<< Chapter < Page Chapter >> Page >

f ( x ) = −2 ( x + 3 ) 2 6

Got questions? Get instant answers now!

f ( x ) = x 2 + 6 x + 4

Domain is ( , ) . Range is [ −5 , ) .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

k ( x ) = 3 x 2 6 x 9

Domain is ( , ) . Range is [ −12 , ) .

Got questions? Get instant answers now!

For the following exercises, use the vertex ( h , k ) and a point on the graph ( x , y ) to find the general form of the equation of the quadratic function.

( h , k ) = ( 2 , 0 ) , ( x , y ) = ( 4 , 4 )

f ( x ) = x 2 4 x + 4

Got questions? Get instant answers now!

( h , k ) = ( −2 , −1 ) , ( x , y ) = ( −4 , 3 )

Got questions? Get instant answers now!

( h , k ) = ( 0 , 1 ) , ( x , y ) = ( 2 , 5 )

f ( x ) = x 2 + 1

Got questions? Get instant answers now!

( h , k ) = ( 2 , 3 ) , ( x , y ) = ( 5 , 12 )

Got questions? Get instant answers now!

( h , k ) = ( 5 , 3 ) , ( x , y ) = ( 2 , 9 )

f ( x ) = 6 49 x 2 + 60 49 x + 297 49

Got questions? Get instant answers now!

( h , k ) = ( 3 , 2 ) , ( x , y ) = ( 10 , 1 )

Got questions? Get instant answers now!

( h , k ) = ( 0 , 1 ) , ( x , y ) = ( 1 , 0 )

f ( x ) = x 2 + 1

Got questions? Get instant answers now!

( h , k ) = ( 1 , 0 ) , ( x , y ) = ( 0 , 1 )

Got questions? Get instant answers now!

Graphical

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.

f ( x ) = x 2 2 x

Graph of f(x) = x^2-2x

Vertex ( 1 ,   1 ) , Axis of symmetry is x = 1. Intercepts are ( 0 , 0 ) ,   ( 2 , 0 ) .

Got questions? Get instant answers now!

f ( x ) = x 2 6 x 1

Got questions? Get instant answers now!

f ( x ) = x 2 5 x 6

Graph of f(x)x^2-5x-6

Vertex ( 5 2 , 49 4 ) , Axis of symmetry is ( 0 , 6 ) , ( 1 , 0 ) , ( 6 , 0 ) .

Got questions? Get instant answers now!

f ( x ) = x 2 7 x + 3

Got questions? Get instant answers now!

f ( x ) = −2 x 2 + 5 x 8

Graph of f(x)=-2x^2+5x-8

Vertex ( 5 4 ,   39 8 ) , Axis of symmetry is x = 5 4 . Intercepts are ( 0 ,   8 ) .

Got questions? Get instant answers now!

f ( x ) = 4 x 2 12 x 3

Graph of f(x)=4x^2-12x-3
Got questions? Get instant answers now!

For the following exercises, write the equation for the graphed quadratic function.

Graph of a positive parabola with a vertex at (2, -3) and y-intercept at (0, 1).

f ( x ) = x 2 4 x + 1

Got questions? Get instant answers now!
Graph of a negative parabola with a vertex at (2, 7).

f ( x ) = −2 x 2 + 8 x 1

Got questions? Get instant answers now!
Graph of a positive parabola with a vertex at (3, -1) and y-intercept at (0, 3.5).

f ( x ) = 1 2 x 2 3 x + 7 2

Got questions? Get instant answers now!

Numeric

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

x –2 –1 0 1 2
y 5 2 1 2 5

f ( x ) = x 2 + 1

Got questions? Get instant answers now!
x –2 –1 0 1 2
y 1 0 1 4 9
Got questions? Get instant answers now!
x –2 –1 0 1 2
y –2 1 2 1 –2

f ( x ) = 2 x 2

Got questions? Get instant answers now!
x –2 –1 0 1 2
y –8 –3 0 1 0
Got questions? Get instant answers now!
x –2 –1 0 1 2
y 8 2 0 2 8

f ( x ) = 2 x 2

Got questions? Get instant answers now!

Technology

For the following exercises, use a calculator to find the answer.

Graph on the same set of axes the functions f ( x ) = x 2 , f ( x ) = 2 x 2 ,  and  f ( x ) = 1 3 x 2 .

