- WELCOME TO OUR LESSON
ON GRAPHING THE SQUARE ROOT FUNCTION
AND VERTICAL AND HORIZONTAL STRETCHES
OF THE SQUARE ROOT FUNCTION.
LET'S START BY GRAPHING THE BASIC SQUARE ROOT FUNCTION
F OF X = THE SQUARED OF X.
SINCE THE SQUARED OF A NEGATIVE NUMBER IS NOT REAL,
THE DOMAIN OF THIS FUNCTION WOULD BE X > OR = TO 0.
WE'LL GRAPH THIS BY COMPLETING A TABLE OF VALUES.
TO DO THIS, WE'LL SELECT X VALUES THAT ARE PERFECT SQUARES.
SO FOR EXAMPLE 0 SQUARED IS 0, 1 SQUARED IS 1, 2 SQUARED IS 4,
AND 3 SQUARED IS 9.
THIS WILL MAKE FINDING Y VERY CONVENIENT.
IF X = 0, Y IS = TO THE SQUARED OF 0, WHICH IS 0.
IF X = 1, Y IS = SQUARED OF 1 OR 1.
WHEN X = 4 Y IS = THE SQUARED OF 4 OR 2
AND WHEN X IS 9 Y IS = TO THE SQUARED OF 9 OR 3.
SO NOW WE'LL PLOT THESE POINTS AND SKETCH OUR GRAPH.
SO WE HAVE (0, 0) (1, 1) (4, 2) AND (9, 3).
SO HERE'S THE GRAPH OF THE BASIC SQUARE ROOT FUNCTION.
NOW LET'S COMPARE THIS
TO THE GRAPH OF G OF X = 2X THE SQUARED OF X,
AND NOTICE OUR COORDINATE PLAN ALREADY CONTAINS A GRAPH
OF BASIC SQUARE ROOT FUNCTION FOR EASY COMPARISON.
NOTICE AGAIN, THE DOMAIN OF THIS FUNCTION
IS STILL GOING TO BE X IS > OR = 0.
BECAUSE WE HAVE X UNDERNEATH THE SQUARE ROOT.
SO WE'LL USE THE SAME X VALUES, 0, 1, 4 AND 9.
WHEN X IS 0 NOTICE Y IS GOING TO BE = 2X THE SQUARED OF 0,
WHICH IS STILL 0.
WHEN X IS 1 Y = 2 x THE SQUARED OF 1
WELL, THAT WOULD BE 2 x 1 OR 2.
WHEN X IS 4 Y = 2 x THE SQUARED OF 4 OR 2 x 2, THAT'S 4.
AND WHEN X IS 9 Y IS = TO 2 x THE SQUARED OF 9
OR 2 x 3 = TO 6.
SO IF WE PLOT THESE FOUR POINTS
WE HAVE (0, 0) (1, 2) (4, 4) AND (9, 6).
SO HERE'S THE GRAPH OF G OF X.
NOTICE FROM THE FUNCTION EVERY Y COORDINATE ON THE GREEN FUNCTION
IS 2 x THE Y COORDINATE OF THE BLUE FUNCTION.
SO WE SAY THE GREEN FUNCTION HAS BEEN STRETCHED VERTICALLY
BY A FACTOR OF 2.
NOW, LET'S CONSIDER THE GRAPH OF H OF X = 1/2 x THE SQUARED OF X.
AGAIN, OUR DOMAIN IS GOING TO BE THE SAME
BECAUSE WE HAVE THE SQUARED OF X SO X > = 0,
WE USE THE SAME X VALUES, SO WE'LL HAVE (0, 1) (4, 9)
AND NOW WE'LL FIND THE CORRESPONDING Y VALUES.
WHEN X IS 0 Y IS 1/2 x THE SQUARED OF 0, STILL 0.
WHEN X IS 1 Y IS = TO 1/2 x THE SQUARED OF 1, WHICH IS 1/2.
