- WE'RE ASKED TO DETERMINE AN EXPONENTIAL FUNCTION
THAT PASSES THROUGH THE POINTS (-1, 5/4) AND (2, 80).
WE ALSO WANT TO FIND THE EXPONENTIAL GROWTH
OR DECAY RATE AND THE FUNCTION VALUE WHEN X EQUALS 3.
IF WE HAVE AN EXPONENTIAL FUNCTION IN THIS FORM HERE
THE BASE B IS EQUAL TO 1 + R
WHERE R IS EQUAL TO THE EXPONENTIAL GROWTH OR DECAY RATE
EXPRESSED AS A DECIMAL.
"A" IS THE INITIAL VALUE OR INITIAL AMOUNT
WHICH IS THE FUNCTION VALUE WHEN X = 0, AND X IS THE EXPONENT.
SO GOING BACK UP TO OUR QUESTION,
IF THE FUNCTION CONTAINS THE POINT (-1, 5/4)
THIS MEANS THAT F OF -1 MUST EQUAL 5/4
AND IF THE FUNCTION CONTAINS THE POINT (2, 80)
THIS MEANS F OF 2 IS EQUAL TO 80.
NOTICE HOW WE'RE NOT GIVEN THE POINT WHERE X = 0
SO WE DON'T KNOW THE INITIAL VALUE OR THE VALUE OF "A"
OR THE VALUE OF B, THE BASE.
SO WHAT WE'LL HAVE TO DO IS SET THIS UP AS A SYSTEM OF EQUATIONS
AND THEN SOLVE FOR "A" AND B.
SO FOR F OF -1 WE'LL REPLACE X WITH -1
AND SET THE FUNCTION VALUE EQUAL TO 5/4.
SO WE WOULD HAVE "A" x B TO THE POWER OF -1 = 5/4
AND THEN IF F OF 2 = 80
THAT MEANS "A" x B TO THE 2nd MUST EQUAL 80.
SO NOW WE'LL SOLVE THIS AS A SYSTEM OF EQUATIONS.
SO LET'S CALL THIS EQUATION 1 AND THIS EQUATION 2.
SO USING EQUATION 1 WE KNOW "A" x B TO THE -1 EQUALS 5/4.
SO WHAT WE'LL DO IS SOLVE THIS EQUATION FOR "A"
AND THEN PERFORM SUBSTITUTION INTO THE SECOND EQUATION.
AND SINCE WE HAVE B TO THE -1 POWER HERE
WE CAN MULTIPLY BOTH SIDES OF THE EQUATION BY B TO THE 1st.
REMEMBER WHEN MULTIPLYING THE BASES ARE THE SAME.
WE ADD OUR EXPONENTS -1 + 1 IS 0.
B TO THE 0 = 1.
SO WE WOULD HAVE "A" = THIS WOULD BE JUST 5/4B.
SO NOW WE'LL SUBSTITUTE THIS FOR "A" IN EQUATION 2.
SO NOW OUR EQUATION 2 IS GOING TO BE 5/4 B
x B TO THE 2nd MUST EQUAL 80.
SO WE WOULD HAVE 5/4,
B TO THE 3rd EQUALS 80 AND NOW WE CAN SOLVE FOR B
TO THE 3rd BY MULTIPLYING BOTH SIDES BY 4/5.
SO LET'S GO AHEAD AND PUT 80/1.
NOTICE ON THE LEFT SIDE THE 4 SIMPLIFIED TO 1,
THE 5 SIMPLIFIED TO ONE.
SO WE HAVE B TO THE 3rd EQUALS--
ON THE RIGHT SIDE THE 5 AND THE 80 SIMPLIFY.
THERE'S ONE 5 IN 5 AND 16 FIVES IN 80.
16 x 4 = 64.
SO WE HAVE B TO THE 3rd = 64
AND NOW TO SOLVE FOR B
WE'LL TAKE THE CUBE ROOT OF BOTH SIDES
OR IF WE WANT WE COULD RAISE BOTH SIDES TO THE 1/3 POWER.
LET'S GO AHEAD AND TAKE THE CUBE ROOT OF BOTH SIDES.
SO THE CUBE ROOT OF B TO THE 3rd IS B.
THE CUBE ROOT OF 64 IS EQUAL TO 4
SINCE 4 TO THE 3rd DOES EQUAL 64.
SO NOW WE KNOW B IS EQUAL TO 4
WE NOW HAVE TO FIND THE VALUE OF "A".
SO WE KNOW THAT F OF X MUST EQUAL "A" x 4 TO THE POWER OF X.
NOW WE STILL HAVE TO FIND THE VALUE OF "A"
OR THE INITIAL VALUE AND WE CAN DO THIS BY USING THE FACT
THAT F OF -1 = 5/4 OR F OF 2 = 80.
SO LET'S GO AHEAD AND DETERMINE THE VALUE OF "A'
ON THE NEXT SLIDE.
SO AGAIN IF WE KNOW THAT F OF 2 EQUALS 80
WE CAN SUBSTITUTE 2 FOR X
AND THEN SET THE FUNCTION VALUE EQUAL TO 80 TO SOLVE FOR "A".
SO FOR F OF 2 WE WOULD HAVE "A" x 4 TO THE 2nd
WHICH MUST EQUAL 80.
WELL 4 SQUARED IS 16 SO WE HAVE 16A = 80,
DIVIDE BOTH SIDES BY 16, WE KNOW THAT "A" MUST = 5.
SO THE EXPONENTIAL FUNCTION
THAT PASSES THROUGH THE 2 GIVEN POINTS WOULD BE F OF X = "A"
WHICH IS 5 x OUR BASE WHICH IS 4 RAISED TO THE POWER OF X.
TO CHECK THIS LET'S GO AHEAD AND GRAPH THIS FUNCTION TO MAKE SURE
IT PASSES THROUGH THESE TWO POINTS.
SO HERE'S OUR FUNCTION GRAPHED IN BLUE.
NOTICE HOW WE HAVE EXPONENTIAL GROWTH
AND IT DOES PASS THROUGH THE TWO GIVEN POINTS
SO OUR FUNCTION IS CORRECT BUT WE'RE STILL ASKED TO DETERMINE
THE EXPONENTIAL GROWTH RATE AND TO DETERMINE THE FUNCTION VALUE
WHEN X = 3.
TO DETERMINE THE EXPONENTIAL GROWTH RATE
WE NEED TO REMEMBER THAT OUR BASE B = 1 + R
WHERE R IS THE GROWTH RATE EXPRESSED AS A DECIMAL.
SO FOR OUR FUNCTION, SINCE OUR BASE = 4
WE WOULD 4 = 1 + R.
SUBTRACTING 1 ON BOTH SIDES WE HAVE R = 3.
WELL 3 AS A PERCENTAGE IS 300%.
SO THE EXPONENTIAL GROWTH RATE IS 300%.
WHICH MEANS EACH TIME X INCREASES BY 1
THE FUNCTION VALUE INCREASES 300%.
AND THEN FOR THE LAST QUESTION
TO DETERMINE THE FUNCTION VALUE WHEN X = 3.
WE NEED TO FIND F OF 3 WHICH IS EQUAL TO 5 x 4
RAISED TO THE 3rd POWER.
WELL 4 TO THE 3rd IS EQUAL TO 64
SO WE HAVE 5 x 64 AND 5 x 64 IS EQUAL TO 320.
SO NOW WE HAVE OUR EXPONENTIAL FUNCTION,
THE EXPONENTIAL GROWTH RATE AND THE FUNCTION VALUE WHEN X = 3.
I HOPE YOU FOUND THIS HELPFUL.