Function Transformations: Horizontal and Vertical Stretches and Compressions

Function Transformations: Horizontal and Vertical Stretches and Compressions


– WELCOME TO THE FIRST
OF SEVERAL VIDEOS ON FUNCTION TRANSFORMATIONS. THIS VIDEO FOCUSES ON HORIZONTAL
AND VERTICAL STRETCHES AND COMPRESSIONS. SO IF WE’RE COMPARING F(X)
TO A FUNCTION IN THIS FORM, WE’LL BE LOOKING AT
HOW THE VALUE OF “A” AND B AFFECT THE GRAPH
OF THE ORIGINAL FUNCTION. SO IF WE HAVE Y=”A” x F(X),
WHERE “A” IS GREATER THAN ONE, THIS WILL STRETCH THE GRAPH
OF F(X) VERTICALLY BY A FACTOR OF “A”. AND IF WE HAVE Y=”A” x F(X)
WHERE “A” IS BETWEEN 0 AND 1, THIS WILL COMPRESS THE GRAPH
OF F(X) VERTICALLY BY A FACTOR OF “A”. SO LET’S COMPARE F(X) TO 1/2
x F(X) AND 2 x F(X) BY COMPLETING T TABLES. SO IF F(X)=X SQUARED,
TO FIND Y WE’LL JUST SQUARE X. SO WE’D HAVE 1 SQUARED,
2 SQUARED, 3 SQUARED, AND 4 SQUARED. LET’S FIRST TAKE A LOOK AT
2 x F(X), WHICH WOULD BE 2 x X SQUARED. WELL, IF THIS IS X SQUARED
AND WE WANT TO FIND 2X SQUARED, WE CAN JUST MULTIPLY 2 x
THESE Y VALUES. SO WE’D HAVE 2 x 1, 2 x 4,
2 x 9, AND 2 x 16. AND THIS WOULD BE
A VERTICAL STRETCH BECAUSE NOTICE FOR THE SAME
X COORDINATES, THESE Y COORDINATES
ARE MUCH LARGER AND THEREFORE
STRETCHING IT VERTICALLY. NOW IF WE TAKE A LOOK AT
1/2 x F(X) OR 1/2X SQUARED, WE CAN JUST MULTIPLY 1/2
x THESE Y VALUES TO OBTAIN 1/2 X SQUARED. SO WE’D HAVE 1/2 x 1, 1/2 x 4,
1/2 x 9, AND 1/2 x 16. AND NOTICE
FOR THE SAME X VALUES, THESE Y VALUES ARE LESS AND THEREFORE WOULD COMPRESS
THE GRAPH VERTICALLY. SO WHAT YOU MAY HAVE NOTICED
IS ONCE YOU IDENTIFY THE VALUE OF “A” TO FIND
THE Y COORDINATES OF A TRANSFORMED FUNCTION
WITH THE SAME X COORDINATES, WHICH WOULD BE JUST TO MULTIPLY
EACH Y COORDINATE BY “A” AND LEAVE THE X COORDINATES
THE SAME. IF WE TAKE A LOOK AT THESE
THREE GRAPHS ON THE SAME COORDINATE PLANE, WE’LL NOTICE THE ORIGINAL
FUNCTION IS IN BLUE AND IF WE STRETCH THE BLUE
FUNCTION VERTICALLY BY A FACTOR OF 2 WE WOULD HAVE H(X),
WHICH=2 x F(X) VERSUS IF WE TAKE
THE BLUE FUNCTION AND COMPRESS IT VERTICALLY
BY A FACTOR OF 1/2 WE WOULD HAVE G(X),
WHICH=1/2 x F(X). LET’S TAKE A LOOK AT
AN ANIMATION OF THIS. SO IF “A” IS GREATER THAN ONE, WE CAN SEE A VERTICAL STRETCH
BY A FACTOR OF “A”.   LET’S TAKE A LOOK
AT WHAT HAPPENS WHEN “A” IS BETWEEN 0 AND 1. SO HERE WE HAVE “A”=1,
“A”=0.9, 0.8, AND SO ON. SO WE HAVE A VERTICAL
COMPRESSION, IN THIS CASE, BY A FACTOR OF 3/10 AND SO ON. LET’S NOW TAKE A LOOK
AT A HORIZONTAL STRETCH AND HORIZONTAL COMPRESSION. IF WE HAVE F(BX)
WHERE B IS GREATER THAN ONE, THIS WILL COMPRESS THE GRAPH
OF F(X) HORIZONTALLY. AND IF WE HAVE Y=F(BX)
WHERE B IS BETWEEN ZERO AND ONE, THIS WILL STRETCH THE GRAPH
OF F(X) HORIZONTALLY. SO LET’S GO AHEAD AND COMPARE
T TABLES FOR F(X), F(2X), AND F(1/2X). SO WE ALREADY KNOW THIS WOULD BE
1, 4, 9, AND 16. NOTICE NOW THAT
WE’RE MULTIPLYING X BY 2 AND THEN SQUARING IT. SO IF X IS 1/2, 2 x 1/2=1,
AND THEN SQUARED WE’D HAVE 1. THEN 2 x 1 SQUARED=4, 2 x 3/2
SQUARED=3 SQUARED OR 9, AND 2 x 2 SQUARED=16. NOTICE WHEN B=2 WE LEAVE
THE Y COORDINATES THE SAME AND THEN EITHER MULTIPLY THE
ORIGINAL X COORDINATES BY 1/B OR JUST DIVIDE BY B. LET’S NOW TAKE A LOOK
AT F(1/2X), WHICH WOULD EQUAL 1/2X SQUARED,
SO B=1/2. WELL, MULTIPLYING BY 1 OVER 1/2 WOULD BE THE SAME AS
MULTIPLYING BY 2. SO IN THIS CASE, IF WE MULTIPLY
