Post on 25-Sep-2020
Chapter 5
Quadratic Equations and Functions
Lesson 5-1
Modeling Data with Quadratic
Functions
Quadratic Function
Standard Form of a Quadratic Function
f x ax bx c2( ) Quadratic
Term
Linear
Term
Constant
Term
Example 1 – Page 237, #2
Determine whether each function is linear or quadratic.
Identify the quadratic, linear and constant terms.
y x x22 (3 5)
y x x22 3 5
quadratic
x x22 , 3 ,5
Example 1 – Page 237, #8
Determine whether each function is linear or quadratic.
Identify the quadratic, linear and constant terms.
y x x x2(1 ) (1 )
y x x x
x
2 21
1
Linear
x, 1none,
Parabola
Axis of symmetry
Vertex
Minimum Value
y x2
Maximum Value
y x2
Parabola
Vertex is the point at which the
parabola intersects the axis of
symmetry
Axis of symmetry is the line that
divides a parabola into two parts that
are mirror images.
Example – Page 237, #12
Identify the vertex and the axis of symmetry of each parabola.
Vertex: ( 1, 4)
Axis of Symmetry: x 1
Example 2 – Page 237, #14
For each parabola, identify points corresponding to P and Q
P '(1,5)
Q'( 2,8)
Lesson 5-2, Part 1
Properties of Parabola
Graphing Parabolas
cxax by 2
Step 1 – Determine the direction of the parabola
a is positive it opens up
a is negative it opens down
Step 2 – Find the y-intercept (0,?).
Substitute x = 0 into the quadratic and find the y-value.
Graphing Parabolas
cxax by 2
xb
a2
Step 3 – Find the axis of symmetry (x = ?)
Step 4 – Find the vertex (x, ?)
Substitute the x-value from step 3 into the quadratic and
find the y-value.
Graphing Parabolas
cxax by 2
Step 5 – Graph the quadratic
Get additional points if needed
Example 1 – Page 244, #6
Graph each function.
y x25 12
Step 1
y
y
25 12
12
(0,12)
0
Opens down
Step 2
Example 1 – Page 244, #6
cxax by 2 xb
a2
y x25 12
Step 3
x
x
0
2 5
0
Example 1 – Page 244, #6
y x25 12
Step 3
x 0
Step 4
y
y
25( ) 12
12
(0,
0
12)
Example 1 – Page 244, #6
y x25 12
Step 5
y
y
25(2) 12
8
(2, 8)
Example 1 – Page 244, #6
y x25 12
Example 2 – Page 244, #16
Graph each function. Label the vertex and axis of symmetry
y x x24 12 9
Step 1
y
y
20 04 12 9
9
(0,9)
Opens up
Step 2
Example 2 – Page 244, #16
cxax by 2 xb
a2
y x x24 12 9
Step 3
x
x
12
2 4
1.5
Example 2 – Page 244, #16
y x x24 12 9
Step 3
x 1.5
Step 4
y
y
21.5 1.4( ) 12 9
0
(1.5,0
5
)
Example 2 – Page 244, #16
y x x24 12 9 Step 5
Vertex (1.5, 0)
Axis of Symmetry
x = 1.5
Lesson 5-2, Part 2
Properties of Parabola
Example 2 – Page 244, #22
Graph each function. If a > 0 find the minimum value.
If a < 0 find the maximum value.
y x x2 2 5
Step 1
Opens down
Step 2
y
y
2( ) 2( ) 5
5
(0,5)
0 0
Example 2 – Page 244, #22
cxax by 2
y x x2 2 5
Step 3
xb
a2
x
x
21
2( 1)
1
Example 2 – Page 244, #22
y x x2 2 5
Step 3
x 1
Step 4
y
y
2( ) 2( ) 5
6
(1,6)
1 1
Example 2 – Page 244, #22
y x x2 2 5
Step 5
y
y
2( ) 2( ) 5
3
4
(4,
4
3)
Example 2 – Page 244, #22
y x x2 2 5
max, 6
Example 4 – Page 244, #28
A model for a company’s revenue is
where p is the price in dollars of the company’s product.
What price will maximize revenue? Find the maximum revenue.
R p p215 300 12,000,
Find the vertex
xb
a2
x
x
30010
2( 15)
10
R
R
215(10) 300(10) 12,000
13500
$10 will maximum the revenue
and the maximum revenue
is $13,500
Lesson 5-3, Part 1
Translating Parabolas
Vertex Form of Quadratic
Function
Parent Function
y ax2
y a x h k2( )
Translated Function
Vertex Form
y x2
Graph of a Quadratic
Function in Vertex Form
h units moves horizontally
h is positive the graph shifts right
h is negative the graph shifts left
k units moves vertically
k is positive the graph shifts up
k is negative the graph shifts down
Vertex is (h, k)
Axis of symmetry is x = h
y a x h k2( )
Example 1 – Page 251, #2
Graph each function
y x 2( 3) 4 y a x h k2( )
Step 1 – Find the vertex.
h 3 k 4
vertex: ( 3, 4)
Example 1 – Page 251, #2
y x 2( 3) 4
y x2
vertex: ( 3, 4)
Example 1 – Page 251, #2
y x 2( 3) 4
Step 2 – Find the y-intercept
y
y
2(0 3) 4
5
(0,5)
Example 1 – Page 251, #12
Graph each function
y x2
4 8 6
Step 1 – Find the vertex
( 8, 6)
Example 1 – Page 251, #12
y x2
4 8 6
Step 2 – Find the y-intercept
Too big
Step 3 – Get additional points
y
y
24( 8) 6
10
( 7,
7
10)
Example 2, Page 251, #16
Write the equation of each parabola in vertex form.
( 2,0)
( 3, 1)
y a x h k2( )
y xa 2( 2) 0
a 21 ( 3 2)
a
a
a
1 1
1
1
y x 2( 2)
Lesson 5-3, Part 2
Translating Parabolas
Example 3, Page 252, #26
Identify the vertex and the y-intercept of the graph of
each function.
y x 2( 125) 125
(125,125)
Vertex
y a x h k2( )
y-intercept
y
y
2(0 125) 125
15750
Example 4 – Page 252, #34
Write each function in vertex form.
y x x22 8 3
Find the vertex.
cxax by 2 xb
a2
x8
22( 2)
y
y
22( ) 8( ) 3
11
(
2 2
2,11)
y x 22( 2) 11
y a x h k2( )
Lesson 5-4, Part 1
Factoring Quadratic Expression
Example 1 – Page 259, #6
Find the GCF of each expression. Then factor the
expression
p p227 9
GCF: p9
p p9 (3 1)
Example 2 – Page 259, #8
Factor each expression.
x x2 65
x xx
x
x
x
x x2
2
2 6
( 3)(
( 3) ( 3)
3
2)
x x2 26 6
x x x
x x x
23 2 6
3 2 5
Example 3 – Page 259, #16
Factor each expression.
x x2 10 24 x x2 224 24
x x x
x x x
26 ( 4 ) 24
6 ( 4 ) 10
x
x
x x
x
x
x
x
2
( 6) 4
( 4)(
( 6
2
)
4 46
6)
Example 4 – Page 259, #22
Factor each expression.
c c2 62 3 C c2 2( 63) 63
c c c
c c c
29 ( 7 ) 63
9 ( 7 ) 2
c
c
c c
c
c
c
c
2
( 7) 9
( 9)(
( 7
6
)
9 37
7)
Example 4 – Page 259, #24
Factor each expression.
t t2 47 4 t t2 2( 44) 44
211 4 44
11 4 7
t t t
t t t
t
t
t t
t
t
t
t
2
( 11)
4
4
( 4)(
4411
( 11)
11)
Factoring
x x2 65 x x( 3)( 2)
x x2 10 24 x x( 4)( 6)
c c2 62 3 c c( 9)( 7)
t t2 47 4 t t( 4)( 11)
Last number is positive – both negative or positive
Last number is negative – a negative and a positive
Example 5 – Page 259, #26
Factor each expression.
xx2 192 24 2 224 42 8xx
216 ( 3 ) 48
16 ( 3 ) 19
x x x
x x x
x
x
x
x
x
x
x
x
2
( 82 3
(2
2 16
3)(
)
4
(
3 2
8)
8)
Lesson 5-4, Part 2
Factoring Quadratic Expression
Example 6 – Page 260, #32
Factor each expression.
yy 2 125 32 2 2( 32) 165 0y y
220 ( 8 ) 160
20 8 12
y y y
y y y
y
y
y
y
y
y
y
y
2
( 45 8
(5
5 20
8)(
)
2
(
8 3
4)
4)
Factoring
Perfect Square Trinomial
Difference of Two Squares
a ab b a b a b a b2 2 22 ( )( ) ( )
a ab b a b a b a b2 2 22 ( )( ) ( )
a b a b a b2 2 ( )( )
Example 7 – Page 260, #40
Factor each expression.
nn2 204 25 n n n2
(2 5)(2 5) 2 5
Example 7 – Page 260, #44
Factor each expression.
c2 64 c c( 8)( 8)
Example 8 – Page 260, #48
Find the area of rectangular cloth is cm².
The length is cm. Find the width.
x x2(6 19 85)
x(2 5)
x26
85
x2
5
x3 17
x cm2(3 17)x34
x15
Example – Page 260, #58
Factor each expression completely
x x212 36 27 x x23(4 12 9)
x x x 23(2 3)(2 3) 3(2 3)
Example – Page 260, #62
2 5 4x x
Factor each expression completely.
21( 5 4)x x 1( 1)( 4)x x
Example – Page 260, #64
Factor each expression completely
x21 1
2 2 x x
11 1
2 x21
12
Lesson 5-5, Part 1
Quadratic Equations
Standard Form
Standard form of a quadratic equation
ax bx c2 0
Example 1 – Page 266, #2
Solve each equation by factoring.
x x2 18 9
x x2 9 18 0
Step 1 – Write in standard form
Example 1 – Page 266, #2
x x2 9 18 0
Step 2 – Factor
x x( 3)( 6) 0
x 3 0 x 6 0
x 3 x 6
Example 2 – Page 266, #8
Solve each equation by finding square roots.
x2 4 0
x2 4
x2 4
x 2
x or x2 2
Example 2 – Page 266, #12
Solve each equation by finding square roots.
x25 40 0
x25 40
x2 8
x 2 2
x2 8
8 2 4 2 2
x or x2 2 2 2
Example 3 – Page 266, #14
Solve each equation by factoring or by taking square roots.
x x26 4 0
x x2 3 2 0
x2 0 x3 2 0
x 0 x
x
3 2
2
3
Example 3 – Page 266, #18
Solve each equation by factoring or by taking square roots.
x24 80 0
x24 80
x2 20
x2 20
20 4 5 2 5
x 2 5
x or x2 5 2 5
Lesson 5-5, Part 2
Quadratic Equations
Solving Quadratics
Not all quadratics can be factored.
Zero of a function is called the solution of the quadratic
It is where the quadratic crosses the x-axis.
Solve quadratics by
Factoring
Square Roots
Graphing (TI-Calculator)
Completing the Square (Lesson 5-7)
Quadratic Formula (Lesson 5-8)
Example – Page 267, #40
Solve each equation by factoring, by taking the square
roots, or by graphing (not factorable)
x x x2 2 6 6
x x x2 2 6 6 0
x x2 8 6 0
Not factorable
Example – Page 267, #52
Solve each equation by factoring, by taking the square
roots, or by graphing (not factorable)
x x22 6 8
x x22 6 8 0
x x2( 1)( 4) 0
x 1 0 x 4 0
x 1 x 4
22( 3 4) 0x x
Lesson 5-6, Part 1
Complex Numbers
Complex Number System
Real Numbers Imaginary
Numbers
Rational Numbers Irrational Numbers
Rational Numbers: 8
5, 0, , 93
Irrational Numbers: 3, 5
Imaginary Numbers: 4 , 3 2 , 2 2i i i
Imaginary Numbers
a i a i a2
Imaginary number is a number whose square is -1. So
that i² = -1. Imaginary number is any form a + bi
Example 1 – Page 274, #10
Simplify each number by using the imaginary number i.
