- IF ASKED TO DETERMINE THE END BEHAVIOR OR LONG RUN BEHAVIOR
OF A POLYNOMIAL FUNCTION,
IT'S POSSIBLE TO DETERMINE THE END BEHAVIOR
JUST BY KNOWING THE DEGREE OF THE POLYNOMIAL
AND THE SIGN OF THE LEADING COEFFICIENT.
BUT BEFORE WE START,
LET'S REVIEW WHAT END BEHAVIOR OR LONG RUN BEHAVIOR IS.
THE END BEHAVIOR DESCRIBES THE VALUE OF THE FUNCTION
OR THE Y VALUE
AS X APPROACHES THE LEFT OR APPROACHES NEGATIVE INFINITY
AND APPROACHES THE RIGHT OR APPROACHES POSITIVE INFINITY.
FIRST LET'S CONSIDER WHEN THE DEGREE IS EVEN,
AND WE'LL ASSUME THAT THE TERMS OF THE POLYNOMIAL
ARE IN DESCENDING ORDER,
SO OUR POLYNOMIAL WILL HAVE DEGREE N
AND THE LEADING COEFFICIENT WILL BE A.
SO IF N IS GREATER THAN OR EQUAL OR 2, AND EVEN,
AND THE LEADING COEFFICIENT IS POSITIVE OR GREATER THAN 0,
THEN AS X APPROACHES INFINITY, OR AS WE MOVE RIGHT,
F OF X APPROACHES POSITIVE INFINITY
OR THE Y VALUES INCREASE WITHOUT BOUND.
SO LOOKING AT THE GRAPH, AS X APPROACHES POSITIVE INFINITY,
WE'D BE MOVING RIGHT ALONG THE FUNCTION.
AS WE MOVE RIGHT ALONG THE FUNCTION,
NOTICE HOW THE Y VALUES OR FUNCTION VALUES ARE GOING UP
OR INCREASE WITHOUT BOUND,
THEREFORE, F OF X APPROACHES POSITIVE INFINITY.
AS X APPROACHES NEGATIVE INFINITY,
F OF X ALSO APPROACHES POSITIVE INFINITY.
SO AS X APPROACHES NEGATIVE INFINITY,
WHERE WE MOVE LEFT ALONG THE FUNCTION,
NOTICE HOW THE Y VALUES OR FUNCTION VALUES
ALSO INCREASE WITHOUT BOUND APPROACHING POSITIVE INFINITY.
AND THIS WILL ALWAYS BE THE CASE
WHEN THE DEGREE IS GREATER THAN OR EQUAL TO 2, EVEN,
AND THE LEADING COEFFICIENT IS POSITIVE.
SO FOR EXAMPLE, IF WE HAVE A FUNCTION LIKE F OF X
EQUALS 5X TO THE EIGHTH - 6X TO THE FIFTH,
PLUS 3X SQUARED - 2,
THE END BEHAVIOR OF THIS FUNCTION WOULD BE THE SAME
AS THE END BEHAVIOR DESCRIBED HERE,
AND WE CAN DETERMINE THIS
JUST BY LOOKING AT THE DEGREE OF THE POLYNOMIAL
AND THE SIGN OF THE LEADING COEFFICIENT.
NEXT, IF N IS GREATER THAN OR EQUAL OR 2 AND EVEN,
BUT THE LEADING COEFFICIENT IS NEGATIVE OR LESS THAN 0,
THEN AS X APPROACHES POSITIVE INFINITY,
F OF X APPROACHES NEGATIVE INFINITY.
SO AS WE MOVE RIGHT ALONG THE GRAPH,
NOTICE HOW THE Y VALUES OR FUNCTIONS VALUES
DECREASE WITHOUT BOUND APPROACHING NEGATIVE INFINITY,
AND AS X APPROACHES NEGATIVE INFINITY,
F OF X ALSO APPROACHES NEGATIVE INFINITY.
SO AS WE MOVE LEFT, OR AS X APPROACHES NEGATIVE INFINITY,
THE FUNCTION VALUES OR Y VALUES ALSO DECREASE WITHOUT BOUND,
APPROACHING NEGATIVE INFINITY.
