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Access these online resources for additional instruction and practice with quadratic equations.

Key equations

general form of a quadratic function f ( x ) = a x 2 + b x + c
the quadratic formula x = b ± b 2 4 a c 2 a
standard form of a quadratic function f ( x ) = a ( x h ) 2 + k

Key concepts

  • A polynomial function of degree two is called a quadratic function.
  • The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
  • The axis of symmetry is the vertical line passing through the vertex. The zeros, or x - intercepts, are the points at which the parabola crosses the x - axis. The y - intercept is the point at which the parabola crosses the y - axis. See [link] , [link] , and [link] .
  • Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See [link] .
  • The vertex can be found from an equation representing a quadratic function. See [link] .
  • The domain of a quadratic function is all real numbers. The range varies with the function. See [link] .
  • A quadratic function’s minimum or maximum value is given by the y - value of the vertex.
  • The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See [link] and [link] .
  • Some quadratic equations must be solved by using the quadratic formula. See [link] .
  • The vertex and the intercepts can be identified and interpreted to solve real-world problems. See [link] .

Section exercises

Verbal

Explain the advantage of writing a quadratic function in standard form.

When written in that form, the vertex can be easily identified.

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How can the vertex of a parabola be used in solving real world problems?

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Explain why the condition of a 0 is imposed in the definition of the quadratic function.

If a = 0 then the function becomes a linear function.

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What is another name for the standard form of a quadratic function?

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What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

If possible, we can use factoring. Otherwise, we can use the quadratic formula.

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Algebraic

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.

f ( x ) = x 2 12 x + 32

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g ( x ) = x 2 + 2 x 3

f ( x ) = ( x + 1 ) 2 2 , Vertex ( 1 , 4 )

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f ( x ) = x 2 + 5 x 2

f ( x ) = ( x + 5 2 ) 2 33 4 , Vertex ( 5 2 , 33 4 )

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h ( x ) = 2 x 2 + 8 x 10

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k ( x ) = 3 x 2 6 x 9

f ( x ) = 3 ( x 1 ) 2 12 , Vertex ( 1 , 12 )

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f ( x ) = 3 x 2 5 x 1

f ( x ) = 3 ( x 5 6 ) 2 37 12 , Vertex ( 5 6 , 37 12 )

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For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

y ( x ) = 2 x 2 + 10 x + 12

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f ( x ) = 2 x 2 10 x + 4

Minimum is 17 2 and occurs at 5 2 . Axis of symmetry is x = 5 2 .

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f ( x ) = x 2 + 4 x + 3

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f ( x ) = 4 x 2 + x 1

Minimum is 17 16 and occurs at 1 8 . Axis of symmetry is x = 1 8 .

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h ( t ) = 4 t 2 + 6 t 1

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

Minimum is 7 2 and occurs at 3. Axis of symmetry is x = 3.

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Questions & Answers

what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
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
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton 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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