THE ORIGINAL X COORDINATES BY 2 WE ARE GOING TO GET
THE SAME Y COORDINATES. LET’S TAKE A LOOK
AT THE FIRST COUPLE. IF X=2, 1/2 x 2=1 SQUARED
=1, 1/2 x 4=2 SQUARED=4, 9, 16. AND THIS WOULD BE CONSIDERED
A HORIZONTAL STRETCH. IF WE TAKE A LOOK
AT THESE THREE GRAPHS, AGAIN, WE HAVE THE ORIGINAL
FUNCTION IN BLUE AND WHEN B=2 WE HAVE
A HORIZONTAL COMPRESSION, SO WE’RE COMPRESSING IT THIS WAY TO OBTAIN THE RED FUNCTION
F(2X). AND IF WE STRETCH THE BLUE
FUNCTION HORIZONTALLY, WHEN B=1/2 WE WOULD HAVE
THE GREEN FUNCTION. LET’S TAKE A LOOK AT OUR
ANIMATION OF THIS AS WELL. SO WHEN B IS GREATER THAN ONE,
WE HAVE A HORIZONTAL COMPRESSION AS WE SEE HERE. AND WHEN B IS BETWEEN ZERO
AND ONE, WE HAVE A HORIZONTAL STRETCH
AS WE SEE HERE. OKAY, LET’S GO AHEAD
AND TAKE A LOOK AT SOME OF OUR OWN EXAMPLES. THE FIRST THING WE NEED TO BE
ABLE TO DO IS RECOGNIZE THE PARENT FUNCTION FOR F(X). IF F(X) IS EQUAL TO THREE TIMES
THE ABSOLUTE VALUE OF X, LET’S LET OUR PARENT FUNCTION
G(X) EQUAL THE ABSOLUTE VALUE OF X. AND WE SHOULD BE ABLE TO MAKE
A NICE GRAPH OF THIS WITHOUT USING OUR
GRAPHING CALCULATORS. WE SHOULD ALL KNOW THIS
FORMS A V SHAPE VERY SIMILAR TO THIS GRAPH. LET’S GO AHEAD AND IDENTIFY
SOME KEY POINTS ON THIS GRAPH, LIKE THE ORIGIN TO POINT (2,2)
AND HOW ABOUT THE POINT (-2,2). NOW, LOOKING
AT THE ORIGINAL FUNCTION, WE SHOULD BE ABLE TO MAKE
THE CONNECTION THAT F(X)=3 x G(X),
WHICH MEANS “A”=3, WHICH MEANS THAT OUR GRAPH
IS GOING TO BE A GRAPH THAT’S VERTICALLY STRETCHED
BY A FACTOR OF 3 FROM THIS ORIGINAL RED GRAPH. SO TO FIND POINTS ON THIS
TRANSFORMED FUNCTION, WE CAN JUST MULTIPLY EACH OF
THESE Y COORDINATES BY 3. SINCE 2 x 3 WOULD BE 6, ONE POINT ON THE TRANSFORMED
GRAPH WOULD BE (-2,6). 0 x 3=0, SO WE HAVE THE ORIGIN AND HERE WE HAVE ANOTHER
Y COORDINATE OF 2 SO THIS WOULD BECOME THE POINT
(2,6). AND SO VERY QUICKLY AND EASILY
WE CAN GRAPH THE TRANSFORMED FUNCTION BASED UPON DETERMINING
THE VALUE OF “A”. LET’S TAKE A LOOK
AT THIS ONE NOW. WE HAVE F(X)=
THE SQUARE ROOT OF 2X. AGAIN, WE SHOULD BE ABLE
TO RECOGNIZE THAT THE PARENT FUNCTION,
WHICH WE’LL CALL G(X), IS EQUAL TO THE SQUARE ROOT
OF X. LET’S GO AHEAD AND GRAPH
THAT FUNCTION. IF X=0, Y=0.
IF X=1, Y=1. IF X=4,
THE SQUARE ROOT OF 4=2. X=9, WE’D HAVE THE SQUARE ROOT
OF 9=3. 16, 4. SO THIS IS THE PARENT FUNCTION AND WHAT WE’RE GOING TO DO IS
GRAPH THE TRANSFORMED FUNCTION. LET’S GO AHEAD AND MAKE A
T TABLE WHERE SOME OF THE KEY POINTS
ON THIS GRAPH, LIKE, (0,0), (1,1), (4,2),
(9,3), AND (16,4). NOW, WE NEED TO BE ABLE
TO RECOGNIZE THAT F(X)=G(2X)=THE SQUARE ROOT
OF 2X. SO THAT TELLS US THAT B=2. SO SINCE B=2, WE HAVE
A HORIZONTAL COMPRESSION, WHICH MEANS TO FIND POINTS
ON THE TRANSFORMED FUNCTION WE CAN JUST MULTIPLY
THE X COORDINATE BY 1/B OR JUST DIVIDE BY B. SO IF B=2, WE’LL JUST TAKE
EACH OF THESE X COORDINATES, DIVIDE THEM BY 2, AND KEEP
THE Y COORDINATES THE SAME. SO THE ORIGIN WOULD STILL BE
ON THE TRANSFORMED FUNCTION. THE NEXT POINT WOULD BE
(1/2,1). 4 DIVIDED BY 2=2,
SO WE HAVE THE POINT (2,2). 9 DIVIDED BY 2=9/2 OR 4.5, 3 AND THEN INSTEAD OF (16,4)
WE’LL HAVE 16 DIVIDED BY 2, THAT’D BE (8,4). THIS SHOULD BE ENOUGH POINTS
TO MAKE A NICE GRAPH OF THE TRANSFORMED FUNCTION
IN GREEN. SO THIS HAS BEEN
HORIZONTALLY COMPRESSED, MEANING PUSHED BACK
IN THIS DIRECTION. OKAY, I HOPE YOU FOUND
THIS VIDEO HELPFUL. THANK YOU FOR WATCHING.  

