Section 31
Quadratic Functions
JMerrill, 05
Revised 08
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.
Parabolas
• The graph of a quadratic
equation is a Parabola.
• Parabolas occur in many
real-life situations
vertex
• 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.
Characteristics
•
Graph of f(x)=ax2, a > 0
•
Domain
− (- ∞, ∞)
•
Range
− [0, ∞)
•
Decreasing
− (- ∞, 0)
•
Increasing
− (0, ∞)
•
Zero/Root/solution
− (0,0)
•
Orientation
− Opens up
Characteristics
• Graph of f(x)=ax2, a > 0
• Domain
− (- ∞, ∞)
• Range
− (-∞, 0]
• Decreasing
− (0, ∞)
• Increasing
− (-∞, 0)
• Zero/Root/solution
− (0,0)
• Orientation
− Opens down
Max/Min
• A parabola has a maximum or a minimum
min
max
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 > o, the parabola opens upward.
the parabola opens downward.
If a < 0,
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
X=5
or
-2 = x – 3
x=1
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
f(x)  x  8x  7
2
 (x  8x
2
)7
 (x  8x  16)  7  16
2
 (x  4)  9
2
Vertex?
(-4, -9) Orientation?
Opens Up
You Do
• Describe the graph of f(x) = x2 - 6x + 7
f(x)  x  6x  7
2
 (x  6x
)7
2
 (x  6x  9)  7  9
2
 (x  3)2  2
Vertex?
(3, -2)
Orientation? Opens Up
Example
• Describe the graph of f(x) =2x2 + 8x + 7
f(x)  2x2  8x  7
 2(x2  4x
)7
 2(x2  4x  4)  7  8
 2(x  2)2  1
Vertex?
(-2, -1)
Orientation? Opens Up
You Do
• Describe the graph of f(x) =3x2 + 6x + 7
f(x)  3x2  6x  7
 3(x2  2x
)7
 3(x2  2x  1)  7  3
 3(x  1)2  4
Vertex?
(-1, 4)
Orientation? Opens Up
Write the vertex form of the equation of
the parabola whose vertex is (1,2) and
passes through (3, - 6)
(h,k) = (1,2)
f(x)  a(x  1)2  2
Since the parabola passes through (3, -6), we know
that f(3) = - 6. So:
f(x)  a(x  1)2  2
6  a(3  1)2  2
6  4a  2
2  a
f(x)  2(x  1)  2
2
Finding Minimums/Maximums
• If a > 0, f has a minimum atx 
b
2a
b
x
• If a < 0, f has a maximum at
2a
• 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?
b
x
2a
1
x
 156.25ft
2( .0032)
Remember the
quadratic model is:
ax2+bx+c
F(x)= - 0.0032(156.25)2+156.25+3
= 81.125 feet
which is the x  coordinate of the vertex
Maximizing Area