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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

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het 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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