72 i 272 i i9 8 3 8 i i3 4 2 3 2 2
i6 2
i 72
Complex Number
Complex numbers are imaginary numbers and real numbers.
a bi
Real Part Imaginary Part
Example 2 – Page 274, #12
Write each number in the form a + bi.
8 8 i8 4 2 i8 2 2 8 8
Absolute Value of a
Complex Number
a bi a b2 2
Example 3 – Page 274, #22
Find the absolute value of each complex number.
i1 4 2 21 ( 4) 1 16 17
Example 4 – Page 274, #26
Find the additive inverse of each number.
i9 i9
Example 5 – Page 274, #30
Simplify each expression
i i3 5 4 2
i i3 4 5 2
i1 7
Example 5 – Page 274, #32
Simplify each expression
i6 8 3
i6 8 3
i2 3
Lesson 5-6, Part 2
Complex Numbers
Example 5 – Page 274, #36
Simplify each expression
i i(4 3 )(5 2 )
i i 220 7 6
i20 7 6( 1)
i20 7 6
i26 7
i i i 220 8 15 6
Example 7 – Page 274, #46
Solve each equation.
x 25 3 0
x 25 3
x 23
5
x 23
5
x i3
5
3 3 15
55 5
5
5
x i15
5
Example – Page 275, #60
Simplify each expression.
10 9 2 25
i i10 3 2 5
i8 2
10 9 2 25
Example – Page 275, #66
Simplify each expression.
1 4 3 25
i i 23 10
i3 10 1
i13
i i
i i i 21 2 3 5
3 5 6 10
Lesson 5-7, Part 1
Completing the Square
Completing the Square
If an quadratic equations is not a perfect square
trinomial (a + b)², you can convert it into a perfect square
trinomial by rewriting the constant term.
Example 1 – Page 281, #4
Solve each equation.
x x2 168 16
9
x 2 16( 4)
9
x2 164
9
x4
43
x4
43
x
x
443
16
3
x
x
443
8
3
Example 2 – Page 281, #10
Complete the square.
x x2 20 ____
2
10
1
20
2
0 100
100
Example 3 – Page 281, #14
Solve each quadratic equation by completing the square.
x x2 3 4 0
Step 1 – Get all terms containing x to one side.
x x2 3 ____ 4 ____
Example 3 – Page 281, #14
x x2 3 ____ 4 ____
Step 2 – Complete the square
23 3 9
2 2 4
9
4
9
4
Example 3 – Page 281, #14
x x2 3 ____ 4 ____
Step 3 – Factor the perfect square.
9
4
9
4
x2
3 25
2 4
Example 3 – Page 281, #14
Step 4 – Solve for x.
x2
3 25
2 4
x2
3 25
2 4
x3 5
2 2
x3 5
2 2
x3 5
12 2
x3 5
42 2
Example 4 – Page 281, #18
Solve each quadratic equation by completing the square.
x x2 6 22
Step 1 – Get all terms containing x to one side.
x x2 6 ____ 22 ____
Example 4 – Page 281, #18
x x2 6 ____ 22 ____
Step 2 – Complete the square
263 9
2
9 9
Example 4 – Page 281, #18
x x2 6 9 22 9
Step 3 – Factor the perfect square.
x23 13
Example 4 – Page 281, #18
Step 4 – Solve for x.
x23 13
x23 13
x i3 13
x i3 13
Lesson 5-7, Part 2
Completing the Square
Example 5 – Page 281, #20
Solve each quadratic equation by completing the square
2 2 5x x
Step 1 – Get all terms containing x to one side.
