Transcript Chapter 5

Chapter 5
Quadratic Functions &
Inequalities
5.1 – 5.2 Graphing Quadratic
Functions

The graph of any Quadratic Function is a Parabola

To graph a quadratic Function always find the following:
•
y-intercept (c - write as an ordered pair)
•
equation of the axis of symmetry x =
•
b
2a
vertex- x and y values (use x value from AOS and plug in for y)
•
roots (factor)
•
minimum or maximum
•
domain and range
These are the solutions to the quadratic function
If a is positive = opens up (minimum)
If a is negative = opens down (maximum)
Ex: 1 Graph by using the vertex, AOS and a table

f(x) = x2 + 2x - 3
Graph Find the y-int, AOS, vertex, roots,
minimum/maximum, and domain and range

f(x) = -x2 + 7x – 14
Graph Find the y-int, AOS, vertex, roots,
minimum/maximum, and domain and range

f(x) = 4x2 + 2x - 3
Graph Find the y-int, AOS, vertex, roots,
minimum/maximum, and domain and range

x2 + 4x + 6 = 0
Graph Find the y-int, AOS, vertex, roots,
minimum/maximum, and domain and range

2x2 – 7x + 5 = 0
5.7 Analyzing graphs of
Quadratic Functions

Most basic quadratic function is

• Axis of Symmetry is x = 0
• Vertex is (0, 0)
A family of graphs is a group of graphs
that displays one or more similar
characteristics!
• y = x2
• y = x2 is called a parent graph

Vertex Form
y = a(x – h)2 + k
•
•
•
•
•
Vertex: (h, k)
Axis of symmetry: x = h
a is positive: opens up, a is negative: opens down
Narrower than y = x2 if |a| > 1, Wider than y = x2 if |a| < 1
h moves graph left and right
•
k moves graph up or down
• - h moves right
• + h moves left
• - k moves down
• + k moves up
Identify the vertex, AOS, and direction of
opening. State whether it will be narrower or
wider than the parent graph

y = -6(x + 2)2 – 1

y = (x - 3)2 + 5

y = 6(x - 1)2 – 4

1
y = - (x + 7)2
2
Graph after identifying the vertex,
AOS, and direction of opening. Make
a table to find additional points.
y = 4(x+3)2 + 1
y = -(x - 5)2 – 3
y = ¼ (x - 2)2 + 4
5.8 Graphing and Solving
Quadratic Inequalities



1. Graph the quadratic equation as
before (remember dotted or solid lines)
2. Test a point inside the parabola
3. If the point is a solution(true) then
shade the area inside the parabola if it is
not (false) then shade the outside of the
parabola
Example 1: Graph
y > x2 – 10x + 25
Example 2: Graph
y < x2 - 16
Example 3: Graph
y < -x2 + 5x + 6
Example 4: Graph
y > x2 – 3x + 2
5.4 Complex Numbers

Let’s see… Can you find the square root of a
number?
A.
25
D.
1
4
B.
E.
400
25
36
C.
F.
75
90
16
G.
200
So What’s new?

To find the square root of negative numbers you
need to use imaginary numbers.
•
•
•
i is the imaginary unit
i2 = -1
i = 1
Square Root Property
For any real number x, if x2 = n, then x = ± n
What about the square root of a
negative number?
A.
32

81
D.
B.
125x
C.
 18
5
E.
1

4
 45
Let’s Practice With i

Simplify
A.
-2i (7i)
B.
(2 – 2i) + (3 + 5i)
C.
 10   15
D. i45
E.
i31
Solve
A.
3x2 + 48 = 0
B.
4x2 + 100 = 0
C.
x2 + 4= 0
5.4 Day #2
More with Complex Numbers

Multiply
• (3 + 4i) (3 – 4i)
• (1 – 4i) (2 + i)
• (1 + 3i) (7 – 5i)
• (2 + 6i) (5 – 3i)
*Reminder: You can’t have i in
the denominator

Divide
3i
2 + 4i
D.
B.
-2i
3 + 5i
E.
C.
2+i
1-i
A.
5+i
2i
4-i
5i
5.6 The Quadratic Formula and
the Discriminant

2

b

b
 4ac
The Quadratic Formula:
2a
•
Use when you cannot factor to find the roots/solutions
The discriminant:
the expression under the radical
sign in the quadratic formula.
*Determines what type and number of roots

b2 – 4ac
Discriminant
Type and Number of Roots
b2 – 4ac > 0
2 rational roots
is a perfect square
b2 – 4ac > 0 is NOT a perfect square
2 irrational roots
b2 – 4ac = 0
1 rational root
b2 – 4ac < 0
2 complex roots
Example 1:

Use the discriminant to determine the
type and number of roots, then find
the exact solutions by using the
Quadratic Formula
x2 – 3x – 40 = 0
Example 2:

Use the discriminant to determine the
type and number of roots, then find
the exact solutions by using the
Quadratic Formula
2x2 – 9x + 15 = 0
Example 3:

x2 + 6x – 9 = 0
Use the discriminant to determine the
type and number of roots, then find
the exact solutions by using the
Quadratic Formula
TOD: Solve using the method of your
choice! (factor or Quadratic Formula)
A. 7x2 + 3 = 0
B. 2x2 – 5x + 7 = 3
C. 2x2 - 5x – 3 = 0
D. -x2 + 2x + 7 = 0