Ch11.2&3(1)

The Characteristics of Parabolas – Points of Interest
Opens Up
x-int
Axis of Sym
x=1

Min (-1,-3)

Axis of Sym
x=1

a) Up if a>o
b) Down if a<o

Quadratic equations in
in this form:
f ( x ) ax 2 bx c

y-int 2) Vertex

Vertex
Max (-1,5)

a) Minimum if a>o

b) Maximum if a<o
3) x-intercepts

a) Where f(x) =0
4) y-intercepts

y-int

Opens
Down

1) Open Up or Down?

a) Where x=0
5) Axis of Symmetry

x-int

a) The x-value at the
vertex

f (x)

a( x

h )2

k

Solving Quadratic
equations in this form:

ax

2

bx

c

0, a

1) Factoring
2) Square Root Property
2
If u
d,
then u

d

3) The Quadratic Formula
x

b

b

2

2a

4ac

0
Which
one to
use?

Ch 11.2 #2,6,9,18

x2

#2

8 x 15

0

a 1b
,
x
x
x

this form:
ax 2 bx c 0, a 0

8, c 15

( 8)

2

8

1) Factoring

4 • 1 • 15


If u 2

2 1
8
-8

then u

64 60
2
4

x

4 1
3 or

d

3) The Quadratic
Formula

2

8 2
x
2
x
4 1
x -5,-3

d,

x

4 1

5

b

b

2

2a

4ac

Ch 11.2 #2,6,9,18

#6

x2

5 x 10

a 1b
,
( 5)

x

5

10
2

4 • 1 • -10

1) Factoring

If u 2

2 1
5

x
x

5,c

0

this form:
ax 2 bx c 0, a 0

-5

25 40
2

d,

then u
3) The Quadratic
Formula

65
2

d

x

b

b

2

2a

4ac

Ch 11.2 #2,6,9,18

# 9 6x

6x
a

2

2x 1

2x 1 0

6, b
( -2)

x

x

2

2, c
-2

2

4
12

24

1) Factoring

If u 2

1
4 • 6 • -1

2 6

2

this form:
ax 2 bx c 0, a 0

2

28
12
2 2 7
12

d,

then u

d

3) The Quadratic
Formula

x

2(1
7)
2 6

b

b

2

2a

1

7
6

4ac

Ch 11.2 #2,6,9,18

this form:
ax 2 bx c 0, a 0

#18 2x( x 4) 3x 3
2x 2 8 x 3 x 3
2x
a
x

x

x

2

5x 3

2, b
(5 )

0

5, c
5

2

1) Factoring

If u 2

3

then u

4• 2 • 3

6
4

25 24
4

3
or x
2

5

5 1
4

1
4
4
4

1

d

3) The Quadratic
Formula

2 2
5

d,

x

x

3
,
2

b

b

2

2a

1

4ac

Ch 11.2 #42,43,46
# 42

4x-1

2

15

4 x -1
4x
x

15

1

Which one
to use?

this form:
ax 2 bx c 0, a 0
1) Factoring

If u 2

15

1

15
4

d,

then u

d

3) The Quadratic
Formula

x

b

b

2

2a

4ac

Ch 11.2 #42,43,46

1
x

# 43

3x x+2

3 x

2

1

1
3 LCD 3 x x

x 2
1
x

1
x 2

3x

3x 6 3x

x

2

1
3

4x 6

x x

x

2

0

this form:
ax 2 bx c 0, a 0
2
1) Factoring

If u 2

d,

then u

2

2x

d

3) The Quadratic
Formula

x

b

b

2

2a

4ac

Ch 11.2 #42,43,46

x

2

4x 6

a 1b
,

Which one
to use?

0

4, c

this form:
ax 2 bx c 0, a 0
1) Factoring

6

If u 2

# 46

7x x-2

7x2 14x

3 2 x

then u

4

3 2x 8

7x

12x 5

0 Which one
to use?

d

3) The Quadratic
Formula

x
2

d,

b

b

2

2a

4ac

Writing an equation when
solutions are given
Ch 11.2 #48,54

# 48
x

2,6
2 or x

this form:
ax 2 bx c 0, a 0
1) Factoring

6

If u 2

Work backwards
x

2
x

x

2

0 or x - 6
2 x 6

4x 12

0

0

0

d,

then u

d

3) The Quadratic
Formula

x

b

b

2

2a

4ac

Writing an equation when
solutions are given
Ch 11.2 #48,54
# 54
x

3, 3

3

0 or x

3

0

this form:
ax 2 bx c 0, a 0
1) Factoring

If u 2

Work backwards
x

3

x

3 =0

then u

x

3

0

d

3) The Quadratic
Formula

x

2

d,

b

b

2

2a

4ac

The Shape of the Graph of a
Quadratic Equation:
a Parabola

in this form:
f ( x ) ax 2 bx c
f (x)

a( x

h )2

k

The Characteristics of Parabolas – Points of Interest
Opens Up
x-int
Axis of Sym
x= -1

Min (-1,-3)