59 Replies to “Function Transformations: Horizontal and Vertical Stretches and Compressions”

  1. Thank you so much. I was learning vertical stretches and compressions from a book, and they were not clicking at all. Now I realize how simple they are!

  2. thank you so much..i have been searching for the explanation about the horizontal stretch..now i finally found it..thanks a lot

  3. Really appreciate. This is by far, the most helpful math tutor I've find in all youtube. Saved my gpa, saved my marriage, saved my life….thanks so much!

  4. Very nice!  You have compressed the information nicely without stretching the truth within it!

  5. This was an exceptionally clear and easy to understand math video – I understand this concept so much better now. Thank you so much.

  6. Thanks a lot for such an amazing, clear, and concise lesson! The quote at the end is also something to think about!

  7. is there any function where horizontal compression are same as vertical strech and what would be the name of that function

  8. a*f(b(x-c))+d
    where :
    a is Horizontal compression and Stretching
    b is Vertical compression and stretching

    it would seem you could achieve the same results through either method ……

  9. Sooooo helpful!! I am reviewing precalc this summer on my own b/c I have to take Calc 1 this fall. I just read the transformation section today and for some reason it wasn't clicking with the horizontal and vertical stretching part, but your video made it click for me! lol thanks!

  10. Thanks! I have a test in 1 hour that includes this and I didn't understand it well until I saw your video! thanks a lot :3

  11. The video goes way too fast and I have to keep stopping a going back, a lot of pausing. And, I still don't get it after watching this video twice. Slow down.

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