Family of Quadratic Functions

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Family of Quadratic Functions
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General Form

• Quadratic functions have the standard form y
  ax2 bx c
• a, b, and c are constants
• a ? 0 (why?)
• Quadratic functions graph as a parabola

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Zeros of the Quadratic

• Zeros are where the function crosses the
• x-axis
• Where y 0
• Consider possible numbers of zeros

?
None (or two complex)
One
Two
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Axis of Symmetry

• Parabolas are symmetric about a vertical axis
• For y ax2 bx c the axisof symmetry is at
• Given y 3x2 8x
• What is the axis of symmetry?

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Vertex of the Parabola

• The vertex is the point of theparabola
• The minimum value
• Can also be a maximum
• What is the x-value of thevertex?
• How can we find the y-value?

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Vertex of the Parabola

• Given f(x) x2 2x 8
• What is the x-value of the vertex?
• What is the y-value of the vertex?
• The vertex is at (-1, -9)

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Vertex of the Parabola

• Given f(x) x2 2x 8
• Graph shows vertex at (-1, -9)
• Note calculators ability to find vertex (minimum
  or maximum)

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Shifting and Stretching

• Start with f(x) x2
• Determine the results of transformations
• ___ f(x a) x2 2ax a2
• ___ f(x) a x2 a
• ___ a f(x) ax2
• ___ f(ax) a2x2

a) horizontal shift
b) vertical stretch or squeeze
c) horizontal stretch or squeeze
d) vertical shift
e) none of these
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Other Quadratic Forms

• Standard formy ax2 bx c
• Vertex formy a (x h)2 k
• Then (h,k) is the vertex
• Given f(x) x2 2x 8
• Change to vertex form
• Hint, use completing the square

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Vertex Form

• Changing to vertex form

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Assignment