Section 2'2: Quadratic Functions Translation and Rotation

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Section 2.2 Quadratic Functions Translation and
Rotation

• Homework 3-19 (page 72)

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First of all, time for questions

• Problems from last nights homework Problems 33,
  35 and 47 (pages 60 and 61.)
• F.A.Q Calculators? Graphing Calculators?

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Quadratic Functions

• In section 2.1, we dealt with functions in
  general.

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Quadratic Functions

• In section 2.1, we dealt with functions in
  general.
• In section 1.2, we dealt with linear functions.
  Namely, functions of the form f(x) bx c

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Quadratic Functions

• In section 2.1, we dealt with functions in
  general.
• In section 1.2, we dealt with linear functions.
  Namely, functions of the form f(x) bx c
• In this section, we will add one term and obtain
  quadratic functions. Namely, functions of the
  form g(x) ax2 bx c.

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How to graph a quadratic function

• The graph of a quadratic function is a parabola.

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How to graph a quadratic function

• If f(x) ax2 bx c, the significance of the
  leading coefficient a / 0 is the following If
  agt0 then the parabola opens upward, if alt0 then
  the parabola opens downward.

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How to graph a quadratic function

• An important piece of information is to figure
  out where the vertex of the parabola.
• The vertex of the parabola is either its highest
  or lowest point (depending on whether alt0 or agt0,
  respectively.)
• Being a point on the plane, the vertex has two
  coordinates, x and y.

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How to graph a quadratic function

• Being a point on the plane, the vertex has two
  coordinates, x and y.
• The abscissa for the vertex is obtained by the
  formula x -(b/2a) (trust me about this, you
  will see why later.)

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How to graph a quadratic function

• Being a point on the plane, the vertex has two
  coordinates, x and y.
• The abscissa for the vertex is obtained by the
  formula x -(b/2a) (trust me about this, you
  will see why later.)
• You can obtain the ordinate for the vertex by
  plugging the x-value above into the function f(x)
  ax2 bx c.

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For example

• Let f(x) 3x2 6x 2
• The abscissa for the vertex is obtained by the
  formula x -(b/2a). So, x -(-6/6) 1.

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For example

• Let f(x) 3x2 6x 2
• The abscissa for the vertex is obtained by the
  formula x -(b/2a). So, x -(-6/6) 1.
• You can obtain the ordinate for the vertex by
  plugging the x-value above into the function f(x)
  ax2 bx c. Therefore, y f(1) 3(1)2
  6(1) 2 -1
• The vertex is the point (1,-1).

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The vertex relates to minimization and
maximization problems

• Let us look at problem 7, page 68.