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

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

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

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When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x -axis from a reflection with respect to the y -axis?

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How can you determine whether a function is odd or even from the formula of the function?

For a function f , substitute ( x ) for ( x ) in f ( x ) . Simplify. If the resulting function is the same as the original function, f ( x ) = f ( x ) , then the function is even. If the resulting function is the opposite of the original function, f ( x ) = f ( x ) , then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

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Algebraic

For the following exercises, write a formula for the function obtained when the graph is shifted as described.

f ( x ) = x is shifted up 1 unit and to the left 2 units.

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f ( x ) = | x | is shifted down 3 units and to the right 1 unit.

g ( x ) = | x - 1 | 3

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f ( x ) = 1 x is shifted down 4 units and to the right 3 units.

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f ( x ) = 1 x 2 is shifted up 2 units and to the left 4 units.

g ( x ) = 1 ( x + 4 ) 2 + 2

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For the following exercises, describe how the graph of the function is a transformation of the graph of the original function f .

y = f ( x + 43 )

The graph of f ( x + 43 ) is a horizontal shift to the left 43 units of the graph of f .

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y = f ( x 4 )

The graph of f ( x - 4 ) is a horizontal shift to the right 4 units of the graph of f .

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y = f ( x ) + 8

The graph of f ( x ) + 8 is a vertical shift up 8 units of the graph of f .

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y = f ( x ) 7

The graph of f ( x ) 7 is a vertical shift down 7 units of the graph of f .

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y = f ( x + 4 ) 1

The graph of f ( x + 4 ) 1 is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of f .

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For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

f ( x ) = 4 ( x + 1 ) 2 5

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g ( x ) = 5 ( x + 3 ) 2 2

decreasing on ( , 3 ) and increasing on ( 3 , )

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k ( x ) = 3 x 1

decreasing on ( 0 , )

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Graphical

For the following exercises, use the graph of f ( x ) = 2 x shown in [link] to sketch a graph of each transformation of f ( x ) .

Graph of f(x).

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

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

Graph of f(t).
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h ( x ) = | x 1 | + 4

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k ( x ) = ( x 2 ) 3 1

Graph of k(x).
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Numeric

Tabular representations for the functions f , g , and h are given below. Write g ( x ) and h ( x ) as transformations of f ( x ) .

x −2 −1 0 1 2
f ( x ) −2 −1 −3 1 2
x −1 0 1 2 3
g ( x ) −2 −1 −3 1 2
x −2 −1 0 1 2
h ( x ) −1 0 −2 2 3

g ( x ) = f ( x - 1 ) , h ( x ) = f ( x ) + 1

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Tabular representations for the functions f , g , and h are given below. Write g ( x ) and h ( x ) as transformations of f ( x ) .

x −2 −1 0 1 2
f ( x ) −1 −3 4 2 1
x −3 −2 −1 0 1
g ( x ) −1 −3 4 2 1
x −2 −1 0 1 2
h ( x ) −2 −4 3 1 0
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For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

Graph of an absolute function.

f ( x ) = | x - 3 | 2

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Graph of an absolute function.

f ( x ) = | x + 3 | 2

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For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

Graph of a parabola.

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

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For the following exercises, determine whether the function is odd, even, or neither.

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function f .

g ( x ) = f ( x )

The graph of g is a vertical reflection (across the x -axis) of the graph of f .

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g ( x ) = 4 f ( x )

The graph of g is a vertical stretch by a factor of 4 of the graph of f .

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g ( x ) = f ( 5 x )

The graph of g is a horizontal compression by a factor of 1 5 of the graph of f .

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g ( x ) = f ( 1 3 x )

The graph of g is a horizontal stretch by a factor of 3 of the graph of f .

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g ( x ) = 3 f ( x )

The graph of g is a horizontal reflection across the y -axis and a vertical stretch by a factor of 3 of the graph of f .

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For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described.

The graph of f ( x ) = | x | is reflected over the y - axis and horizontally compressed by a factor of 1 4 .

g ( x ) = | 4 x |

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The graph of f ( x ) = x is reflected over the x -axis and horizontally stretched by a factor of 2.

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The graph of f ( x ) = 1 x 2 is vertically compressed by a factor of 1 3 , then shifted to the left 2 units and down 3 units.

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

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The graph of f ( x ) = 1 x is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

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The graph of f ( x ) = x 2 is vertically compressed by a factor of 1 2 , then shifted to the right 5 units and up 1 unit.

g ( x ) = 1 2 ( x - 5 ) 2 + 1

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The graph of f ( x ) = x 2 is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

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For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

g ( x ) = 4 ( x + 1 ) 2 5

The graph of the function f ( x ) = x 2 is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

Graph of a parabola.
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g ( x ) = 5 ( x + 3 ) 2 2

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h ( x ) = 2 | x 4 | + 3

The graph of f ( x ) = | x | is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

Graph of an absolute function.
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m ( x ) = 1 2 x 3

The graph of the function f ( x ) = x 3 is compressed vertically by a factor of 1 2 .

Graph of a cubic function.
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n ( x ) = 1 3 | x 2 |

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p ( x ) = ( 1 3 x ) 3 3

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

Graph of a cubic function.
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q ( x ) = ( 1 4 x ) 3 + 1

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a ( x ) = x + 4

The graph of f ( x ) = x is shifted right 4 units and then reflected across the vertical line x = 4.

Graph of a square root function.
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For the following exercises, use the graph in [link] to sketch the given transformations.

Graph of a polynomial.

Questions & Answers

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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