANING WE CAN DISREGARD THE REMAINDER PART OF OUR ANSWER.
LET'S LOOK AT AN EXAMPLE.
HERE WE WANT TO DETERMINE ALL OF THE ASYMPTOTES
AND THEN GRAPH THE FUNCTION.
SO THE FIRST STEP IN ALL THESE TYPES OF PROBLEMS
IS TO FIRST FACTOR THE GIVEN FUNCTION.
SO OUR NUMERATOR IS THE DIFFERENCE OF SQUARES,
OUR DENOMINATOR HAS A COMMON FACTOR OF 2.
NOTICE THERE'S ONLY ONE ZERO OF THE DENOMINATOR
AND IT'S NOT A ZERO OF THE NUMERATOR
SO WE DO HAVE A VERTICAL ASYMPTOTE AT X = 2.
LET'S GO AHEAD AND SKETCH THAT.
NEXT, FOR THE HORIZONTAL ASYMPTOTES,
SINCE THE DEGREE OF THE NUMERATOR IS EQUAL TO 2
AND THE DEGREE OF THE DENOMINATOR IS EQUAL TO 1,
THE NUMERATOR HAS A HIGHER DEGREE
AND THEREFORE, THERE IS NO HORIZONTAL ASYMPTOTE.
THE REASON FOR THIS IS IF WE ANALYZE THIS
AS X APPROACHES POSITIVE INFINITY,
THE NUMERATOR INCREASES FASTER THAN THE DENOMINATOR.
BUT SINCE THE DEGREE OF THE NUMERATOR IS ONE DEGREE HIGHER
THAN THAT OF THE DENOMINATOR, WE DO HAVE A SLANT ASYMPTOTE
SO WE HAVE TO TAKE X SQUARED - 1 AND DIVIDE BY 2X - 4.
SO LET'S GO AHEAD AND PERFORM LONG DIVISION HERE.
WE'LL HAVE X SQUARED,
NOW WE NEED OUR TERMS IN DESCENDING ORDER
SO WE'RE GOING TO PUT IN A ZERO X TERM
AND WE'LL DIVIDE THIS BY 2X - 4.
AND NOW WE HAVE TO ASK OURSELVES WHAT TIMES 2X
WOULD EQUAL X SQUARED
AND THAT WOULD BE 1/2X.
1/2X x 2X WOULD GIVE US X SQUARED, 1/2X x -4 = -2X
AND WE'RE SUBTRACTING HERE
BUT WE NORMALLY ADD THE OPPOSITE INSTEAD
SO WE CHANGE THIS TO ADDITION, CHANGE THIS TO A NEGATIVE,
CHANGE THIS TO PLUS.
THE NEXT TERM WOULD BE 2X, BRING DOWN THE -1,
WHAT TIMES 2X WOULD GIVE US 2X.
THAT WOULD BE A +1.
(1 x 2X) - 4 IS 2X - 4.
BE CAREFUL HERE, WE'RE SUBTRACTING
SO WE'LL ADD THE OPPOSITE INSTEAD.
WE HAVE A REMAINDER OF 3.
SO WHAT WE'D NORMALLY DO NOW IS PUT +3/2X - 4.
SO THIS TELLS US THAT WE CAN REWRITE THE GIVEN FUNCTION
AS (1/2X + 1) + (3/2X - 4).
BUT IF WE TAKE A LOOK AT WHAT HAPPENS TO THIS FUNCTION
AS X APPROACHES, LET'S SAY POSITIVE INFINITY
THE FRACTION PART OF THIS QUOTIENT
IS GOING TO APPROACH ZERO.
SO THE REMAINING PART TELLS US
THE EQUATION OF OUR SLANT ASYMPTOTE.
OUR SLANT ASYMPTOTE WILL BE Y = 1/2X + 1.
LET'S GO AHEAD AND GRAPH THAT.
Y INTERCEPT OF +1, SLOPE OF 1/2, UP 1 RIGHT 2.
AND NOW TO MAKE A NICE GRAPH OF THIS FUNCTION
WE JUST NEED TO SELECT A FEW POINTS
NEAR THE VERTICAL ASYMPTOTE.
SO WE LET X = 3 AND X = 1 THAT MIGHT BE ENOUGH INFORMATION.
WHEN X IS EQUAL TO 3 WE'RE GOING TO HAVE 9 - 1 THAT'S 8
DIVIDED BY 6 - 4 THAT'S 2.
8 DIVIDED BY 2 WOULD EQUAL 4.
X IS EQUAL TO 1 WE'D HAVE 0.
SO WE HAVE THE POINT (3,4) HERE AND THE POINT (1,0).
AND BECAUSE WE KNOW IT APPROACHES THE ASYMPTOTE
THIS IS ENOUGH INFORMATION TO MAKE A NICE GRAPH.
AND HERE'S THE GRAPH USING SOME SOFTWARE
AND YOU CAN SEE OUR SKETCH IS PRETTY GOOD.
AND AGAIN, WE COULD ALSO SKETCH IN OUR VERTICAL ASYMPTOTE HERE.
LET'S TAKE A LOOK AT ONE MORE EXAMPLE.
AGAIN, OUR FIRST STEP IS GOING TO BE TO FACTOR
BOTH THE NUMERATOR AND DENOMINATOR.
NOTICE OUR NUMERATOR HAS A COMMON FACTOR OF X
AND THEN THIS TRINOMIAL FACTORS AGAIN.
SO WE'D HAVE X(X - 2)(X + 1).
