View
240
Download
15
Category
Preview:
Citation preview
Chp 4.notebook
1
October 23, 2012
4.1 Graphing Quadratic Functions in Standard forma quadratic function in standard form is written y = ax2 + bx + c, where a ≠ 0A quadratic Function creates a U-shaped graph called a parabola The vertex is the lowest/highest point of the parabola
Axis of symmetry is the line that divides the graph into two symmetric parts
The roots of the function (answers) are the two points that cross the x-axis (x-intercepts), when y = 0
VOCAB:
Chapter 4: Quadratic Functions and Factoring
Chp 4.notebook
2
October 23, 2012
Practice Graphing Graph of a quadratic functionif a is positive, the parabola opens up
if a is negative, the parabola opens down
The vertex has an xcoordinate of
The axis of symmetry is written x =
b2a
b2a
Chp 4.notebook
3
October 23, 2012
Practice Graphing Graph of a quadratic function
STEP 1: find the xcoordinate of the vertex by using
STEP 2: Make a table of values
STEP 3: Plot the points and create the parabola
b2avertex =
Chp 4.notebook
4
October 23, 2012
Graph: y = x2 2x 3
Chp 4.notebook
5
October 23, 2012
Graph: y = 2x2 x +2
Chp 4.notebook
6
October 23, 2012
Graph: y = 2x2 7x8
Chp 4.notebook
7
October 23, 2012
4.2 Graphing Quadratic Functions in Vertex or Intercept form
vertex form: standard form:
The graph of y = a(xh)2 + k is the parabola y = ax2 translated horizontally h units and vertically k units
Characteristics of the graph:• the vertex is (h,k)• The axis of symmetry is x = h• The graph opens up if a>0 and down if a<0
y
x
y = ax2
(0,0)
y = a(xh)2+k
(h,k)
hk
Chp 4.notebook
8
October 23, 2012
Step 1: Identify h, a and k. Use a to determine if the parabola opens up or down
Graph a quadratic function in vertex form: y = a(x-h) 2+k
Step 2: Plot the vertex (h,k) and draw the axis of symmetry
Step 3: Evaluate the function for two values of x.
Step 4: Draw parabola through plotted points
STEP 1
STEP 2
STEP 3
STEP 4
Chp 4.notebook
9
October 23, 2012
Example 2
Step 1: Identify h, a and k. Use a to determine if the parabola opens up or down
Graph a quadratic function in vertex form: y = a(x-h) 2+k
Step 2: Plot the vertex (h,k) and draw the axis of symmetry
Step 3: Evaluate the function for two values of x.
Step 4: Draw parabola through plotted points
STEP 1
STEP 2
STEP 3
STEP 4
Chp 4.notebook
10
October 23, 2012
Characteristics of the graph y=a(xp)(xq):• The xintercepts are p and q• The axis of symmetry is halfway between (p,0) and (q,0) it has equation• The graph opens up if a>0 and down if a<0
y
x
y = ax2
(p,0) (q,0)
Intercept form: y=a(xp)(xq)
Chp 4.notebook
11
October 23, 2012
Example 1
Step 1: Identify the x-intercepts
Graph a quadratic function in intercept form: y = a(x-p)(x-q)
Step 2: Find the coordinates of the vertex
Step 3: Draw parabolua through the vertex and x-intercepts
STEP 1
STEP 2
STEP 3
STEP 4
Chp 4.notebook
12
October 23, 2012
Example 1
Step 1: Identify the x-intercepts
Graph a quadratic function in intercept form: y = a(x-p)(x-q)
Step 2: Find the coordinates of the vertex
Step 3: Draw parabolua through the vertex and x-intercepts
STEP 1
STEP 2
STEP 3
STEP 4
Chp 4.notebook
13
October 23, 2012
Example: Change intercept form to standard form
Chp 4.notebook
14
October 23, 2012
Example: Change intercept form to standard form
Chp 4.notebook
15
October 23, 2012
Example: Change vertex form to standard form
Chp 4.notebook
16
October 23, 2012
Example: Change vertex form to standard form
Chp 4.notebook
17
October 23, 2012
4.3 Solve x2 + bx + c = 0 by factoring
ac
b
Chp 4.notebook
18
October 23, 2012
8
6
Chp 4.notebook
19
October 23, 2012
x2 5x + 6Factor
Chp 4.notebook
20
October 23, 2012
x2 + 7x + 12
Factor
Chp 4.notebook
21
October 23, 2012
x2 1x 6
Factor
Chp 4.notebook
22
October 23, 2012
Solve
x2 + 3x 4 = 0
Chp 4.notebook
23
October 23, 2012
Solvex2 + 5x 14 = 0
Chp 4.notebook
24
October 23, 2012
Solvex2 + 12x = 27
Chp 4.notebook
25
October 23, 2012
Example 1:
Special Factoring Patterns
Difference of Squares (DOS):
Perfect Square Trinomial:
a2 b2 = (ab)(a+b)
a2 + 2ab + b2 = (a+b)2
a2 2ab + b2 = (ab)2
4x2 9
Example 2:
Example 3:
x2 + 6x + 9
x2 8x + 16
Chp 4.notebook
26
October 23, 2012
You try:
1. x2 + 16x + 64
2. 9x2 49
3. x2 18x + 81
Chp 4.notebook
27
October 23, 2012
4.4 Solve ax2 + bx + c = 0 by factoring
2x2 3x 20Factor
Chp 4.notebook
28
October 23, 2012
3x2 + 11x 4Factor
Chp 4.notebook
29
October 23, 2012
7x2 31x + 12Factor
Chp 4.notebook
30
October 23, 2012
6x2 11x + 3Factor
Chp 4.notebook
31
October 23, 2012
5x2 8x + 3 = 0Solve
Chp 4.notebook
32
October 23, 2012
3x2 + 14x + 15 = 0Solve
Chp 4.notebook
33
October 23, 2012
7x2 + 11x 30 = 0Solve
Chp 4.notebook
34
October 23, 2012
2x2 + 3x 5 = 0Solve
Chp 4.notebook
35
October 23, 2012
Factor out monomials first
Chp 4.notebook
36
October 23, 2012
4.5 Solve Quadratic Equations by Finding Square RootsSQUARE ROOT OF A NUMBERif b2 = a, then b is a square root of a
example: if 32 = 9, then 3 is a square root of 9.
VOCAB: positive square root: √ negative square root: -√ positive/negative square root: ±√ radicand: the number or expression inside a radical symbol.
MATH IS √ DUDE
Chp 4.notebook
37
October 23, 2012
Simplifying radicalsProperties of Radicals
Product property √ab = √a√b
Quotient property √ = a
b√a
√b
Chp 4.notebook
38
October 23, 2012
An expression with radicals is insimplest form if the following are true 1.) No perfect square factors other
than 1 are in the radicand
2.) No fractions are in the radicand
3.) No radicals appear in the denominator of a fraction
Chp 4.notebook
39
October 23, 2012
Examples
Chp 4.notebook
40
October 23, 2012
Examples
Chp 4.notebook
41
October 23, 2012
Examples:rationalize denominators of fractions
Chp 4.notebook
42
October 23, 2012
Examples: solve a quadratic
Chp 4.notebook
43
October 23, 2012
Examples: solve a quadratic
Chp 4.notebook
44
October 23, 2012
4.6 Perform Operations with Complex Numbers
Not all quadratic equations have real-number solutions. For example: x2 = -1 has no real-number solutions because the square of any real number x is never a negative number
To overcome this problem mathematicians created an expanded system of numbers using the imaginary unit i. Defined as not that i2 = -1. The imaginary unit can be used to write the square root of any negative number.
The square root of a negative numberProperty Example
1. If r is a positive real number, then
2. By Property (1), it follows that
Chp 4.notebook
45
October 23, 2012
example:
Chp 4.notebook
46
October 23, 2012
example:
Chp 4.notebook
47
October 23, 2012
Complex Numbers:A complex number written in standard form is a number a+bi where a and b are real numbers. The number a is the real part of the complex number and the bi is imaginary part.
If b≠0 then a + bi is an imaginary number. If a = 0 and b ≠ 0, then a+bi is a pure imaginary number.
RealNumbers
ImaginaryNumbers
Pure ImaginaryNumbers
Complex Numbers: a + bi
Chp 4.notebook
48
October 23, 2012
The sum and difference of complex numbersTo add (subtract) two complex numbers, add (subtract) their real parts and their imaginary parts separately.
1. Sum of complex numbers:
2. Difference of complex numbers:
examples:a:
b:c:
Chp 4.notebook
49
October 23, 2012
examples:1:
2:
3:
Chp 4.notebook
50
October 23, 2012
Multiply Complex Numbers:
a:
b:
Chp 4.notebook
51
October 23, 2012
Multiply Complex Numbers:
1:
2:
Chp 4.notebook
52
October 23, 2012
Complex Conjugates:Two complex numbers of the form a+bi and abi are called complex conjugates. The product of conjugates is always a real number.
Divide Complex Numbers: use complex conjugates
Chp 4.notebook
53
October 23, 2012
Divide Complex Numbers: use complex conjugates
1:
2:
Chp 4.notebook
54
October 23, 2012
Plot Complex NumbersPlot the complex numbers in the same complex plane.
a.
b.
c.
d.
Chp 4.notebook
55
October 23, 2012
Find absolute values of complex numbersAbsolute Vale of a Complex Number:
a.
b.
