- WELCOME TO A VIDEO ON GRAPHING PIECEWISE DEFINED FUNCTIONS,
AND THE GOALS OF THIS VIDEO ARE TO GRAPH
A PIECEWISE DEFINED FUNCTION BY HAND
AS WELL AS ON THE GRAPHING CALCULATOR.
SO A PIECEWISE DEFINED FUNCTION IS A FUNCTION WHOSE RULE CHANGES
DEPENDING ON THE VALUE OF THE INPUT,
OR IN THIS CASE THE VALUE OF THE X.
HERE'S AN EXAMPLE OF A PIECEWISE DEFINED FUNCTION.
YOU CAN KIND OF SEE WHY IT'S CALLED
A PIECEWISE DEFINED FUNCTION
BECAUSE IT HAS 2 DIFFERENT PIECES OR 2 DIFFERENT RULES
BASED UPON THE GIVEN VALUE OF X.
IF X IS LESS THAN 1, WE'RE GOING TO USE THIS FUNCTION RULE,
AND IF X IS GREATER THAN OR EQUAL TO 1,
WE'LL USE THIS FUNCTION RULE.
SO TO COMPLETE THIS TABLE WHEN X = 1,
WE'RE GOING TO USE THE FIRST RULE, BECAUSE -1 IS LESS THAN 1.
SO WE'LL REPLACE X WITH -1, 2 x -1 - 1 WOULD BE -3.
WHEN X IS 0, 0 IS STILL LESS THAN 1,
SO WE'LL USE THIS RULE AGAIN, 2 x 0 - 1 WOULD BE -1.
NOW WHEN X = 1, WE'RE GOING TO USE THE SECOND RULE
BECAUSE 1 IS GREATER THAN OR EQUAL TO 1,
SO WE'LL HAVE -1 + 2 WHICH = 1.
AND THEN WHEN X IS 2, 2 IS GREATER THAN OR EQUAL TO 1,
SO WE'LL USE THE SECOND RULE AGAIN, -2 + 2 WOULD BE 0.
LET'S GO AHEAD AND SEE IF WE CAN GRAPH THIS NOW.
NOW, THERE ARE A COUPLE TECHNIQUES
FOR GRAPHING PIECEWISE DEFINED FUNCTIONS.
I THINK IT'S HELPFUL WHEN YOU'RE FIRST LEARNING HOW TO DO THIS
IS JUST GO AHEAD AND GRAPH THE FIRST RULE AS Y = 2X - 1.
ONCE WE GRAPH THIS LINE,
WE'LL GO AHEAD AND ERASE THE PART THAT WE DON'T WANT.
SO TO GRAPH THIS LINE, WE HAVE A Y INTERCEPT OF -1
AND A SLOPE OF +2,
SO WE'LL GO UP 2, RIGHT 1.
NOW BECAUSE I KNOW I'M GOING TO HAVE TO ERASE PART OF THIS LINE,
I'M GOING TO GO AHEAD AND MAKE IT DASH TEMPORARILY.
I ONLY WANT THIS LINE WHEN X IS LESS THAN 1.
HERE'S WHERE X = 1,
SO WE WANT EVERYTHING TO THE LEFT OF THIS POINT.
AND SINCE 1 IS NOT INCLUDED IN THIS INEQUALITY,
WE'RE GOING TO CHANGE THIS POINT TO AN OPEN POINT.
AND AGAIN, WE'LL GRAB EVERYTHING TO THE LEFT
BECAUSE THAT'S WHEN X IS LESS THAN 1.
SO WE WANT THIS PIECE OF THE LINE
BUT WE DON'T WANT THE PIECE ON THE RIGHT,
SO WE'LL ERASE THIS.
NOW, WE'LL GRAPH Y = -X + 2, SO WE HAVE A Y INTERCEPT OF +2
AND OUR SLOPE WOULD BE -1/1.
SO WE'LL GO DOWN 1, RIGHT 1, DOWN 1, RIGHT 1,
AND WE'LL SKETCH A LINE THROUGH THESE 3 POINTS.
AGAIN, WE'LL MAKE IT DASH TEMPORARILY.
AND NOW, WE'LL DETERMINE WHICH PIECE OF THE LINE WE WANT.
