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- NOW WE'LL LOOK AT TWO EXAMPLES
OF DETERMINING END BEHAVIOR OR LONG RUN BEHAVIOR
OF EXPONENTIAL FUNCTIONS.
END BEHAVIOR DESCRIBES THE VALUE OF THE FUNCTION
OR THE Y VALUE AS X APPROACHES THE LEFT
OR AS X APPROACHES NEGATIVE INFINITY
AND AS X APPROACHES THE RIGHT
OR X APPROACHES POSITIVE INFINITY,
SO LET'S FIRST DETERMINE THE END BEHAVIOR OF F OF X = 3 x 2
TO THE POWER OF X - 10.
WE'LL FIRST CONSIDER AS X APPROACHES POSITIVE INFINITY.
WELL AS X APPROACHES POSITIVE INFINITY
LOOKING AT THE FUNCTION NOTICE HOW THE EXPONENT ON BASE 2
IS GETTING LARGER AND LARGER
AND THEREFORE THE FUNCTION VALUES
ARE GETTING LARGER AND LARGER,
THEREFORE F OF X WILL APPROACH POSITIVE INFINITY.
WE CAN EASILY VERIFY THIS GRAPHICALLY.
AS X APPROACHES POSITIVE INFINITY
WE'RE MOVING TOWARD THE RIGHT ALONG THE FUNCTION
SO AS X APPROACHES POSITIVE INFINITY
WE'D BE MOVING IN THIS DIRECTION HERE
AND NOTICE HOW THE FUNCTION VALUES OR THE Y VALUES
INCREASE WITHOUT BOUND
AND THEREFORE F OF X DOES APPROACH POSITIVE INFINITY.
NOW LET'S CONSIDER X APPROACHING NEGATIVE INFINITY.
WELL AS X APPROACHES NEGATIVE INFINITY
NOTICE HOW THE EXPONENTS ON THE BASE 2 ARE NOW NEGATIVE
AND GETTING SMALLER AND SMALLER
WHICH MEANS 3 x 2 TO THE POWER OF X IS APPROACHING ZERO
AND SINCE 0 - 10 = -10,
AS X APPROACHES NEGATIVE INFINITY
F OF X APPROACHES -10.
IF WE LOOK AT THIS GRAPHICALLY,
AS X APPROACHES NEGATIVE INFINITY
WE'RE MOVING LEFT ALONG THE GRAPH.
SO AS WE MOVE LEFT ALONG THE GRAPH
WE CAN SEE THE FUNCTION VALUES ARE APPROACHING -10
AND THEREFORE AS X APPROACHES NEGATIVE INFINITY
F OF X DOES APPROACH -10.
WHICH ALSO MEANS THIS EXPONENTIAL FUNCTION
HAS A HORIZONTAL ASYMPTOTE OF Y = -10.
LET'S TAKE A LOOK AT A SECOND EXAMPLE.
HERE WE HAVE F OF X = -2 x 0.5 RAISED TO THE POWER OF X + 5.
AGAIN, WE'LL FIRST CONSIDER
AS X APPROACHES POSITIVE INFINITY.
SO AS X APPROACHES POSITIVE INFINITY
NOTICE HOW WE WOULD HAVE 0.5
RAISED TO A HIGHER AND HIGHER EXPONENT,
BUT EACH TIME WE MULTIPLY BY 0.5
THE VALUE GETS SMALLER AND SMALLER
SO THIS FIRST PRODUCT APPROACHES ZERO
AND THEREFORE THE FUNCTION VALUE WILL APPROACH 0 + 5
OR 5.
LOOKING AT THIS GRAPHICALLY,
AS X APPROACHES POSITIVE INFINITY
WE'RE MOVING TOWARD THE RIGHT ALONG THE FUNCTION
AND WE CAN SEE AS WE APPROACH THE RIGHT
THE FUNCTION VALUES DO APPROACH POSITIVE 5.
WHICH DOES MEAN WE HAVE A HORIZONTAL ASYMPTOTE OF Y = 5
FOR THIS EXPONENTIAL FUNCTION.
AND NOW WHEN X APPROACHES NEGATIVE INFINITY
IT MIGHT BE A LITTLE MORE DIFFICULT TO TELL
BUT IF RAISED A FRACTION TO A NEGATIVE EXPONENT
AND THAT EXPONENT GETS SMALLER AND SMALLER
THE VALUE OF THAT GETS LARGER AND LARGER.
SO THIS NUMBER IS GETTING LARGER AND LARGER
BUT WE'RE MULTIPLYING IT BY A NEGATIVE NUMBER.
SO THIS IS APPROACHING NEGATIVE INFINITY.
LOOKING AT IT ON THE GRAPH AS X APPROACHES NEGATIVE INFINITY
WE'RE MOVING LEFT ALONG THE GRAPH.
AS WE MOVE LEFT, WE CAN SEE THE Y VALUES
OR FUNCTION VALUES ARE DECREASING WITHOUT BOUND
AND THEREFORE, AS X APPROACHES NEGATIVE INFINITY
F OF X APPROACHES NEGATIVE INFINITY.
SO WHENEVER IN DOUBT ABOUT END BEHAVIOR OR LONG RUN BEHAVIOR
I ALWAYS RECOMMEND ANALYZING THE GRAPH
TO BE SURE OF THE BEHAVIOR.
I HOPE YOU FOUND THIS HELPFUL.