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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

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
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100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
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Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
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10y
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tan hyperbolic inverse (x+iy)=alpha +i bita
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prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
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what is complex numbers
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Solve 2cos x + 3sin x = 0.5
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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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