What appears to be the effect of changing the coefficient?

Got questions? Get instant answers now!

Graph on the same set of axes f ( x ) = x 2 , f ( x ) = x 2 + 2 and f ( x ) = x 2 , f ( x ) = x 2 + 5 and f ( x ) = x 2 3. What appears to be the effect of adding a constant?

The graph is shifted up or down (a vertical shift).

Got questions? Get instant answers now!

Graph on the same set of axes f ( x ) = x 2 , f ( x ) = ( x 2 ) 2 , f ( x 3 ) 2 ,  and  f ( x ) = ( x + 4 ) 2 .

What appears to be the effect of adding or subtracting those numbers?

Got questions? Get instant answers now!

The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function h ( x ) = 32 ( 80 ) 2 x 2 + x where x is the horizontal distance traveled and h ( x ) is the height in feet. Use the TRACE feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally.

50 feet

Got questions? Get instant answers now!

A suspension bridge can be modeled by the quadratic function h ( x ) = .0001 x 2 with −2000 x 2000 where | x | is the number of feet from the center and h ( x ) is height in feet. Use the TRACE feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.

Got questions? Get instant answers now!

Extensions

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

Vertex ( 1 , −2 ) , opens up.

Domain is ( , ) . Range is [ −2 , ) .

Got questions? Get instant answers now!

Vertex ( −1 , 2 ) opens down.

Got questions? Get instant answers now!

Vertex ( −5 , 11 ) , opens down.

Domain is ( , ) Range is ( , 11 ] .

Got questions? Get instant answers now!

Vertex ( −100 , 100 ) , opens up.

Got questions? Get instant answers now!

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains ( 1 , 1 ) and has shape of f ( x ) = 2 x 2 . Vertex is on the y - axis.

f ( x ) = 2 x 2 1

Got questions? Get instant answers now!

Contains ( −1 , 4 ) and has the shape of f ( x ) = 2 x 2 . Vertex is on the y - axis.

Got questions? Get instant answers now!

Contains ( 2 , 3 ) and has the shape of f ( x ) = 3 x 2 . Vertex is on the y - axis.

f ( x ) = 3 x 2 9

Got questions? Get instant answers now!

Contains ( 1 , −3 ) and has the shape of f ( x ) = x 2 . Vertex is on the y - axis.

Got questions? Get instant answers now!

Contains ( 4 , 3 ) and has the shape of f ( x ) = 5 x 2 . Vertex is on the y - axis.

f ( x ) = 5 x 2 77

Got questions? Get instant answers now!

Contains ( 1 , −6 ) has the shape of f ( x ) = 3 x 2 . Vertex has x-coordinate of −1.

Got questions? Get instant answers now!

Real-world applications

Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.

50 feet by 50 feet. Maximize f ( x ) = x 2 + 100 x .

Got questions? Get instant answers now!

Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.

Got questions? Get instant answers now!

Find the dimensions of the rectangular corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.

125 feet by 62.5 feet. Maximize f ( x ) = −2 x 2 + 250 x .

Got questions? Get instant answers now!

Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?

Got questions? Get instant answers now!

Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?

6 and −6 ; product is –36; maximize f ( x ) = x 2 + 12 x .

Got questions? Get instant answers now!

Suppose that the price per unit in dollars of a cell phone production is modeled by p = $ 45 0.0125 x , where x is in thousands of phones produced, and the revenue represented by thousands of dollars is R = x p . Find the production level that will maximize revenue.

Got questions? Get instant answers now!

A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by h ( t ) = −4.9 t 2 + 229 t + 234. Find the maximum height the rocket attains.

2909.56 meters

Got questions? Get instant answers now!

A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by h ( t ) = 4.9 t 2 + 24 t + 8. How long does it take to reach maximum height?

Got questions? Get instant answers now!

A soccer stadium holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?

$10.70

Got questions? Get instant answers now!

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

Got questions? Get instant answers now!

Questions & Answers

what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
what is ecology
Philip Reply
study of interaction between living things and their environment
AI-Robot
the same
Menseh
ecology
Ahmed
so what are the sub topics under it
Lukmon
the draw of microscope and lebel it
Saidu Reply
What's atomic
John Reply
atomic? or atom🙄
joyce
related to or concerning an atom.
Nakita
what is biology?
Wine Reply
what are the important of biology?
Wine
what is Biology
Akech Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 7

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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