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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

The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
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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