WE WANT THE RED LINE WHEN X IS GREATER THAN OR EQUAL TO 1.
SO HERE'S WHERE X = 1,
AND IT IS INCLUDED ON THIS RED LINE
SO WE'LL CLOSE THE POINT IN HERE.
AND FROM HERE, WE WANT THE LINE ON THE RIGHT SIDE
WHERE X IS GREATER THAN OR EQUAL TO 1.
SO WE'LL GO AHEAD AND ERASE THE PIECE ON THE LEFT,
AND THERE'S THE GRAPH OF THAT PIECEWISE DEFINED FUNCTION.
LET'S GO AHEAD AND VERIFY THIS ON THE GRAPHING CALCULATOR.
IF WE PRESS Y =, WHAT WE CAN DO IS ENTER IN (2X - 1)
AND THEN JUST MULTIPLY THIS BY X LESS THAN 1,
AND IT WILL ONLY GRAPH THE APPROPRIATE PIECE.
SO WE'LL HAVE (X,
TO ACCESS THE INEQUALITIES, WE'LL PRESS 2nd MATH,
AND WE WANT OPTION 5, PUT A 1), PRESS ENTER.
AND WE'LL DO A SIMILAR PROCEDURE FOR THE SECOND RULE.
SO IN PARENTHESES, WE'LL HAVE (-X + 2)
x X GREATER THAN OR EQUAL TO 1.
SO 2nd MATH, OPTION 4, 1),
AND NOW WE'LL PRESS GRAPH.
SO THERE'S THE FIRST FUNCTION RULE
AND THERE'S THE SECOND WHICH MATCHES OUR GRAPH.
WE CAN ALSO PRESS 2nd GRAPH AND DETERMINE FUNCTION VALUES
IF WE WANTED TO.
THE INITIAL TABLE WE HAD WAS -1, 0, 1 AND 2 FOR X.
THE ONLY THING WE HAVE TO BE CAREFUL ABOUT
HERE WITH THE TABLE
IS THAT THE 0s SHOW THAT WHEN X IN -1, WE'RE USING FUNCTION Y1,
THIS 0 IS AN INDICATION THAT THIS FUNCTION WOULD BE FALSE
FOR THESE GIVEN X VALUES BASED UPON THE INEQUALITY.
LET'S GO AHEAD AND TRY ANOTHER ONE.
NOW THE MORE YOU KNOW ABOUT FUNCTIONS,
THE EASIER THESE ARE GOING TO BE,
MEANING IF WE WANT TO GRAPH Y = -X SQUARED - 1
FOR THE FIRST RULE,
IF YOU KNOW THAT'S A PARABOLA THAT OPENS DOWN
AND SHIFTED DOWN 1 UNIT,
WE CAN QUICKLY MAKE AN ACCURATE GRAPH
THAT WOULD LOOK SOMETHING LIKE THIS.
THIS WOULD BE THE VERTEX. IT'S GOING TO OPEN DOWN.
AND AGAIN IF WE DIDN'T KNOW HOW TO DO THIS,
WE COULD JUST PICK FUNCTION VALUES.
FOR EXAMPLE, WHEN X IS 2, WE WOULD HAVE -4 - 1,
THAT WOULD BE -5.
AND THERE'S A MIRROR IMAGE OF THIS POINT ON THE LEFT.
SO WE'LL GRAPH THIS ENTIRE PARABOLA.
WE'LL MAKE IT DASH THOUGH,
BECAUSE WE KNOW WE DO HAVE TO ERASE PART OF IT
BASED UPON THE GIVEN INTERVAL FOR X.
AND WE WANT THIS ONLY WHEN X IS LESS THAN 0,
SO WE ONLY WANT THIS BLUE GRAPH TO THE LEFT OF 0.
IT DOES NOT INCLUDE X = 0
SO WE WANT TO CHANGE THIS TO AN OPEN CIRCLE,
AND THEN WE'LL GRAPH THE PIECE TO THE LEFT
AND ERASE THE PIECE TO THE RIGHT.
SO AGAIN, I THINK IT'S HELPFUL
TO JUST GRAPH THE ENTIRE FUNCTION FIRST
AND THEN DETERMINE WHICH PIECE YOU WANT.
