<< Chapter < Page Chapter >> Page >

Shifts of the parent function f ( x ) = b x

For any constants c and d , the function f ( x ) = b x + c + d shifts the parent function f ( x ) = b x

  • vertically d units, in the same direction of the sign of d .
  • horizontally c units, in the opposite direction of the sign of c .
  • The y -intercept becomes ( 0 , b c + d ) .
  • The horizontal asymptote becomes y = d .
  • The range becomes ( d , ) .
  • The domain, ( , ) , remains unchanged.

Given an exponential function with the form f ( x ) = b x + c + d , graph the translation.

  1. Draw the horizontal asymptote y = d .
  2. Identify the shift as ( c , d ) . Shift the graph of f ( x ) = b x left c units if c is positive, and right c units if c is negative.
  3. Shift the graph of f ( x ) = b x up d units if d is positive, and down d units if d is negative.
  4. State the domain, ( , ) , the range, ( d , ) , and the horizontal asymptote y = d .

Graphing a shift of an exponential function

Graph f ( x ) = 2 x + 1 3. State the domain, range, and asymptote.

We have an exponential equation of the form f ( x ) = b x + c + d , with b = 2 , c = 1 , and d = 3.

Draw the horizontal asymptote y = d , so draw y = −3.

Identify the shift as ( c , d ) , so the shift is ( 1 , −3 ) .

Shift the graph of f ( x ) = b x left 1 units and down 3 units.

Graph of the function, f(x) = 2^(x+1)-3, with an asymptote at y=-3. Labeled points in the graph are (-1, -2), (0, -1), and (1, 1).

The domain is ( , ) ; the range is ( 3 , ) ; the horizontal asymptote is y = −3.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Graph f ( x ) = 2 x 1 + 3. State domain, range, and asymptote.

The domain is ( , ) ; the range is ( 3 , ) ; the horizontal asymptote is y = 3.

Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).
Got questions? Get instant answers now!

Given an equation of the form f ( x ) = b x + c + d for x , use a graphing calculator to approximate the solution.

  • Press [Y=] . Enter the given exponential equation in the line headed “ Y 1 = ”.
  • Enter the given value for f ( x ) in the line headed “ Y 2 = ”.
  • Press [WINDOW] . Adjust the y -axis so that it includes the value entered for “ Y 2 = ”.
  • Press [GRAPH] to observe the graph of the exponential function along with the line for the specified value of f ( x ) .
  • To find the value of x , we compute the point of intersection. Press [2ND] then [CALC] . Select “intersect” and press [ENTER] three times. The point of intersection gives the value of x for the indicated value of the function.

Approximating the solution of an exponential equation

Solve 42 = 1.2 ( 5 ) x + 2.8 graphically. Round to the nearest thousandth.

Press [Y=] and enter 1.2 ( 5 ) x + 2.8 next to Y 1 =. Then enter 42 next to Y2= . For a window, use the values –3 to 3 for x and –5 to 55 for y . Press [GRAPH] . The graphs should intersect somewhere near x = 2.

For a better approximation, press [2ND] then [CALC] . Select [5: intersect] and press [ENTER] three times. The x -coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for Guess? ) To the nearest thousandth, x 2.166.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve 4 = 7.85 ( 1.15 ) x 2.27 graphically. Round to the nearest thousandth.

x 1.608

Got questions? Get instant answers now!

Graphing a stretch or compression

While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function f ( x ) = b x by a constant | a | > 0. For example, if we begin by graphing the parent function f ( x ) = 2 x , we can then graph the stretch, using a = 3 , to get g ( x ) = 3 ( 2 ) x as shown on the left in [link] , and the compression, using a = 1 3 , to get h ( x ) = 1 3 ( 2 ) x as shown on the right in [link] .

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask