# Ex: Find the Equation of a Transformed Exponential Function From a Graph

– WE WANT TO DETERMINE
THE EQUATION FOR THE EXPONENTIAL FUNCTION
FROM THE GRAPH USING A FORM OF F OF X=”A” x B
RAISED TO THE POWER OF X. BUT THIS ONE’S A LITTLE MORE
INVOLVED THAN THE OTHER EXAMPLES BECAUSE NOTICE HOW THIS GRAPH
DOES HAVE A HORIZONTAL ASYMPTOTE OF Y=+4. WHICH MEANS
THE EXPONENTIAL FUNCTION WOULD HAVE TO BE SHIFTED UP
4 UNITS IN ORDER TO HAVE THIS
HORIZONTAL ASYMPTOTE. THIS GRAPH ALSO APPEARS TO BE
EXPONENTIAL DECAY THAT’S BEEN REFLECTED
ACROSS THE X AXIS. SO WE SHOULD RECOGNIZE
THAT THE VALUE OF “A” IS GOING TO BE NEGATIVE, BUT IF NOTHING ELSE
WE DO HAVE TO RECOGNIZE THAT THE EXPONENTIAL FUNCTION
HAS BEEN SHIFTED UP 4 UNITS BECAUSE OF THIS
HORIZONTAL ASYMPTOTE, WHICH MEANS OUR EXPONENTIAL
FUNCTION WILL BE IN THE FORM OF F OF X=”A” x B RAISED
TO THE POWER OF X + 4. SO ONCE WE RECOGNIZE THIS, THEN WE CAN DETERMINE THE VALUES
OF “A” AND B BY IDENTIFYING TWO POINTS
ON THE FUNCTION THAT WILL GIVE US
TWO FUNCTION VALUES. SO NOTICE HOW THIS POINT HERE
OR THE Y INTERCEPT IS (0, 2) AND THIS POINT HERE
IS THE POINT (-1, -4) WHICH MEANS IF THE FUNCTION
CONTAINS THE POINT (0, 2) F OF 0 MUST=2 AND IF THE FUNCTION CONTAINS
THE POINT (-1, -4) THEN F OF -1 MUST=-4. NOW THAT WE KNOW F OF 0=2 WE SHOULD BE ABLE TO USE THIS
TO DETERMINE THE VALUE OF “A” AND THEN WE CAN USE
THE SECOND FUNCTION VALUE TO DETERMINE THE VALUE OF B. SO IF F OF 0=2
WE’LL REPLACE X WITH 0. SO WE WOULD HAVE “A” x B
TO THE 0 + 4 MUST EQUAL +2. WELL, B TO THE 0 IS EQUAL TO 1 SO WE HAVE “A” + 4=2,
SUBTRACT 4 ON BOTH SIDES. SO WE HAVE “A”=-2. SO NOW WE KNOW THAT F OF X MUST EQUAL -2 x B
TO THE POWER OF X + 4. NOTICE HOW “A” IS NEGATIVE
WHICH IS WHAT WE THOUGHT BECAUSE IT DOES LOOK LIKE THIS
IS EXPONENTIAL DECAY REFLECTED ACROSS THE X AXIS. SO NOW WE’LL USE THIS FORM
OF OUR FUNCTION AND OUR SECOND FUNCTION VALUE
TO DETERMINE THE VALUE OF B AND LET’S GO AHEAD AND DO THAT
ON THE NEXT SLIDE. SO WE WERE ABLE TO USE F OF 0=
2 TO DETERMINE THE VALUE OF “A” AND NOW WE’LL USE F OF -1=-4
TO DETERMINE THE VALUE OF B. SO F OF -1 IS GOING TO BE EQUAL
TO -2 x B TO THE -1 + 4 WHICH MUST EQUAL -4. SO THE FIRST STEP, LET’S GO AHEAD AND SUBTRACT 4
ON BOTH SIDES. SO WE HAVE -2 B TO THE -1=-8. I’M GOING TO MAKE THIS
INTO A FRACTION AND REMEMBER IF I MOVE THE B
DOWN TO THE DENOMINATOR IT’S GOING TO CHANGE THE SIGN
OF THE EXPONENT. SO WE WOULD HAVE -2/B=-8 AND NOW WE’RE GOING TO MULTIPLY
BOTH SIDES OF THE EQUATION BY B. THIS IS B/1
SO THE Bs SIMPLIFY OUT. WE HAVE -2=-8B. NOW WE CAN DIVIDE BOTH SIDES
BY -8. SO WE HAVE B
=THIS SIMPLIFIES TO +1/4. SO NOW THAT WE HAVE B
WE HAVE OUR FUNCTION. OUR FUNCTION IS F OF X
=-2 x THE BASE OF 1/4 RAISED TO THE POWER OF X + 4. NOW HOPEFULLY WE RECOGNIZE THAT BECAUSE THE BASE IS BETWEEN
0 AND 1 WE HAVE EXPONENTIAL DECAY. BECAUSE “A” IS NEGATIVE IT’S
REFLECTED ACROSS THE X AXIS. BECAUSE OF THE 2 IT’S BEEN
VERTICALLY STRETCHED AND BECAUSE OF THE +4
IT’S BEEN SHIFTED UP 4 UNITS AND THESE TRANSFORMATIONS DO DESCRIBE THE PROVIDED GRAPH
IN RED. I HOPE YOU FOUND THIS HELPFUL.

### 21 thoughts on “Ex: Find the Equation of a Transformed Exponential Function From a Graph”

• November 12, 2013 at 7:27 am

Bro, I'm pretty sure you just saved me for my math test today.

• April 15, 2014 at 1:02 am

thnx that helps alotย

• February 8, 2015 at 5:08 pm

Thank you, you have no idea how much this helped me out!

• April 20, 2015 at 8:16 am

thank you so much this is wonderful

• October 30, 2015 at 3:49 pm

THANK YOU!!!

• April 26, 2016 at 1:49 am

MY BRAIN IS STRONG NOW!!!

• August 22, 2016 at 2:25 am

THANKS YOU SO MUCHHHH!!! I've been looking for this for soo looongggg

• September 12, 2016 at 3:32 am

thankss

• October 11, 2016 at 8:19 pm

Thank you so much.

• October 13, 2016 at 10:57 pm

Nice ๐

• October 29, 2016 at 4:46 pm

Thanks!

• April 21, 2017 at 4:56 am

wow thanks! I'm so ready for my math test now

• April 26, 2018 at 2:29 am

what do you do if your first f(x) doesn't have f(0)? Then you have both an a and a b in the equation. ๐

• July 25, 2018 at 7:09 am

i need help still when the exponential graph is shifted horizontally

• September 20, 2018 at 10:04 pm

What if the y-intercept is below 1??

• September 28, 2018 at 5:04 pm

this was also a great video

• November 8, 2018 at 2:25 am

You're actually a lifesaver

• November 9, 2018 at 12:24 am

Legend

• April 23, 2019 at 2:09 am