THE SECOND PIECE WILL COME FROM THE GRAPH OF Y = X SQUARED + 1
WHICH IS Y = X SQUARED SHIFTED UP 1 UNIT,
SO THAT SHOULD BE FAIRLY EASY TO GRAPH, SHOULD BE THE VERTEX.
WE'D HAVE THESE POINTS HERE.
AND THEN WHEN X IS 2, WE'D HAVE +5,
AND WHEN X IS -2, WE'D STILL HAVE +5.
WE'LL MAKE THIS GRAPH DASH TEMPORARILY.
NOW, WE WANT THE 2nd GRAPH WHEN X IS GREATER THAN OR EQUAL TO 0,
SO WE ARE GOING TO INCLUDE 0 ON THIS GRAPH
AND THEN THE PIECE ON THE RIGHT,
BUT WE DON'T WANT THE PIECE ON THE LEFT.
LET'S GO AHEAD AND MAKE SURE
IT'S OBVIOUS THIS IS A CLOSED POINT HERE.
LET'S VERIFY THIS ON THE GRAPHING CALCULATOR.
SO WE'LL TYPE IN THE FIRST RULE, -X SQUARED - (1).
AND THEN WE'LL MULTIPLY THIS BY X LESS THAN 0,
2nd MATH, OPTION 50.
AND THEN FOR Y2, X SQUARED + 1 x X GREATER THAN OR EQUAL TO 0,
AND GRAPH.
THE ONLY THING THAT WE CAN'T TELL FROM THE CALCULATOR
IS IT'S NOT VERY CLEAR THAT THIS POINT HERE
WOULD BE AN OPEN CIRCLE,
SO YOU DO HAVE TO BE CAREFUL ABOUT THAT
WHEN YOU'RE USING THE GRAPHING CALCULATOR TO CHECK YOU WORK.
I THINK WE HAVE TIME FOR ONE MORE.
NOW, THIS ONE HAS 3 PIECES.
NOTICE THE SECOND RULE THOUGH IS A POINT
WHEN X IS -1, Y WILL BE +3.
SO LET'S GO AHEAD AND GRAPH EACH FUNCTION,
AND THEN WE'LL ERASE THE PIECE WE DON'T WANT.
SO WE'LL GRAPH Y = -X SQUARED
WHICH A PARABOLA THAT OPENS DOWN.
AGAIN, WE COULD ALWAYS MAKE A TABLE
TO FIND THESE FUNCTION VALUES
BUT THE MORE YOU KNOW ABOUT YOUR FUNCTIONS,
THE EASIER THIS BECOMES.
WE ONLY WANT THIS GRAPH WHEN X IS LESS THAN -1.
SO HERE'S WHERE X = -1.
BECAUSE OF THE INEQUALITY, IT'D BE AN OPEN POINT.
CHANGE THAT OPEN POINT HERE,
AND WE ONLY WANT THE PIECE ON THE LEFT OF -1.
SO THIS SMALL PIECE HERE,
AND WE'LL ERASE THE PIECE ON THE RIGHT.
NEXT, WHEN X = -1, Y = 3, THAT'S JUST THE POINT (-1,3).
SO THIS POINT HERE IS PART OF THE FUNCTION.
AND NOTICE THAN WHEN X IS GREATER THAN -1,
WE HAVE THIS HAVE A Y INTERCEPT OF +2 AND WE HAVE A SLOPE OF +1,
SO WE'LL GO UP 1, RIGHT 1 AND MAKE OUR LINE.
AGAIN, WE'LL MAKE IT DASH TEMPORARILY.
AND WE WANT THE LINE WHEN X IS GREATER THAN -1.
WELL, WHEN X = -1, Y WOULD BE +1.
SO RIGHT HERE, WE'RE GOING TO HAVE AN OPEN CIRCLE
THAT IS ON THAT LINE,
AND THEN WE WANT THE GRAPH TO THE RIGHT.
WE'LL ERASE THE PIECE ON THE LEFT.
NOW, YOU CAN GO AHEAD AND CHECK THIS ON THE GRAPHING CALCULATOR.
IT'S GOING TO BE VERY DIFFICULT TO SEE THIS POINT
ON THE GRAPHING CALCULATOR,
BUT YOU CAN EASILY VERIFY THE PIECE ON THE LEFT
AND THE PIECE ON THE RIGHT.
I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING.