<< Chapter < Page Chapter >> Page >

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?

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

g ( x ) = | x - 1 | 3

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

y = f ( x 4 )

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

Got questions? Get instant answers now!

y = f ( x ) + 8

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

Got questions? Get instant answers now!

y = f ( x ) 7

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

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

k ( x ) = 3 x 1

decreasing on ( 0 , )

Got questions? Get instant answers now!

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).
Got questions? Get instant answers now!

h ( x ) = | x 1 | + 4

Got questions? Get instant answers now!

k ( x ) = ( x 2 ) 3 1

Graph of k(x).
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!
Graph of an absolute function.

f ( x ) = | x + 3 | 2

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

g ( x ) = 4 f ( x )

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

Got questions? Get instant answers now!

g ( x ) = f ( 5 x )

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

Got questions? Get instant answers now!

g ( x ) = f ( 1 3 x )

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

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!

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 |

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

n ( x ) = 1 3 | x 2 |

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

q ( x ) = ( 1 4 x ) 3 + 1

Got questions? Get instant answers now!

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.
Got questions? Get instant answers now!

For the following exercises, use the graph in [link] to sketch the given transformations.

Graph of a polynomial.

Questions & Answers

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask