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f ( x ) = −2 ( x + 3 ) 2 6

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

Domain is ( , ) . Range is [ −5 , ) .

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

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k ( x ) = 3 x 2 6 x 9

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

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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

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

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

f ( x ) = x 2 + 1

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

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

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

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

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

f ( x ) = x 2 + 1

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

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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 ) .

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f ( x ) = x 2 6 x 1

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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 ) .

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f ( x ) = x 2 7 x + 3

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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 ) .

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f ( x ) = 4 x 2 12 x 3

Graph of f(x)=4x^2-12x-3
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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

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Graph of a negative parabola with a vertex at (2, 7).

f ( x ) = −2 x 2 + 8 x 1

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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

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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

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x –2 –1 0 1 2
y 1 0 1 4 9
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x –2 –1 0 1 2
y –2 1 2 1 –2

f ( x ) = 2 x 2

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x –2 –1 0 1 2
y –8 –3 0 1 0
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x –2 –1 0 1 2
y 8 2 0 2 8

f ( x ) = 2 x 2

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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?

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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).

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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?

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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

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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.

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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 , ) .

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Vertex ( −1 , 2 ) opens down.

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Vertex ( −5 , 11 ) , opens down.

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

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Vertex ( −100 , 100 ) , opens up.

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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

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

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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

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

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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

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

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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 .

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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.

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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 .

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

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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 .

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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.

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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

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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?

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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

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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?

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Questions & Answers

what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
what's microbiome?
john Reply
Microbiology is the scientific study of microorganisms
Ibra
the microorganisms in a particular environment (including the body or a part of the body).
Ibra
describe the bacterial cell
Akello
The biggest populations of microbes reside in the gut.other popular habitats include the skin genitals.The microbial cells and their genetic material ,the microbiome ,live with human from birth .
Zahreen
can agriculture be integrated into biology
David
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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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