WHEN X IS 4 Y IS = TO 1/2 x THE SQUARED OF 4
OR 1/2 x 2, THAT'S 1.
AND WHEN X IS 9 Y IS = TO 1/2 x THE SQUARED OF 9
WHICH IS 1/2 x 3 OR 3/2.
NOW, LET'S PLOT THESE POINTS.
WE HAVE (0, 0) (1, 1/2) (4, 1) AND (9 3/2).
SO HERE'S THE GRAPH OF H OF X.
AGAIN, COMPARING THIS TO OUR PARENT FUNCTION F OF X
= THE SQUARED OF X
NOTICE HOW THE Y COORDINATES ARE 1/2
THE Y COORDINATES OF THE PARENT FUNCTION.
THIS IS CALLED A VERTICAL COMPRESSION BY A FACTOR OF 1/2.
LET'S GO AHEAD AND SUMMARIZE THE VERTICAL STRETCH
AND COMPRESSION.
IF WE HAVE Y = A x F OF X WHERE A IS > THAN 1,
WE HAVE A VERTICAL STRETCH BY A FACTOR OF A.
THIS WAS OUR FUNCTION G OF X = 2 x THE SQUARED OF X
AND IF WE HAVE Y = A x F OF X WHERE A IS BETWEEN 0 AND 1
WE HAVE A VERTICAL COMPRESSION BY A FACTOR OF A,
THIS WAS OUR FUNCTION H OF X = 1/2 x THE SQUARED OF X.
SO THE VERTICAL STRETCH IN GREEN
AND THE VERTICAL COMPRESSION IN RED.
NOW, LET'S CONSIDER G OF X = THE SQUARE ROOT OF 2X.
NOTICE HOW NOW THE 2 IS UNDERNEATH THE SQUARE ROOT.
NOW FOR THE DOMAIN OF THIS
WE KNOW THAT 2X MUST BE > OR = TO 0,
IF WE DIVIDE BOTH SIDES BY 2
OUR DOMAIN IS STILL GOING TO BE X IS > OR = TO 0.
HOWEVER, BECAUSE WE WANT THE RADICAND OF THE NUMBER
UNDERNEATH THE SQUARE ROOT TO BE A PERFECT SQUARE
WE ARE GOING TO USE DIFFERENT X VALUES.
IF X IS 0 OUR RADICAND WILL STILL BE 0,
IF WE WANT THE RADICAND TO BE 1 WE'LL HAVE TO LET X = 1/2.
IF WE WANT THE RADICAND TO BE 4 WE'LL LET X = 2,
IF WE WANT THE RADICAND TO BE 9 WE'LL HAVE TO LET X = 9/2.
AND NOW LET'S FIND THE CORRESPONDING Y VALUES.
WHEN X IS 0 Y IS = TO THE SQUARED OF 2 x 0
WHICH IS THE SQUARED OF 0 OR 0.
WHEN X IS = TO 1/2 Y IS = TO THE SQUARED OF 2 x 1/2
WHICH IS THE SQUARED OF 1 OR 1.
WHEN X IS = TO 2 Y IS = TO THE SQUARED OF 2 x 2
OR THE SQUARED OF 4 WHICH IS 2
AND WHEN X IS = TO 9/2 Y IS = TO THE SQUARED OF 2 x 9/2
WHICH IS = TO THE SQUARED OF 9 OR 3.
LET'S PLOT OUR POINTS WE HAVE (0, 0) (1/2, 1) (2, 2) (9/2, 3)
OR (4 1/2, 3)
AND HERE'S A GRAPH OF OUR FUNCTION.
NOW, THIS DOES KIND OF LOOK LIKE A VERTICAL STRETCH UPWARD
BUT IF YOU LOOK AT OUR TABLE OF VALUES
NOTICE HOW THE X COORDINATES HAVE BEEN COMPRESSED
AND IT'S BEEN COMPRESSED HORIZONTALLY IN THIS DIRECTION.
