- WE WANT TO DESCRIBE THE TYPE OF TRANSLATION THAT WOULD OCCUR
TO THE BASIC FUNCTION F(X).
WE'RE GIVEN F(X) + 4, F(X) - 4, F(X + 4), AND F(X - 4).
LOOKING AT THESE FIRST TWO EXAMPLES
NOTICE HOW WE'RE ADDING OR SUBTRACTING 4 FROM F(X),
WHICH WOULD BE THE FUNCTION VALUE OR THE Y VALUE.
SO BECAUSE WE'RE ADDING AND SUBTRACTING IN CONSTANT
TO THE Y VALUE,
THESE TWO WILL SHIFT THE GRAPH UP OR DOWN.
BUT NOTICE FOR THESE LAST TWO EXAMPLES
WE'RE ADDING OR SUBTRACTING 4 FROM X,
WHICH WOULD BE ALONG THE HORIZONTAL AXIS,
AND THEREFORE THESE TWO WILL SHIFT THE GRAPH LEFT OR RIGHT.
LET'S LOOK AT AN ANIMATION TO ILLUSTRATE THESE TRANSLATIONS.
LET'S START BY LOOKING AT WHAT HAPPENS
WHEN WE ADD OR SUBTRACT A CONSTANT
FROM F(X), THE FUNCTION VALUE.
NOTICE AS WE ADD A CONSTANT, THE GRAPH IS SHIFTED UP.
AND IF WE SUBTRACT A CONSTANT FROM F(X),
WE CAN SEE THE GRAPH IS SHIFTED DOWN.
SO THIS ANSWERS OUR FIRST TWO QUESTIONS.
F(X) + 4 WILL SHIFT F(X) UP 4 UNITS.
AND F(X) - 4 WILL SHIFT F(X) DOWN 4 UNITS.
NOW LET'S GO BACK TO THE ANIMATION.
NOW LET'S ADD AND SUBTRACT A CONSTANT FROM THE INPUT X.
NOTICE AS WE SUBTRACT A CONSTANT FROM THE INPUT X
THE GRAPH IS ACTUALLY SHIFTED RIGHT,
WHICH MIGHT BE THE OPPOSITE DIRECTION THAT WE WOULD THINK,
BUT IF WE PAUSE HERE FOR A MOMENT IT SHOULD MAKE SENSE.
IF WE TAKE A LOOK AT THE RED FUNCTION FOR A MOMENT,
NOTICE THAT WHEN X IS ZERO
THE FUNCTION VALUE OR Y VALUE WOULD BE ZERO.
BUT NOW LOOKING AT THE BLUE FUNCTION, F(X - 1),
NOTICE HOW IT WOULD TAKE A LARGER X VALUE,
OR IN THIS CASE AN X VALUE OF + 1,
TO GET THE SAME FUNCTION VALUE OR THE FUNCTION VALUE OF ZERO.
BECAUSE WE'RE SUBTRACTING 1 FROM X
IT'LL TAKE LARGER VALUES OF X TO GET THE SAME FUNCTION VALUES,
AND THEREFORE SHIFTING THE GRAPH RIGHT.
AND IF WE ADD A CONSTANT TO X
THE GRAPH IS ACTUALLY SHIFTED LEFT.
AND IF WE PAUSE HERE FOR A MOMENT,
NOTICE HOW IT'S GOING TO TAKE SMALLER VALUES OF X
TO GET THE SAME INPUTS AS F(X),
AND THEREFORE RESULTING IN THE SAME OUTPUTS.
NOTICE FOR THE RED FUNCTION F(0) WOULD BE ZERO,
BUT FOR THE BLUE FUNCTION IT WOULD TAKE X = -1 HERE
TO GET A FUNCTION VALUE OF ZERO.
SO IF WE HAVE F(X + A CONSTANT),
THE GRAPH IS ACTUALLY SHIFTED LEFT.
OF COURSE ASSUMING, C IS POSITIVE.
SO F(X + 4) WILL SHIFT F(X) LEFT 4 UNITS,
AND F(X - 4) WILL SHIFT F(X) RIGHT 4 UNITS.
NOW BEFORE WE GO LET'S CONSIDER THE BASIC FUNCTION
F(X) = X SQUARED AS AN EXAMPLE.
IF F(X) = X SQUARED
THEN F(X + 4) WOULD BE Y = X SQUARED + 4.
F(X - 4) WOULD BE Y = X SQUARED - 4.
AND F(X + 4), NOTICE HOW X + 4 BECOMES THE INPUT
INTO OUR FUNCTION,
WE WOULD HAVE Y = (X + 4) SQUARED,
AND THEN FINALLY FOR F(X - 4) WE WOULD HAVE Y = (X - 4) SQUARED.
I HOPE YOU FOUND THIS HELPFUL.
I'LL LEAVE YOU WITH SOME NOTES THAT WILL SUMMARIZE
WHAT WE JUST DISCOVERED.