- LETS TAKE A LOOK AT ONE MORE EXAMPLE.
NOW AS I MENTIONED BEFORE, OFTEN IT'S HELPFUL
TO LOOK TO AT THESE RATIONAL FUNCTIONS IN FACTORED FORM.
SO I WILL GO AHEAD AND FACTOR THE DENOMINATOR HERE.
NOW NOTICE THERE'S A COMMON FACTOR OF X
AND THEN WE'D HAVE OVER MEANING QUADRATIC
WHICH WOULD FACTOR AS WELL.
I'LL PUT THIS IN PARENTHESES.
NOW THE REASON IT'S HELPFUL TO FACTOR
IS REMEMBER FROM OUR PREVIOUS STUDIES
THE ZEROS OF THESE COMMON FACTORS,
IN THIS CASE X - 2 DO NOT GIVE US A VERTICAL ASYMPTOTE
THEY GIVE US A HOLE.
SO RIGHT AWAY I'M GOING TO MAKE A NOTE OVER WE HAVE A HOLE
AT X = 2.
SO ESSENTIALLY OUR FUNCTION WILL LOOK LIKE THE FUNCTION F OF X
IS EQUAL TO THE SIMPLIFIED VERSION 1/X x X - 3
WITH THIS HOLE.
SO I'M GOING TO GO AHEAD AND USE THE SIMPLIFIED FUNCTION
TO FIND THE HORIZONTAL AND VERTICAL ASYMPTOTES.
LET'S FIND OUR HORIZONTAL ASYMPTOTE
IF THERE IS ONE BY TAKING THE LIMIT OF THE FUNCTION
AS X APPROACHED INFINITY OR NEGATIVE INFINITY.
NOW I NOTICED RIGHT AWAY
THAT THE DEGREE OF THE DENOMINATOR IS DEGREE 2.
THE DEGREE OF THE NUMERATOR IS ZERO.
THIS LIMIT IS EQUAL TO ZERO.
THEREFORE, THERE'S A HORIZONTAL ASYMPTOTE AT Y EQUALS ZERO.
LET'S SKETCH THAT.
NEXT, LETS TAKE A LOOK AT OUR VERTICAL ASYMPTOTES.
NOW LET'S GO AHEAD AND USE THE ALGEBRAIC METHOD
AND THEN WE'LL VERIFY THE LIMITS THAT ARE REQUIRED FROM THE GRAPH
ON THE GRAPHING CALCULATOR.
AGAIN, LOOKING AT THIS FUNCTION HERE X = 0 AND X = 3
ARE ZEROS OF THE DENOMINATOR
THEREFORE WE HAVE VERTICAL ASYMPTOTES AT THOSE VALUES.
LET'S SKETCH THOSE.
HERE'S X = 0.
HERE'S X = 3.
NOW REMEMBER, TO USE LIMITS TO VERIFY THESE
WE NEED TO FIND SOME LIMITS.
LET'S FIND THE LIMIT AS X APPROACHES ZERO
FROM THE POSITIVE SIDE OR THE RIGHT SIDE OF THE FUNCTION.
NOW REMEMBER THIS HAS TO BE EITHER POSITIVE
OR NEGATIVE INFINITY
AND LET'S ALSO VERIFY THE OTHER VERTICAL ASYMPTOTE
BY FINDING THE LIMIT AS X APPROACHES 3
FROM THE LEFT SIDE THIS TIME.
NOW TO FIND THESE LIMITS WE ARE JUST GOING TO GO AHEAD
AND LOOK AT THE GRAPH ON THE GRAPHING CALCULATOR.
SO LET'S GO AHEAD AND GET OUR CALCULATORS OUT AGAIN.
GO TO Y = AND LET'S TYPE IN OUR ORIGINAL FUNCTION.
WELL WE CAN EITHER TYPE IN THE ORIGINAL
OR WE CAN TYPE IN THE SIMPLIFIED FUNCTION
AS LONG AS WE REMEMBER THE HOLE.
I'M GOING TO GO AHEAD AND JUST TYPE IN THE SIMPLIFIED FUNCTION.
LET'S LOOK AT THE GRAPH.
NOW AS WE ANTICIPATED THERE ARE THREE PIECES TO THIS FUNCTION
BECAUSE THERE ARE TWO VERTICAL ASYMPTOTES.
NOW LETS TAKE A LOOK AT THESE LIMITS THAT WE WROTE OUT HERE
THESE ONE-SIDED LIMITS.
AS X APPROACHES ZERO FROM THE RIGHT
OR THE POSITIVE SIDE YOU CAN SEE THAT THE GRAPH IS GOING DOWN.
THAT'S GOING TO GIVE US NEGATIVE INFINITY HERE.
NOW LET'S TAKE A LOOK AT THAT GRAPH AGAIN.
LET'S APPROACH THREE FROM THE LEFT
AND IF I APPROACH +3 FROM THE LEFT SIDE AGAIN
(AUDIO SKIP)
IT WILL ALSO GIVE US A LIMIT OF NEGATIVE INFINITY
VERIFYING WE DO HAVE OUR VERTICAL ASYMPTOTES.
NOW TO MAKE A COMPLETE GRAPH
WE DO NEED TO GO BACK TO THE CALCULATOR
AND FIND SOME ADDITIONAL POINTS.
AGAIN, BY HITTING 2nd GRAPH WE CAN SCROLL UP AND DOWN HERE
AND TRY TO FIND SOME POINTS THAT WOULD BE EASY TO PLOT
AND MAKE A NICE ACCURATE GRAPH.
I'VE ALREADY GRAPHED THIS FUNCTION
USING A DIFFERENT GRAPHING TECHNOLOGY
SO LET'S GO AHEAD AND TAKE A LOOK AT THAT.
HERE IT IS.
AGAIN, HERE WE HAVE A VERTICAL ASYMPTOTE X = 0, X = 3.
HORIZONTAL ASYMPTOTE Y = 0.
NOW THE ONE THING THAT WE HAVE TO ADD IN HERE
REMEMBER THERE WAS A HOLE AT X = 2.
IT'S NOT SHOWING ON THIS GRAPH
SO WHAT WE HAVE TO DO TO MAKE AN ACCURATE GRAPH
IS GO HERE TO THE VALUE WHERE X = 2
AND WE SHOULD HAVE AN OPEN CIRCLE
TO SHOW THERE IS A HOLE IN THAT GRAPH THERE.
NOW SOMETIMES A FUNCTION CAN HAVE A LINEAR ASYMPTOTE
THAT IS NEITHER VERTICAL NOR HORIZONTAL,
WHICH IS CALLED A SLANT, OR OBLIQUE ASYMPTOTE.
FOR ANY RATIONAL FUNCTION OF THE FORM F OF X = P OF X
DIVIDED BY Q OF X
A SLANT ASYMPTOTE OCCURS WHEN THE DEGREE OF P OF X
OR THE NUMERATOR IS EXACTLY ONE MORE
THAN THE DEGREE OF THE DENOMINATOR Q OF X
A GRAPH CAN CROSS A SLANT ASYMPTOTE.
IN GENERAL, TO FIND THE EQUATION OF A SLANT ASYMPTOTE
WE NEED TO PERFORM LONG DIVISION
AND THEN TAKE THE LIMIT OF THAT
AS X APPROACHES POSITIVE OR NEGATIVE INFINITY.
NOW THERE ARE MANY STRATEGIES TO FIND KEY COMPONENTS OF GRAPHS.
I'M NOT GOING TO GO THROUGH EVERY SINGLE ONE OF THESE.
A LOT OF THESE ARE REVIEW
BUT I WILL HIGHLIGHT A COUPLE POINTS ON EACH OF THESE.
YOU MAY WANT TO FIND INTERCEPTS, ASYMPTOTES, DERIVATIVES,
AND DOMAIN.
CRITICAL VALUES OF F.
DETERMINE WHETHER THE FUNCTION IS INCREASING AND OR DECREASING
AS WELL AS RELATIVE EXTREMA, INFLECTION POINTS, CONCAVITY,
AND THE SKETCH OF THE GRAPH.
YOU CAN SEE THE NEXT SEVERAL VIDEOS FOR ADDITIONAL EXAMPLES
ON THESE TOPICS.
JUST REMEMBER THIS SECTION IS JUST A COMBINATION
OF SEVERAL TOPICS COVERED IN PREVIOUS SECTIONS
SO TRY NOT TO GET OVERWHELMED
AND JUST TAKE IT ONE STEP AT A TIME.
HAVE A GREAT DAY.