# 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”

Thank you for the video really explained a lot.

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!

thankes alot for the good explen ^-^

This video was really good!. Thank you so much!

thanks for the help!

would have been better if intercepts were included

this did help explain it a little better i think thanks

Very helpful!

how do I know when to stretch a function vertical and when to stretch it horizontally?

where does logic behind the rule: 'multiply each x-coordinate by 1/b' derive from?

thanks

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

Thank you! You saved me!

The quote is by Khalil Gibran, not kahlil gibran

Thank you! Thank you, thank you!!

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!

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

Im confused what if the factor for the vertical shift is 3/2 ?? ughh

Thank you very much.

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

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

which software did you use to illustrate these diffrent functions?

Thank you very much, this was a great lesson, that helped me a lot, you are a very good teacher.

there are so many easier ways

Hate learning via slides.

where can i get those slides please?????

thank you so much!!!!

very nice, thanks

Super Helpful! especially the way you explained the horizontal stretch and compressions

Well done. Thank you sir.

Thank you.

Great Video!

thank you

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

what is the difference between a horizontal compression and a vertical stretch

Thanks for the help! I was stumped over my hw for the past hour and this really helped clear things up!

What grade are y'all in ?

So A is the vertical stretch and compression? and B is the Horizontal stretch and compression?

So can a compression of 1/2 be thought of as a vertical shrink of 2?

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 ……

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!

not clear at all 🙁

This video was extremely helpful! Saved my gpa, saved my marriage, saved my life! @Ivan Herrera

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

I loved this thank you for sharing!

very well explained.

wait so a vertical stretch is the parabola narrowing, and the vertical shrink is the parabola widening?

Still don’t get it . My test is gonna be less than 12% for a fact

NERD

Makes no sense

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.

This helped me a lot thank you

I have never understood this until now, thank you!

YOU SUCK ASS

super awful

What if a or b is less than 0?

you are too fast, your so shit

were are you getting the x values the function of h(x)

It is really helpful thanks