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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.

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Questions & Answers

a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark 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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