Axis of Sym
x= -1

a) Up if a>o
b) Down if a<o

in this form:
f ( x ) ax 2 bx c

y-int 2) Vertex

Vertex
Max (-1,5)

a) Minimum if a>o

b) Maximum if a<o
3) x-intercepts

a) Where f(x) =0
4) y-intercepts

y-int

Opens
Down

1) Open Up or Down?

a) Where x=0
5) Axis of Symmetry

x-int

a) The x-value at the
vertex

f (x)

a( x

h )2

k

Graphing Parabolas in form
Ch 11.3 #10,11,14,16 Find the Vertex
#10

f ( x ) -3 x - 2
h 2
vertex

k

2

12

(2,12)

12

ax 2

bx

c

a( x
Rules

h )2

k

f (x)
f (x)

1) Open Up or Down?

a) Up if a>o
b) Down if a<o
2) Find the Vertex

a( x h )2 k
Vertex= ( h , k )
for f ( x ) a 2 b
ax bx c
b
b ,f
Vertex
2a
2a

for f ( x )

3) Find x-intercepts

Set f(x) =0, solve for x
4) Find y-intercepts

Set x=0, find f(0)
5) Axis of Symmetry

x

h or x

b
2a

Graphing Parabolas in form
#11 f ( x ) -2 x 1

2

1
f ( x ) -2 x ( 1)

ax 2

bx

c

a( x
Rules

h )2

k

f (x)
f (x)

1) Open Up or Down?

a) Up if a>o
b) Down if a<o

5
2

h
1 k 5
vertex
(-1,5)

5

2) Find the Vertex

for f ( x )

a( x h )2 k
Vertex= ( h , k )
for f ( x ) a 2 b
ax bx c
b
b ,f
Vertex
2a
2a


Set x=0, find f(0)
5) Axis of Symmetry

x

h or x

b
2a

#14

f ( x ) 3 x 2 12 x 1
12
a 3 b

( 12)
2(3)
f ( 2)
Vertex

3 2

12
6
2

1) Open Up or Down?

a) Up if a>o
b) Down if a<o

2

12 2

b
2 ,
2a

Rules

11
f ( 2)

1 12 24 1

11

2) Find the Vertex

for f ( x )

k

for f ( x )

a 2 b
ax bx c
b
b ,f
2a
2a

a( x h )2
Vertex= ( h , k )

Vertex



Set x=0, find f(0)
5) Axis of Symmetry

x

h or x

b
2a

#16

2x 2 8x 1
2 b 8

f (x)
a

(8 )
2( 2)
f ( 2)
Vertex

8
4

2 2

2

b
2 ,
2a

Rules
1) Open Up or Down?

a) Up if a>o
b) Down if a<o

2

8 2
f (7 )
2

1

8 16 1 7

2) Find the Vertex

for f ( x )

k

for f ( x )

a 2 b
ax bx c
b
b ,f
2a
2a

a( x h )2
Vertex= ( h , k )

Vertex



Set x=0, find f(0)
5) Axis of Symmetry

x

h or x

b
2a

Ch 11.3 #18,34 Graph Opens
2
#18
f x
x 1
2 up
a=1 >0
h 1 k
2
vertex

x-intercepts

y-intercepts

(1

2,0) (1

2,0)

f (0 )

1) Open Up or Down?

a) Up if a>o
b) Down if a<o

(1, 2)

2

x 1
2 0
2
x 1
2
x 1 ± 2
x 1± 2

Rules

2

0 1
2
2
1
2
1 2
1

(0, 1)

2) Find the Vertex

for f ( x )

a( x h )2 k
Vertex= ( h , k )
for f ( x ) a 2 b
ax bx c
b
b ,f
Vertex
2a
2a


Set x=0, find f(0)
5) Axis of Symmetry

x

x

h or x

1

b
2a

# 34

6 4 x x 2 Opens
x 2 4 x 6 up >0
a=1
4
a 1 b

f x
f x

Vertex

4

b

2(1)

2(a )
f ( 2)
Vertex

4
2

2

2

4 2

b
2a

2 ,

Rules
1) Open Up or Down?

a) Up if a>o
b) Down if a<o
2) Find the Vertex

2

for f ( x )

6
f (2 )

4 8 6 2

a( x h )2 k
Vertex= ( h , k )
for f ( x ) a 2 b
ax bx c
b
b ,f
Vertex
2a
2a


Set x=0, find f(0)
5) Axis of Symmetry

x

h or x

b
2a

6 4 x x 2 Opens
x 2 4 x 6 up
a=1 >0
Vertex = (2,2)

# 34

f x
f x

x-intercepts
x

2

4x

6

y-intercepts

0

Use Quad. Form.
OR
Look at the graph
so far....