OUR DENOMINATOR WOULD BE (X x X) - 1.
SO ONE THING WE SHOULD NOTICE RIGHT AWAY
IS THAT THERE'S A COMMON FACTOR HERE OF X.
SO THIS CAN SIMPLIFY OUT
BUT SINCE X = 0 IS A 0 OF BOTH THE NUMERATOR
AND THE DENOMINATOR
WE HAVE A HOLE AT X = 0.
LET'S GO AHEAD AND WRITE THAT DOWN.
AND THE OTHER 0 OF THE DENOMINATOR X = 1
WOULD BE A VERTICAL ASYMPTOTE.
AND THEN, FOR THE HORIZONTAL ASYMPTOTE
SINCE THE DEGREE OF THE NUMERATOR IS EQUAL TO 3
THE DEGREE OF THE DENOMINATOR IS EQUAL TO 2,
THERE IS NO HORIZONTAL ASYMPTOTE.
HOWEVER, SINCE THE DEGREE OF THE NUMERATOR
IS ONE DEGREE HIGHER THAN THE DENOMINATOR
WE DO HAVE A SLANT ASYMPTOTE
SO NOW WE DO HAVE TO PERFORM LONG DIVISION
TO DETERMINE THE EQUATION OF THE SLANT ASYMPTOTE.
WE DO HAVE TO DIVIDE THE NUMERATOR BY THE DENOMINATOR.
TO SET UP THE DIVISION
WE COULD EITHER USE THE ORIGINAL FUNCTION FORM
OR WE COULD USE THE FUNCTION
AFTER WE SIMPLIFIED OUT THE COMMON FACTOR
BECAUSE WE ALREADY NOTED THAT WE HAVE TO HAVE A HOLE AT X = 0.
WHAT I MEAN BY THAT IS INSTEAD OF USING THIS FORM
FOR LONG DIVISION,
ONCE WE SIMPLIFY OUT THE COMMON FACTOR OF X
WE COULD PERFORM LONG DIVISION IN THIS FORM
AND THAT WOULD BE A LITTLE BIT EASIER.
SO LET'S GO AHEAD AND DO THAT.
SO WE'RE GOING TO TAKE X SQUARED - X - 2
AND DIVIDE BY X - 1.
SO WHAT TIMES X WOULD GIVE US X SQUARED, WELL THAT WOULD BE X.
WE'RE GOING TO MULTIPLY.
IT WOULD BE X SQUARED - X
AND THEN WE'RE GOING TO SUBTRACT BY ADDING THE OPPOSITE.
SO THAT WOULD BE 0, BRING DOWN THE -2
AND THIS WOULD BE THE REMAINDER.
SO WE'RE GOING TO HAVE X - 2/X - 1.
BUT ONCE AGAIN, AS X APPROACHES
EITHER POSITIVE OR NEGATIVE INFINITY,
THIS FRACTION HERE IS GOING TO APPROACH ZERO,
SO THE EQUATION OF THE SLANT ASYMPTOTE
IS JUST GOING TO BE Y = X.
NOW LET'S GO AHEAD AND GRAPH OUR ASYMPTOTES.
WE HAVE Y = X AS OUR SLANT ASYMPTOTE
AND WE HAVE X = 1 AS OUR VERTICAL ASYMPTOTE.
NOW LET'S GO AND SELECT A COUPLE POINTS TO MAKE A NICE GRAPH.
AGAIN, SIMPLE TO PICK VALUES OF X
THAT ARE NEAR THE VERTICAL ASYMPTOTE
SO WE'LL SELECT X = 2 AND WE CAN SELECT X = 0
AS LONG AS WE USE THE SIMPLIFIED FORM OF THE FUNCTION
BECAUSE REMEMBER THERE IS A HOLE THERE.
SO LET'S GO AHEAD AND DO THAT.
SO IF X = 2 WE WOULD HAVE 2 - 2 THAT'S GOING TO BE 0
AND WHEN X = 0 WE'RE GOING TO HAVE -2/-1
THAT'S GOING TO BE POSITIVE 2.
BUT REMEMBER THAT'S ACTUALLY A HOLE IN THE FUNCTION.
SO IF WE PLOT THE POINT (2,0) WE'D BE HERE.
IF WE PLOT THE POINT (0,2) WE'D BE HERE
BUT BECAUSE IT'S A HOLE WE'LL MAKE AN OPEN CIRCLE HERE
OR AN OPEN POINT
AND AGAIN, WE KNOW THE FUNCTION
WILL BE APPROACHING OUR ASYMPTOTES
SO WE CAN SKETCH A NICE GRAPH THAT LOOKS SOMETHING LIKE THIS
AND THIS PIECE WOULD LOOK SOMETHING LIKE THIS.
LET'S TAKE A LOOK AT THIS WITH SOME SOFTWARE.
AND WHAT YOU'LL NOTICE WITH A LOT OF SOFTWARE
IS BECAUSE THIS SCREEN DOES NOT HAVE ENOUGH DEFINITION
YOU CAN'T SEE THE HOLES IN THE FUNCTION.
SO IF YOU'RE CHECKING THESE ON YOUR GRAPHING CALCULATOR
THAT'S GOOD
BUT MAKE SURE THAT YOU LOOK AT THE TABLE TO IDENTIFY
WHERE THE HOLES SHOULD OCCUR.
WE SHOULD HAVE A HOLE RIGHT HERE WHEN X = 0.
AND THAT'S GOING TO DO IT FOR THIS VIDEO.
I HOPE YOU FOUND THIS HELPFUL.