Step 2 – Leading coefficient x² is 1.
2 2 5x x
2 51 2x x
2 2 5x x
Example 5 – Page 281, #20
Step 3 – Complete the square.
2 2 5x x
2 2 ____ 5 ____x x
221 1
2
1 1
2 2 1 4x x
Step 4 – Factor the perfect Square.
21 4x
Example 5 – Page 281, #20
Step 5 – Solve for x.
21 4x
21 4x
1 2x i
1 2x i
Example 5 – Page 281, #26
Solve each quadratic equation by completing the square
29 12 5 0x x
Step 1 – Get all terms containing x to one side.
Step 2 – Leading coefficient x² is 1.
29 12 5x x
219 12 5
9x x
2 12 5
9 9x x
Example 5 – Page 281, #26
Step 3 – Complete the square.
2 4 5
3 9x x
2 4 5____ ____
3 9x x
24
2 432 3 9
4
9
2 4 4 1
3 9 9x x
4
9
Example 5 – Page 281, #26
Step 4 – Factor the perfect Square.
2 4 4 1
3 9 9x x
22 1
3 9x
Example 5 – Page 281, #26
22 1
3 9x
Step 5 – Solve for x.
22 1
3 9x
2 1
3 3x i
2 1
3 3x i
Example 6 – Page 281, #28
Rewrite each equation in vertex form.
2 4 7y x x
2 ____( 4 ) ___7 _y x x
242 4
2
4 4
2 4 4 11y x x
22 11y x
Lesson 5-8, Part 1
Quadratic Formula
Quadratic Formula
x xa cb2 0
x
a
cb ab 2 4
2
Example 1 – Page 289, #10
Solve each equation using the Quadratic Formula.
x x28 2 3 0
a
b
c
8
2
3
b b acx
a
2 4
2
x
22 2 4 8 3
2 8
x2 4 96
16
Example 1 – Page 289, #10
x2 4 96
16
x2 100 2 10
16 16
x2 10 12 3
16 16 4
x2 10 8 1
16 16 2
Example 2 – Page 289, #20
Solve each equation using the Quadratic Formula.
x x22 7 8
a
b
c
2
7
8
b b acx
a
2 4
2
x
27 7 4 2 8
2 2
x7 49 64
4
x x22 7 8 0
Example 2 – Page 289, #20
x7 49 64
4
ix7 15 7 15
4 4
ix7 15
4 4
Lesson 5-8, Part 2
Quadratic Formula
Example 3 – Page 289, #24
Solve each quadratic using the Quadratic Formula. Find the
exact solutions. Then approximate any radical solutions.
Round to the nearest hundredth
x x23 4 3 0
a
b
c
3
4
3
b b acx
a
2 4
2
x
24 4 4 3 3
2 3
x
4 16 36
6
Example 3 – Page 289, #24
x
4 16 36
6
x4 52
6
x4 4 13 4 2 13
6 6
x4 2 13 2 13 2 13
6 6 3 3 3
x2 13
0.543
x2 13
1.873
Discriminant
x xa cb2 0
x
a
cb ab 2 4
2
Discriminant
2 Real Solutions
2 x-intercepts
b ac2 4 0
2 Imaginary Solutions
no x-intercepts
1 Real Solutions
1 x-intercepts
b ac2 4 0 b ac2 4 0
Example 4 – Page 289, #34
Evaluate the discriminant of each equation. Tell how many
solutions each equation has and whether the solutions are
imaginary or real.
x x22 28 0
a
b
c
2
1
28
b ac2 4
21 4 2 28
1 224
223 0
2 imaginary solutions
Example 4 – Page 289, #38
Evaluate the discriminant of each equation. Tell how many
solutions each equation has and whether the solutions are
imaginary or real.
x x2 12 36 0
a
b
c
1
12
36
b ac2 4
2
12 4 1 36
144 (144)
0 0
1 real solution