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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
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] .
  • 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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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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Questions & Answers

can anyone recommend an app or. website to help me refresh my knowledge and abilities? I stopped learning more about calc and trig in the 90's
scott Reply
if 6x=-2 find value of 6/x
issaya Reply
l don't know
Aman
I don't know
Aman
l don't know
Aman
I don't know
Aman
x=.333333333334, so 6/x= 18±
scott
x=.33333333334± 6/x=18±
scott
that is strange, I don't remember inserting the À. the result I got was x= 3333333334± 6/x=18±
scott
I need glasses
scott
X=(-1÷3) X equals minus one third
Melvin
if sin15°=√p, express the following in terms of p
Norman Reply
I am learning and does someone have the time to tell me what or where this particular equation would be used?
scott
prove sin²x+cos²x=3+cos4x
Kiddy Reply
the difference between two signed numbers is -8.if the minued is 5,what is the subtrahend
jeramie Reply
the difference between two signed numbers is -8.if the minuend is 5.what is the subtrahend
jeramie
what are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
7%2=1, 679%2=1, 866245%2=1
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
ask
Ans
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
Practice Key Terms 7

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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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