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Finding a triple transformation of a graph

Use the graph of f ( x ) in [link] to sketch a graph of k ( x ) = f ( 1 2 x + 1 ) 3.

Graph of a half-circle.

To simplify, let’s start by factoring out the inside of the function.

f ( 1 2 x + 1 ) 3 = f ( 1 2 ( x + 2 ) ) 3

By factoring the inside, we can first horizontally stretch by 2, as indicated by the 1 2 on the inside of the function. Remember that twice the size of 0 is still 0, so the point (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0). See [link] .

Graph of a vertically stretch half-circle.

Next, we horizontally shift left by 2 units, as indicated by x + 2. See [link] .

Graph of a vertically stretch and translated half-circle.

Last, we vertically shift down by 3 to complete our sketch, as indicated by the 3 on the outside of the function. See [link] .

Graph of a vertically stretch and translated half-circle.
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Access this online resource for additional instruction and practice with transformation of functions.

Key equations

Vertical shift g ( x ) = f ( x ) + k (up for k > 0 )
Horizontal shift g ( x ) = f ( x h ) (right for h > 0 )
Vertical reflection g ( x ) = f ( x )
Horizontal reflection g ( x ) = f ( x )
Vertical stretch g ( x ) = a f ( x ) ( a > 0 )
Vertical compression g ( x ) = a f ( x ) ( 0 < a < 1 )
Horizontal stretch g ( x ) = f ( b x ) ( 0 < b < 1 )
Horizontal compression g ( x ) = f ( b x ) ( b > 1 )

Key concepts

  • A function can be shifted vertically by adding a constant to the output. See [link] and [link] .
  • A function can be shifted horizontally by adding a constant to the input. See [link] , [link] , and [link] .
  • Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See [link] .
  • Vertical and horizontal shifts are often combined. See [link] and [link] .
  • A vertical reflection reflects a graph about the x - axis. A graph can be reflected vertically by multiplying the output by –1.
  • A horizontal reflection reflects a graph about the y - axis. A graph can be reflected horizontally by multiplying the input by –1.
  • A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See [link] .
  • A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See [link] .
  • A function presented as an equation can be reflected by applying transformations one at a time. See [link] .
  • Even functions are symmetric about the y - axis, whereas odd functions are symmetric about the origin.
  • Even functions satisfy the condition f ( x ) = f ( x ) .
  • Odd functions satisfy the condition f ( x ) = f ( x ) .
  • A function can be odd, even, or neither. See [link] .
  • A function can be compressed or stretched vertically by multiplying the output by a constant. See [link] , [link] , and [link] .
  • A function can be compressed or stretched horizontally by multiplying the input by a constant. See [link] , [link] , and [link] .
  • The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See [link] and [link] .

Section exercises

Verbal

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

Got questions? Get instant answers now!

Questions & Answers

can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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