JMerrill, 05

1
Section 31Quadratic Functions

• JMerrill, 05
• Revised 08

2
Definition of a Quadratic Function

• Let a, b, and c be real numbers with a ? 0. The
  function given by f(x) ax2 bx c is called a
  quadratic function
• Your book calls this another form, but this is
  the standard form of a quadratic function.

3
Parabolas

• The graph of a quadratic equation is a Parabola.
  
• Parabolas occur in many real-life situations
• All parabolas are symmetric with respect to a
  line called the axis of symmetry.
• The point where the axis intersects the parabola
  is the vertex.

vertex
4
Characteristics

• Graph of f(x)ax2, a gt 0
• Domain
• (- 8, 8)
• Range
• 0, 8)
• Decreasing
• (- 8, 0)
• Increasing
• (0, 8)
• Zero/Root/solution
• (0,0)
• Orientation
• Opens up

5
Characteristics

• Graph of f(x)ax2, a gt 0
• Domain
• (- 8, 8)
• Range
• (-8, 0
• Decreasing
• (0, 8)
• Increasing
• (-8, 0)
• Zero/Root/solution
• (0,0)
• Orientation
• Opens down

6
Max/Min

• A parabola has a maximum or a minimum

min
max
7
Vertex Form

• The vertex form of a quadratic function is given
  by f(x) a(x h)2 k, a ? 0
• In this parabola
• the axis of symmetry is x h
• The vertex is (h, k)
• If a gt o, the parabola opens upward. If a lt 0,
  the parabola opens downward.

8
Example

• In the equation f(x) -2(x 3)2 8, the graph
• Opens down
• Has a vertex at (3, 8)
• Axis of Symmetry x 3
• Has zeros at
• 0 -2(x 3)2 8
• -8 -2(x 3)2
• 4 (x 3)2
• 2 x 3 or -2 x 3
• X 5 x 1

9
Vertex Form from Standard Form

• Describe the graph of f(x) x2 8x 7
• In order to do this, you have to complete the
  square to put the problem in vertex form

Opens Up
Vertex?
(-4, -9)
Orientation?
10
You Do

• Describe the graph of f(x) x2 - 6x 7

Vertex?
Opens Up
(3, -2)
Orientation?
11
Example

• Describe the graph of f(x) 2x2 8x 7

Vertex?
Opens Up
(-2, -1)
Orientation?
12
You Do

• Describe the graph of f(x) 3x2 6x 7

Vertex?
Opens Up
(-1, 4)
Orientation?
13
Write the vertex form of the equation of the
parabola whose vertex is (1,2) and passes through
(3, - 6)

• (h,k) (1,2)
• Since the parabola passes through (3, -6), we
  know that f(3) - 6. So

14
Finding Minimums/Maximums

• If a gt 0, f has a minimum at
• If a lt 0, f has a maximum at
• Ex a baseball is hit and the path of the
  baseball is described by
• f(x) -0.0032x2 x 3. What is the maximum
  height reached by the baseball?

Remember the quadratic model is ax2bxc
F(x) - 0.0032(156.25)2156.253 81.125
feet
15
Maximizing Area