JMerrill, 05
Revised 08
Section 31Quadratic Functions
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
• 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
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
max
min
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. If a < 0, the parabola opens downward.
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
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
2
2
2
2
f (x) x 8x 7
(x 8x ) 7
(x 8x 16) 7 16
(x 4) 9
Vertex? (-4, -9) Orientation? Opens Up
You Do• Describe the graph of f(x) = x2 - 6x + 7
2
2
2
2
f (x) x 6x 7
(x 6x ) 7
(x 6x 9) 7 9
(x 3) 2
Vertex? (3, -2) Orientation? Opens Up
Example• Describe the graph of f(x) =2x2 + 8x + 7
2
2
2
2
f (x) 2x 8x 7
2(x 4x ) 7
2(x 4x 4) 7 8
2(x 2) 1
Vertex? (-2, -1) Orientation? Opens Up
You Do• Describe the graph of f(x) =3x2 + 6x + 7
2
2
2
2
f (x) 3x 6x 7
3(x 2x ) 7
3(x 2x 1) 7 3
3(x 1) 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)
Since the parabola passes through (3, -6), we know that f(3) = - 6. So:
2f (x) a(x 1) 2
2
2
f (x) a(x 1) 2
6 a(3 1) 2
6 4a 2
2 a
2f (x) 2(x 1) 2
Finding Minimums/Maximums• If a > 0, f has a minimum at
• If a < 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?
bx
2a
b
x2a
bx
2a1
x 156.25f t2( .0032)
which is the x coordinate of the vertex
Remember the quadratic model is: ax2+bx+c
F(x)= - 0.0032(156.25)2+156.25+3
= 81.125 feet
Maximizing Area
