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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

what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
Practice Key Terms 6

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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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