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)
b
2a
– vertex- x and y values (use x value from AOS and solve for y)
– equation of the axis of symmetry x =
– roots (factor)
These are the solutions to the quadratic function
– minimum or maximum
– domain and range
If a is positive = opens up (minimum) – y coordinate of the vertex
If a is negative = opens down (maximum) – y coordinate of the
vertex
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 = f(x)
Graph Find the y-int, AOS, vertex, roots, minimum/maximum,
and domain and range
• 2x2 – 7x + 5 = f(x)
5.7 Analyzing graphs of Quadratic
Functions
• Most basic quadratic function is
• y = x2
– 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 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
• - h moves right
• + h moves left
– k moves graph up or down
• - 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
Graph after identifying the vertex, AOS, and direction
of opening. Make a table to find additional points.
y = -(x - 5)2 – 3
Graph after identifying the vertex, AOS, and direction
of opening. Make a table to find additional points.
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.
D.
25
25
36
B.
E.
400
90
16
C.
75
F.
G.
200
 18
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
n
For any real number x, if x2 = n, then x = ±
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.
D. i45
E. i31
 10   15
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
A.
3i
2 + 4i
B.
-2i
3 + 5i
C.
2+i
1-i
D.
5+i
2i
E.
4-i
5i
5.5 Completing the Square
5.5 Completing the Square
Let’s try some:
Solve:
5.6 The Quadratic Formula and the
Discriminant
The discriminant:
the expression under the radical sign in
the quadratic formula.
*Determines what type and number of roots
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
5.6 The Quadratic Formula and the
Discriminant
• The Quadratic Formula:
 b  b 2  4ac
2a
» Use when you cannot factor to find the roots/solutions
Example 1:
• x2 – 3x – 40 = 0
Use the discriminant to determine the
type and number of roots, then find
the exact solutions by using the
Quadratic Formula
Example 2:
• 2x2 – 8x + 11 = 0
Use the discriminant to determine the
type and number of roots, then find
the exact solutions by using the
Quadratic Formula
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
C. 2x2 - 5x – 3 = 0
B. 2x2 – 5x + 7 = 3
D. -x2 + 2x + 7 = 0