SO WE SAY WE HAVE A HORIZONTAL COMPRESSION.
NOW WE MAY BE THINKING THIS IS A HORIZONTAL COMPRESSION
BY A FACTOR OF 2 BECAUSE OF THIS 2
BUT IT'S ACTUALLY BY A FACTOR OF 1/2.
IF WE TAKE A LOOK AT OUR TABLE OF VALUES
NOTICE HOW THE X COORDINATES ARE 1/2
THE X COORDINATES OF THE PARENT FUNCTION
WITH THE SAME FUNCTION VALUES.
SO THE SCALE FACTOR IS ACTUALLY THE RECIPROCAL OF THIS 2 HERE
OR 1/2.
NOW, LET'S TAKE A LOOK AT H OF X = THE SQUARED OF 1/2 X.
AGAIN, IN THIS CASE OUR DOMAIN IS STILL GOING TO BE X
IS > OR = TO 0
BUT WE ARE GOING TO BE CAREFUL ABOUT WHAT X VALUES
WE SELECT FOR OUR TABLE.
WE'LL STILL START WITH 0
BUT IF WE WANT THE NUMBER UNDERNEATH THE SQUARE ROOT
WITH THE RADICAND TO BE 1, WE'LL HAVE TO LET X = 2.
IF WE WANT THE RADICAND TO BE 4, WELL HAVE TO LET X = 8,
IF WE WANT THE RADICAND TO BE 9 WE'LL HAVE TO LET X = 18.
SO IF X IS 0 Y IS = TO THE SQUARED OF 1/2 X 0 OR 0.
WHEN X IS 2 Y IS = TO THE SQUARED OF 1/2 X 2
OR THE SQUARED OF 1 THAT'S 1.
WHEN X IS 8 Y IS = TO THE SQUARED OF 1/2 X 8
OR THE SQUARED OF 4 THAT'S 2.
AND WHEN X IS EQUAL TO 18 Y IS EQUAL TO THE SQUARED OF 1/2 X 18
OR THE SQUARED OF 9 WHICH IS 3.
LET'S PLOT OUR POINTS.
WE HAVE (0, 0) WE HAVE (2, 1) AND WE HAVE (8, 2).
OF COURSE, (18, 3) IS OFF THE SCREEN
SO HERE'S THE GRAPH OF H OF X.
AGAIN, THE HORIZONTAL STRETCH AND COMPRESSIONS
ARE A LITTLE BIT TRICKIER.
NOTICE HOW THE RED GRAPH WOULD BE A HORIZONTAL STRETCH
OF THE BLUE GRAPH.
SO WE HAVE A HORIZONTAL STRETCH
AND BECAUSE THIS COEFFICIENT IS 1/2 IT'S BY A FACTOR OF 2.
NOTICE HOW THE X COORDINATES ARE TWICE THE X COORDINATES
OF THE PARENT FUNCTION
WITH THE SAME Y VALUES OR FUNCTION VALUES.
NOW, LET'S SUMMARIZE THE HORIZONTAL STRETCH
AND COMPRESSION.
IF Y IS = TO F OF BX WHERE B IS > 1
THE GRAPH WILL BE HORIZONTAL COMPRESSION
WHICH WE HAD AS G OF X = THE SQUARED OF 2X
AND IF Y IS = TO F OF BX WHERE B IS BETWEEN 0 AND 1
WE HAVE A HORIZONTAL STRETCH
WHICH WE HAD H OF X = THE SQUARED OF 1/2 X.
IN EACH CASE, FROM THE PARENT FUNCTION
WE MULTIPLY THE X COORDINATE BY ONE OVER B
WHICH IS THE REASON WHY OUR SCALE FACTORS
WERE THE RECIPROCAL OF WHAT WE MIGHT HAVE THOUGHT.
OKAY. HOPE YOU FOUND THIS HELPFUL.