The absolute value of a complex number z = a + bi, denoted
is a nonnegative real number defined as
This is the distance between z and the origin in the complex plane
examples:
Chp 4.notebook
56
October 23, 2012
4.7 Complete the Squarex2 + bx +( )2 = (x + )2
b2
b2
To complete the square for the expression x2 + bx, add
Diagram:
x
x
b
xx2 bxx
x
Chp 4.notebook
57
October 23, 2012
example 1: Solve a quadratic by finding square roots
step 1: Write left side as binomial squared
step 2: Take square root of each side.
step 3: Solve for x
Chp 4.notebook
58
October 23, 2012
example 2: Make a perfect square trinomial
Step 1: Find half of the coefficient of x
Step 2: square the result of step 1
Step 3: Replace c with the result from step 2
Chp 4.notebook
59
October 23, 2012
example 3: Solve x 2 + bx + c (a = 1)
step 1: write left side in the form x 2+bx
step 2: Complete the square and add it to both sides
step 3: Write left side as a binomial squared
step 4: Take square root of each side
step 5: Solve for x (simplify if necessary)
Chp 4.notebook
60
October 23, 2012
example 4: Solve ax 2 + bx + c (a ≠ 1)
step 1: divide everything by a
step 2: write left side in the form x2 + bx
step 3: complete the square and add it to both sides
step 4: write the left side as a binomial squared
step 5: take square root of both sides
step 6: solve for x, simplify if necessary
Chp 4.notebook
61
October 23, 2012
Extra ExamplesSolve the equation by completing the square
Chp 4.notebook
62
October 23, 2012
Write a quadratic function in vertex form
step 1: complete the square and add it to both sides
step 2: write beginning part as binomial squared
step 3: solve for y
Chp 4.notebook
63
October 23, 2012
Write a quadratic function in vertex form
Chp 4.notebook
64
October 23, 2012
4.8 Use the quadratic formula and the discriminant
Quadratic Formula: The solutions of thequadratic equation ax2+bx+c=0 are
Chp 4.notebook
65
October 23, 2012
Proving the Quadratic Formula!
ax2+bx+c = 0 where a≠0
Chp 4.notebook
66
October 23, 2012
Example 1:x2 8x + 15 = 0
Chp 4.notebook
67
October 23, 2012
Example 2:
2x2 + 6x + 2 = 1
Chp 4.notebook
68
October 23, 2012
Example 3: One solution
Chp 4.notebook
69
October 23, 2012
Example 4:One solution
Chp 4.notebook
70
October 23, 2012
Example 5: Imaginary Solution
Chp 4.notebook
71
October 23, 2012
Example 6: Imaginary Solution
Chp 4.notebook
72
October 23, 2012
Word ProblemsObjects Dropped:
Objects Launched/Thrown:initial velocity
initial height
A juggler tosses a ball into the air. The ball leaves the juggler's hand 4 feet above the ground and has an initial vertical velocity of 40 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air?
Chp 4.notebook
73
October 23, 2012
Word ProblemsObjects Dropped:
Objects Launched/Thrown:initial velocity
initial height
A basketball player passes the ball to a teammate. The ball leaves the player's hand 5 feet above the ground and has an initial vertical velocity of 55 feet per second. The teammate catches the ball when it returns to a height of 5 feet. How long is the ball in the air?
Chp 4.notebook
74
October 23, 2012
Word ProblemsThe equation h = 16t2 + 20t + 6 gives the height, h, in feet of a basketball as a function of t, in seconds
a) What is the maximum height the ball reaches?
b) At what time does the ball hit the ground?
Chp 4.notebook
75
October 23, 2012
Applications of the discriminantIn the quadratic formula the expression inside the radical is the discriminant
Discriminantb2-4ac
Chp 4.notebook
76
October 23, 2012
Number of solutions of a quadraticConsider the quadratic equation ax2 + bx + c = 0:
• If b24ac is positive, then the equation has two solutions• if b24ac is zero, then the equation has one solution• if b24ac is negative, then the equation has no real solutions
Chp 4.notebook
77
October 23, 2012
Example:a. x2 3x 4
b. x2 + 2x 1
c. 2x22x + 3
Chp 4.notebook
78
October 23, 2012
Example:a. 3x2 + 5x 1
b. x2 + 10x 25
c. x2 2x + 4
Chp 4.notebook
79
October 23, 2012
Match the graph with the discriminant:
a. b24ac = 2
b. b24ac = 0
c. b24ac = 3
c
A
b
Chp 4.notebook
80
October 23, 2012
4.10 Write Quadratic Functions and Models
Example 1: Write a quadratic function in vertex form y = a(x-h)2+K
Vertex: (1, 2) Point: (3, 2)
Chp 4.notebook
81
October 23, 2012
Example 2: Write a quadratic function in vertex form y = a(x-h)2+K
Vertex: (4, 5) Point: (2, 1)
Chp 4.notebook
82
October 23, 2012
Example 3: Write a quadratic function in intercept form y = a(x-p)(x-q)
xintercepts: 2 and 5 Point: (6, 2)
Chp 4.notebook
83
October 23, 2012
Example 4: Write a quadratic function in intercept form y = a(x-p)(x-q)
xintercepts: 1 and 4 Point: (3, 2)
Chp 4.notebook
84
October 23, 2012
Example 5: Write a quadratic function in Standard form y = ax 2 + bx + c
points: (1, 3), (0, 4), (2, 6)
Chp 4.notebook
85
October 23, 2012
Example 6: Write a quadratic function in Standard form y = ax 2 + bx + c
points: (1, 5), (0, 1), (2, 11)
Recommended