AND AGAIN, THIS WILL ALWAYS BE TRUE
WHEN WE HAVE A POLYNOMIAL FUNCTION WITH AN EVEN DEGREE
AND A NEGATIVE LEADING COEFFICIENT.
SO FOR EXAMPLE, THE FUNCTION F OF X EQUALS -0.25X TO THE FOURTH
+ 5X TO THE THIRD - 200
WILL HAVE THE SAME END BEHAVIOR AS DESCRIBED HERE,
AND WE KNOW THIS JUST BECAUSE THE DEGREE IS EVEN
AND THE LEADING COEFFICIENT IS NEGATIVE.
NOW, LET'S CONSIDER END BEHAVIOR
WHEN WE HAVE A POLYNOMIAL FUNCTION AND THE DEGREE IS ODD.
IF N IS GREATER THAN OR EQUAL TO 1, AND ODD,
WITH A POSITIVE LEADING COEFFICIENT,
THEN AS X APPROACHES POSITIVE INFINITY,
F OF X APPROACHES POSITIVE INFINITY.
SO AS WE MOVE RIGHT ALONG THE GRAPH,
OR AS X APPROACHES POSITIVE INFINITY,
NOTICE HOW THE FUNCTION VALUES OR Y VALUES
ARE INCREASING WITHOUT BOUND,
THEREFORE, F OF X IS APPROACHING POSITIVE INFINITY,
AND AS X APPROACHES NEGATIVE INFINITY,
F OF X APPROACHES NEGATIVE INFINITY.
SO AS WE MOVE LEFT ALONG THE GRAPH,
NOTICE HOW THE Y VALUES OR FUNCTION VALUES
ARE DECREASING WITHOUT BOUND, APPROACHING NEGATIVE INFINITY.
THIS WILL ALWAYS BE THE CASE WHEN THE DEGREE IS ODD
AND THE LEADING COEFFICIENT IS POSITIVE.
SO FOR EXAMPLE, IF WE HAVE THE FUNCTION F OF X EQUALS
2X TO THE SEVENTH - 2X TO THE SIXTH
+ 7X TO THE FIFTH - 3X SQUARED - 1,
AGAIN, BECAUSE THE DEGREE IS ODD
AND THE LEADING COEFFICIENT IS POSITIVE,
THE END BEHAVIOR WILL BE THE SAME AS DESCRIBED HERE.
AND THE LAST CASE
IS WHEN N IS GREATER THAN OR EQUAL TO 1 AND ODD
AND A IS NEGATIVE OR LESS THAN 0,
THAN AS X APPROACHES POSITIVE INFINITY,
F OF X APPROACHES NEGATIVE INFINITY,
SO AS WE MOVE RIGHT ALONG THE GRAPH,
NOTICE HOW THE FUNCTION VALUES OR Y VALUES
ARE DECREASING WITHOUT BOUND APPROACHING NEGATIVE INFINITY,
AND AS X APPROACHES NEGATIVE INFINITY,
F OF X APPROACHES POSITIVE INFINITY.
SO AS WE MOVE LEFT ALONG THE GRAPH,
NOTICE HOW THE FUNCTION VALUES OR Y VALUES
ARE INCREASING WITHOUT BOUND APPROACHING POSITIVE INFINITY.
THIS WILL ALWAYS BE THE CASE WHEN THE DEGREE IS ODD
AND THE LEADING COEFFICIENT IS NEGATIVE.
SO IF WE HAVE F OF X = -10X TO THE FIFTH
+ 4X TO THE FOURTH - 3,
THIS FUNCTION WILL HAVE THE SAME END BEHAVIOR
AS DESCRIBED HERE BECAUSE THE DEGREE IS ODD,
THE LEADING COEFFICIENT IS NEGATIVE.
OF COURSE, WHENEVER YOU'RE ASKED TO DESCRIBE THE END BEHAVIOR
AND YOU'RE NOT SURE,
IT'S BEST TO GRAPH THE FUNCTION AND ANALYZE THE GRAPH.
I HOPE YOU FOUND THIS SUMMARY HELPFUL.