6'1 Graphing Quadratic Functions 12207

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6.1 Graphing Quadratic Functions 1/22/07

• A Quadratic Function is described by an equation
  of the following form
• f(x) ax2 bx c, where a ? 0.
• ax2 quadratic term
• bx linear term
• c constant term
• The graph of any quadratic function is called a
  parabola.

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Graph a Quadratic Function

• Graph f(x) 2x2 8x 9 by making a table of
  values.

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Parabolas

• All parabolas have an axis of symmetry the
  portions of the parabola on either side of this
  line would match.
• The point at which the axis of symmetry
  intersects a parabola is called the vertex.

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Graph of a Quadratic Function

• Consider the graph of y ax2 bx c, where a ?
  0.
• The y intercept is a(0)2 b(0) c, or c.
• The equation of the axis of symmetry is x - b
  .
• 
  2a
• The x-coordinate of the vertex is - b .
• 2a

Axis of symmetry x - b 2a
y-intercept c
vertex
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Axis of Symmetry, y-intercept, and vertex

• Consider the quadratic function f(x) x2 9
  8x.
• Find the y-intercept, the equation of the axis of
  symmetry, and the x-coordinate of the vertex.
• f(x) ax2 bx c
• f(x) 1x2 8x 9
• a 1, b 8, c 9
• The y intercept is 9. You can find the
  equation of the axis of symmetry using a and b.
• x - b - 8 -4 .
• 2a 2(1)
• The equation of the axis of symmetry is x -4.
  Therefore, the x-coordinate of the vertex is -4.

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• B. Make a table of values that includes the
  vertex.
• Choose some values that are less than -4 and some
  that are greater than -4. This ensures that
  points on each side of the axis of symmetry are
  graphed.

Vertex
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• c. Use the information to graph the function.
• Graph the vertex and y-intercept. Then graph the
  points from your table connecting them and the
  y-intercept with a smooth curve. As a check,
  draw the axis of symmetry,
• x -4, as a dashed line. The graph of the
  function should be symmetrical about this line.

(0 , 9)
x -4
(-4 , -7)
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Maximum and Minimum Values

• The y-coordinate of the vertex of a quadratic
  function is the maximum value or minimum value
  obtained by the function.
• The graph of f(x) ax2 bx c, where a ? 0,
• Opens up and has a minimum value where a gt 0, and
• Opens down and has a maximum value when a lt 0.
  a is positive
  a is negative

f(x)
f(x)
maximum
minimum
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Maximum or Minimum Value

• Consider the function f(x) x2 4x 9.
• Determine whether the function has a maximum or
  minimum value.
• For this function, a 1, b -4, and c 9.
  Since a gt 0, the graph opens up and the function
  has a minimum value.
• b. State the maximum or minimum value of the
  function.
• The minimum value of the function is the
  y-coordinate of the vertex. The x-coordinate of
  the vertex is - 4 or 2.
• 
  2(1)
• Find the y-coordinate of the vertex by evaluating
  the function for x 2.
• f(x) x2 4x 9
• f(2) 22 4(2) 9 5
• Therefore, the minimum value of the function is 5.

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f(x) x2 4x 9
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More Practice!!!

• Textbook p. 291 14 24 even.
• Homework worksheet 6.1