- WE WANT TO COMPLETE THE TABLE USING THE FUNCTION RULE F OF X
= THE ABSOLUTE VALUE OF X + 2,
AND THEN USE A TABLE TO GRAPH THE FUNCTION.
AND IF WE FOUND IT HELPFUL, WE COULD REPLACE F OF X WITH Y
AND WRITE THIS AS Y = ABSOLUTE VALUE OF X + 2.
BUT OVER TIME WE SHOULD GET MORE COMFORTABLE
USING FUNCTION NOTATION HERE.
SO WHAT WE'LL DO IS SELECT X VALUES, PERFORM SUBSTITUTION,
AND THEN FIND THE CORRESPONDING FUNCTION VALUE OR Y VALUE.
AND EVERY TIME WE DO THIS WE'LL BE FINDING A POINT
THAT WOULD BE ON THE GRAPH OF THE GIVEN FUNCTION.
NOW THE MORE WE KNOW ABOUT THE ABSOLUTE VALUE FUNCTION
THE BETTER.
THIS IS GOING TO BE A V SHAPED GRAPH.
ONE THING THAT'S HELPFUL TO KNOW THOUGH
WHEN SKETCHING THE GRAPH OF AN ABSOLUTE VALUE FUNCTION IS
IF WE CAN DETERMINE WHICH X VALUE
WOULD MAKE THE ABSOLUTE VALUE 0,
WE WANT TO SELECT VALUES
THAT ARE LESS THAN AND GREATER THAN THAT VALUE.
NOTICE WHEN X = -2 WE'D HAVE THE ABSOLUTE VALUE OF 0.
SO I'M GOING TO PUT -2 IN THE MIDDLE OF THE TABLE,
AND THEN I'LL SELECT VALUES THAT ARE < -2 HERE AND > -2 HERE.
SO I'LL GO AHEAD AND USE -3 AND -4.
THESE ARE SMALLER THAN -2.
AND THEN I'LL ALSO USE -1 AND 0, TWO VALUES THAT ARE > -2.
NOW LET'S SEE WHAT HAPPENS WHEN WE USE THESE VALUES.
TO DETERMINE THE CORRESPONDING FUNCTION VALUE OR Y VALUE
WHEN X = -4,
WE'D HAVE TO SUBSTITUTE -4 FOR X,
SO WE'D HAVE THE ABSOLUTE VALUE OF -4 + 2.
THIS IS THE ABSOLUTE VALUE OF -2 WHICH IS 2.
SO THE GRAPH OF OUR FUNCTION WOULD CONTAIN THE POINT
WITH AN X COORDINATE OF -4 AND A Y COORDINATE OF 2,
WHICH WOULD BE THIS POINT HERE.
NEXT WE'LL SUBSTITUTE -3 FOR X.
SO WE'LL HAVE THE ABSOLUTE VALUE OF -3 + 2.
WELL, -3 + 2 = -1, AND THE ABSOLUTE VALUE OF -1 IS 1.
SO THE GRAPH OF OUR FUNCTION MUST CONTAIN THE POINT
WITH AN X COORDINATE OF -3 AND A Y COORDINATE OF 1,
WHICH IS THIS POINT HERE.
AND THEN WHEN X = -2, Y WOULD BE THE ABSOLUTE VALUE OF -2 + 2,
THE ABSOLUTE VALUE OF 0,0.
SO OUR FUNCTION CONTAINS THE POINT WITH AN X COORDINATE OF -2
AND A Y COORDINATE OF 0,
OR THIS POINT HERE.
AND THEN FOR X = -1 THE FUNCTION VALUE, OR Y VALUE,
WOULD BE THE ABSOLUTE VALUE OF -1 + 2.
WELL, -1 + 2 = 1.
THE ABSOLUTE VALUE OF 1 IS 1,
SO OUR FUNCTION MUST ALSO CONTAIN THE POINT -1,1,
WHICH IS THIS POINT HERE.
AND THEN LASTLY, WHEN X = 0
WE'D HAVE THE ABSOLUTE VALUE OF 0 + 2.
THE ABSOLUTE VALUE OF 2 IS 2,
SO OUR FUNCTION CONTAINS THE POINT 0,2,
WHICH WOULD BE THIS POINT HERE.
NOW, IF WE GO BACK AND TAKE A LOOK Y IN EXAMPLE TWO,
SELECT THE NEXT VALUE HERE THAT MADE THE ABSOLUTE VALUE = 0,
NOTICE THIS CREATED THE POINT -2,0,
WHICH WAS THIS POINT HERE
WHERE THE FUNCTION ACTUALLY CHANGES DIRECTION.
SO WHEN WE SELECTED POINTS THAT WERE < -2
WE FOUND POINTS ON THIS HALF OF THE GRAPH.
AND THEN WHEN WE SELECTED X VALUES THAT WERE > -2
WE FOUND POINTS ON THIS PART OF THE GRAPH.
AND NOW WE CAN SEE THE GRAPH OF OUR FUNCTION
IS GOING TO BE A V SHAPE,
AND IT WOULD LOOK SOMETHING LIKE THIS.
NOW I DO WANT TO MENTION
THAT WE DIDN'T HAVE TO SELECT THESE EXACT X VALUES.
WE CAN SELECT ANY X VALUES THAT WE WANT.
BUT WHEN IT COMES TO THE GRAPH OF AN ABSOLUTE VALUE FUNCTION,
WE WOULD HAVE TO DETERMINE ENOUGH POINTS
TO DETERMINE THE V SHAPE OF ALL ABSOLUTE VALUE FUNCTIONS.
NOTICE IF WE ONLY FOUND THESE THREE POINTS
WITHOUT THESE TWO POINTS HERE,
WE MIGHT MISTAKE THIS FOR A LINE,
WHEN IN FACT ALL ABSOLUTE VALUE FUNCTIONS DO FORM A V SHAPE.
I HOPE YOU FOUND THIS HELPFUL.