No x intercepts

f (0)

0

2

6

(0,6)

4(0) 6

Rules
1) Open Up or Down?

a) Up if a>o
b) Down if a<o
2) Find the Vertex

for f ( x )

a( x h )2 k
Vertex= ( h , k )
for f ( x ) a 2 b
ax bx c
b
b ,f
Vertex
2a
2a


Set x=0, find f(0)
5) Axis of Symmetry

x

x

h or x

2

b
2a

Finding Minimum and Maximum
Ch 11.3 #60

Definition

A person standing close to the edge
on the top of a 200-foot building
throws a baseball vertically upward.
The quadratic function
s (t )

16t 2

64t

200

If parabola opens………

a) Up (a>o)
Vertex = Min

b) Down (a<o)
Vertex = Max

models the ball’s height above the
ground, s(t), in feet, t seconds after it Rules
1) Determine what needs to
was thrown.
be maximized or
a) After how many seconds does the ball
reach max height? What is the max height?
b) How many seconds does it take until the
ball finally hits the ground?
c) Find s(0) and describe what it means
d) Use results to graph the parabola

minimized
2) Express quantity as
function

3) Rewrite in form:

f (x)

ax 2

bx c

4) Calculate Max or
Min by finding the
vertex

Max point…

Vertex = (t, s(t))

s (t )

16t 2

a

is the Vertex

64t

200

16 b

64

Feet from ground
at max

Seconds it took to
reach max
a) After how many seconds
does the ball reach max
height? What is the max
height?
Find the vertex to find max
time, t, and max height, s(t)

200 ft

Vertex

Vertex

b
64 , s
16)
2( a
2 sec ,

b
2a

s(2)

Max point…

MAX at (2 sec, s(t))feet)
Vertex = (t, 264

s (t )

16t 2

a

is the Vertex

64t

200

16 b

64

a) After how many seconds
does the ball reach max
height? What is the max
height?
Vertex

s( 2 )
s ( 2)

200 ft

s ( 2)

2 sec , 264 (2)
s feet
16( 2 )2

64( 2 ) 200

64 128 200
264 feet

It took 2 seconds to reach a
maximum height of 264 feet

Max point…

MAX at (2 sec, 264 feet)

16t 2

s (t )

a

is the Vertex

64t

200

16 b

64

b) How many seconds does it
take until the ball finally hits
the ground?

s(t)

16t 2
Factor out -2
8t 2

0

64t

200

32t 100

0
0

Use Quadratic Formula to slove

200 ft
What is the height , s(t) , when
the ball hits the ground?

s(t)

0

Max point…


s (t )

16t 2

a

is the Vertex

64t

200

16 b

64

the ground?

8t 2 32t 100 0
32 c
a 8 b

t

( 32) ±

t

32

32

2

100

4(8)( 100)

2(8)
1024 3200
16

200 ft
What is the height , s(t) , when
the ball hits the ground?

s(t)

0

Max point…


s (t )

16t 2

a

is the Vertex

64t

200

16 b

64

the ground?

8t 2 32t 100 0
32 c
a 8 b

t

( 32) ±

t

200 ft

32

2

4(8)( 100)

2(8)
32
1024 3200
16
32 64.99
16

6.1 seconds

100

Max point…


s (t )

16t 2

a

is the Vertex

64t

200

16 b

64

c) Find s(0) and describe what
it means

t =0 means zero seconds
Where is the baseball at zero
seconds?

s (0 )
s (0 )

200 ft

16(0)2

64(0) 200

200 feet

The ball is in the thrower’s
hand at the top of the
building, 200 feet up.


s (t )
Vertex

16t 2

64t

200 Opens down

2, 264

a=-16 <0

x-intercepts

Set t=0,
find s(0)
s(o)= 200

1) Open Up or Down?

a) Up if a>o
b) Down if a<o

y-intercepts

Set s(t) =0,
solve for t
t = 6.1

Rules

2) Find the Vertex

for f ( x )

a( x h )2 k
Vertex= ( h , k )
for f ( x ) a 2 b
ax bx c
b
b ,f
Vertex
2a
2a


Set x=0, find f(0)
5) Axis of Symmetry

x

h or x

b
2a


16t 2

s (t )

64t

200 Opens down

a=-16 <0

2, 264

Vertex

x-intercepts

Set t=0,
find s(0)
s(o)= 200

1) Open Up or Down?

a) Up if a>o
b) Down if a<o

y-intercepts

Set s(t) =0,
solve for t
t = 6.1

Rules

H
e
i
g
h
t

250
200

2) Find the Vertex

for f ( x )

a( x h )2 k
Vertex= ( h , k )
for f ( x ) a 2 b
ax bx c
b
b ,f
Vertex
2a
2a

150


100

Set x=0, find f(0)

50
2

5) Axis of Symmetry
4

6

Time

x

h or x

b
2a

•Tonight’s Lecture portion Ch 11.2 & 11.3: •DONE
•5 points for Attendance given at:

•8:30 PM

•Homework Assignments due by 8:30PM:•Ch 11.2 & 11.3
•Tonight’s Assignments already done? :
•Turn them in BEFORE you leave
•Receive 5 points for attendance
•You may leave anytime AFTER the LECTURE portion

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Ch11.2&3(1)

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