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

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For the following exercises, determine the domain and range of the quadratic function.

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

Domain is ( , ) . Range is [ 2 , ) .

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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, solve the equations over the complex numbers.

x 2 = 8

{ 2 i 2 , 2 i 2 }

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x 2 + 27 = 0

{ 3 i 3 , 3 i 3 }

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x 2 4 x + 5 = 0

{ 2 + i , 2 i }

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x 2 4 x + 13 = 0

{ 2 + 3 i , 2 3 i }

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x 2 10 x + 26 = 0

{ 5 + i , 5 i }

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x ( x 4 ) = 20

{ 2 + 2 6 ,   2 2 6 }

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2 x 2 + 2 x + 5 = 0

{ 1 2 + 3 2 i ,   1 2 3 2 i }

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5 x 2 + 6 x + 2 = 0

{ 3 5 + 1 5 i ,   3 5 1 5 i }

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x 2 + x + 2 = 0

{ 1 2 + 1 2 i 7 ,   1 2 1 2 i 7 }

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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 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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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