- IN THIS QUESTION WE'RE ASKED TO DETERMINE
AN EXPONENTIAL FUNCTION THAT PASSES THROUGH THE POINTS
(0, 15) AND (5, 480).
WE ALSO WANT TO KNOW
WHAT THE EXPONENTIAL GROWTH OR DECAY RATE IS.
WE ALSO WANT TO FIND THE FUNCTION VALUE WHEN X = 2.
IF AN EXPONENTIAL FUNCTION IS IN THIS FORM HERE
THEN THE BASE B IS EQUAL TO 1 + R
WHERE R IS EQUAL TO OUR DECAY RATE EXPRESSED AS A DECIMAL,
"A" IS THE INITIAL VALUE OR THE FUNCTION VALUE WHEN X = 0,
AND X IS THE EXPONENT.
SO IF OUR FUNCTION CONTAINS THE POINT (0, 15)
THIS MEANS F OF 0 IS EQUAL TO 15.
SO 15 IS THE INITIAL VALUE OR EQUAL TO "A"
AND IF THE FUNCTION CONTAINS THE POINT (5, 480)
THEN F OF 5 MUST EQUAL 480.
AND TO EMPHASIS THAT "A" IS EQUAL TO 15
WE CAN REPLACE X WITH 0 IN THE EXPONENTIAL FUNCTION.
IF WE DID THIS WE WOULD HAVE
"A" x B RAISED TO THE 0 POWER = 15.
BUT B TO THE 0 IS EQUAL TO 1,
SO WE KNOW "A" MUST EQUAL 15.
NOW WE CAN USE "A" = 15 AND THE FACT THAT F OF 5 EQUALS 480
TO DETERMINE THE VALUE OF B OUR BASE.
SO WE'LL SUBSTITUTE 5 FOR X.
SO WE KNOW "A" IS 15 x B TO THE 5th POWER
MUST EQUAL THE FUNCTION VALUE OF 480.
SO NOW WE'LL SOLVE FOR B SO WE'LL DIVIDE BOTH SIDES BY 15.
WE HAVE B TO THE 5th = 32.
WE WANT B NOT B TO THE 5th.
SO NOW WE'LL TAKE THE 5th ROOT OF BOTH SIDES OF THE EQUATION.
THE 5th ROOT OF B TO THE 5th SIMPLIFIES TO 1 FACTOR OF B
AND THE 5th ROOT OF 32 IS EQUAL TO 2
BECAUSE 32 IS EQUAL TO 2 TO THE 5th.
SO THIS SIMPLIFIES NICELY TO 1 FACTOR OF 2.
WHICH MEANS THE EXPONENTIAL FUNCTION
THAT WE'RE LOOKING FOR THAT PASSES THROUGH THESE TWO POINTS
IS F OF X = "A" WHICH IS 15 x B WHICH IS 2
RAISED TO THE POWER OF X.
NOW IF WANT TO CHECK THIS
WE COULD GRAPH THIS FUNCTION
TO MAKE SURE IT PASSES THROUGH THESE TWO POINTS.
LET'S GO AHEAD AND DO THAT.
HERE'S A GRAPH OF OUR FUNCTION IN BLUE
AND HERE ARE THE TWO POINTS THAT WE WERE GIVEN.
NOTICE HOW OUR FUNCTION DOES PASS THROUGH THESE TWO POINTS.
THIS VERIFIES THAT OUR FUNCTION IS CORRECT
BUT WE NOW WE STILL WANT TO FIND THE EXPONENTIAL GROWTH
OR DECAY RATE AND ALSO THE FUNCTION VALUE AT X = 2.
WE CAN DETERMINE THE EXPONENTIAL GROWTH RATE
SINCE THE BASE B IS EQUAL TO 1 + R
WHERE R IS OUR EXPONENTIAL GROWTH OR DECAY RATE
EXPRESSED AS A DECIMAL.
SINCE THE BASE IS 2 WE KNOW THAT 2 MUST EQUAL 1 + R,
SUBTRACTING 1 ON BOTH SIDES, R = 1.
WELL 1 EXPRESSED AS A PERCENTAGE WOULD BE 100%
WHICH MEANS THE EXPONENTIAL GROWTH RATE IS 100%.
WHICH MEANS EACH TIME X INCREASES BY 1
THE FUNCTION VALUE WILL INCREASE BY 100%.
BUT I ALSO WANT TO POINT OUT THAT NOTICE HOW THE BASE IS 2,
SO AS X INCREASES BY 1
WE'RE MULTIPLYING BY ANOTHER FACTOR OF 2
WHICH MEANS THE FUNCTION VALUE WILL DOUBLE EACH TIME
X INCREASES BY 1.
SO HAVING AN EXPONENTIAL GROWTH RATE OF 100%
IS THE SAME AS THE FUNCTION VALUE DOUBLING
EACH TIME X INCREASES BY 1.
AND NOW FOR THE LAST QUESTION
WE WANT TO FIND THE FUNCTION VALUE WHEN X = 2
SO WE NEED TO DETERMINE THE VALUE OF F OF 2.
WELL F OF 2 IS GOING TO BE EQUAL TO 15 x 2 SQUARED.
WELL 2 SQUARED IS 4, 4 x 15 = 60.
NOTICE FROM THE INITIAL VALUE OF 15 THE X VALUE INCREASES BY 2
WHICH MEANS THE FUNCTION VALUE OF 15
WOULD DOUBLE TWICE OR INCREASE BY 100% TWICE.
SO IF WE START WITH 15 AND WE DOUBLE THIS
OR INCREASE IT BY 100% THAT WOULD BE 30.
IF WE TAKE 30 AND DOUBLE IT OR INCREASE IT BY 100%
THAT WOULD GIVE US A VALUE OF 60
WHICH IS OUR FUNCTION VALUE F OF 2.
OKAY, I HOPE YOU FOUND THIS EXAMPLE HELPFUL.