This document covers key concepts about quadratic functions including: - The standard form and vertex form of quadratics - Finding the vertex, axis of symmetry (line of symmetry), maximum/minimum values, and x-intercepts of quadratic functions - Graphing quadratic functions on a graphing calculator and using transformations - Writing quadratic functions in standard form given the vertex and a point

1. Quadratics
Parts of a Parabola and Vertex
Form
Section 2.1

2. Objective
Vertex and Standard Form of
Quadratic
Graphing and
Transformations of
Quadratic
Use GDC to find the
intercepts.

3. Relevance
Learn how to evaluate
data from real world
applications that fit into
a quadratic model.

4. Warm Up #1
 Find the equation in the form
Ax + By + C = 0 of the line that passes
through the given point and has the
indicated slope.
  undefined
is
slope
99
,
287


5. Warm Up #2
 Find the equation in the form y = mx + b
of the line that passes through the given
point and has the indicated slope.
  0
9821
,
3482 is
slope



6. Warm Up #3
 Find the equation in the form y = mx + b
of the line that passes through the given
point and has the indicated slope.
 
3
,
1
4
10
2 


 through
y
x
to
parallel

7. Warm Up #4
 Find the equation in the form y = mx + b
of the line that passes through the given
point and has the indicated slope.
 
4
,
1
2
3
2


 through
x
y
to
lar
perpendicu

8. Find the following.
      6
2
5
2
3
10 2






 x
x
x
h
x
x
g
x
x
f
 
x
f
Find 1


9. Find the following.
      6
2
5
2
3
10 2






 x
x
x
h
x
x
g
x
x
f
 
x
fg
Find

10. Find the following.
      6
2
5
2
3
10 2






 x
x
x
h
x
x
g
x
x
f
  
x
g
h
Find 

11. Quadratic Functions
 Function:
 Standard Form (Vertex Form):
 Graphs a parabola
 All are symmetric to a line called the axis of
symmetry or line of symmetry (los)
2
( )
f x ax bx c
  
2
( ) ( )
f x a x h k
  

12. Show that g represents a quadratic
function. Identify a, b, and c.
  
7
2
3
)
( 

 x
x
x
g
  
14
19
3
14
2
21
3
7
2
3
)
(
2
2










x
x
x
x
x
x
x
x
g
14
19
3





c
b
a

13. Show that g represents a quadratic
function. Identify a, b, and c.
  
2
5
8
2
)
( 


 x
x
x
g
  
  
32
76
10
32
80
4
10
2
5
16
2
2
5
8
2
)
(
2
2

















x
x
x
x
x
x
x
x
x
x
g
32
76
10




c
b
a

14. Show that g represents a quadratic
function. Identify a, b, and c.
  4
6
)
(
2


 x
x
g
 
  
32
12
4
36
6
6
4
6
6
4
6
)
(
2
2
2















x
x
x
x
x
x
x
x
x
g
32
12
1



c
b
a

15. Parts of a Parabola
 Axis of Symmetry (Line of
Symmetry) LOS:
The line that divides the parabola into two
parts that are mirror images of each other.
 Vertex:
Either the lowest or highest point.

16. Let’s look at a transformation
2
)
( x
x
f 
Up
Concave
Parent
  2
3
)
(
2


 x
x
f
Up
Concave
Form
Vertex
3
.
Rt
2
Up
What is the vertex, max or min, and los?

17.   2
3
)
(
2


 x
x
f
 Vertex Form:
)
2
,
3
(
:
V ertex
2
:
min 
y
3
: 
x
los

18. Let’s look at a transformation
2
)
( x
x
f 
Up
Concave
Parent
 2
4
)
( 
 x
x
f
Up
Concave
Form
Vertex
4
Left
What is the vertex, max or min, and los?

19.  2
4
)
( 
 x
x
f
 Vertex Form:
)
0
,
4
(
: 
V ertex
0
:
min 
y
4
: 

x
los

20. Let’s look at a transformation
2
)
( x
x
f 
Up
Concave
Parent
2
)
0
(
)
( 2



 x
x
f
Down
Concave
Form
Vertex
Flips
2
Down
What is the vertex, max or min, and los?

21. 2
)
( 2


 x
x
f
 Vertex Form:
)
2
,
0
(
: 
V ertex
2
:
min 
y
0
: 
x
los

22. Finding the Vertex and los on the
GDC
 Put equation into y1.
 Press Zoom 6 – fix if necessary by changing
the window.
 Press 2nd Trace (Calc).
 Press max/min.
 Left of it; Then right of it; Then ENTER.

23. Let’s Try It….
Find the vertex, min, and los.
4
7
3
)
( 2


 x
x
x
f
 
17
.
1
:
08
.
8
:
min
08
.
8
,
17
.
1
:






x
los
y
Vertex

24. Use your GDC to find the zeros
(x-intercepts)….
 Press 2nd Trace (Calc)
 Press zero.
 Again, to the left, to the right, ENTER.

25. Find the zeros for the last
example.
4
7
3
)
( 2


 x
x
x
f
47
.
0
8
.
2



x
x

26. Example:
 Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
8
3
2
)
( 2



 x
x
x
f
 
Down
Concave
x
x
zeros
y
x
los
Vertex
89
.
2
,
39
.
1
:
13
.
9
:
max
75
.
0
:
13
.
9
,
75
.
0
:






27. Example:
 Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
5
2
3
)
( 2


 x
x
x
f
 
Up
Concave
x
x
zeros
y
x
los
Vertex
1
,
70
.
1
:
3
.
5
:
min
3
.
0
:
3
.
5
,
3
.
0
:









These are the solutions.
What are the x-intercepts?

28. Example:
 Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
7
6
2
)
( 2



 x
x
x
f
 
Down
Concave
Solutions
REAL
No
Solutions
NONE
x
y
x
los
Vertex
:
:
int
5
.
2
:
max
5
.
1
:
5
.
2
,
5
.
1
:





These are the solutions.
What are the x-intercepts?

29. Example
 Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
2
3
5
)
( 2


 x
x
x
f
 
Up
Concave
None
zeros
y
x
los
Vertex
:
55
.
1
:
min
30
.
0
:
55
.
1
,
30
.
0
:



30. Example:
 Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
x
x
x
f 7
)
( 2



 
Down
Concave
x
x
zeros
y
x
los
Vertex
7
,
0
:
25
.
12
:
max
5
.
3
:
25
.
12
,
5
.
3
:





31. Example:
 Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
20
2
3
)
( 2


 x
x
x
f
 
Up
Concave
x
x
zeros
y
x
los
Vertex
3
.
2
,
9
.
2
:
3
.
20
:
min
3
.
0
:
3
.
20
,
3
.
0
:










32. How do I know if it is concave up or down just
by looking at the function?
 In the following examples, state
whether the parabola is concave
up or down and whether the
vertex is a max or a min by just
looking at the function.

33. m ax
,
D o w n
C o n ca v e
1
7
8
)
( 2


 x
x
x
f
1
4
7
)
( 2



 x
x
x
f
x
x
x
f 

 2
3
8
)
(
2
6
2
)
( x
x
x
f 


m in
,
U p
C o n ca v e
m ax
,
D o w n
C o n ca v e
min
,
U p
C on ca v e

34. Write the equation in standard form of the
parabola whose vertex is (1, 2) and passes
through the point (3, -6).
k
h
x
a
y 

 2
)
(
(h, k)
(x, y)
 
2
4
8
2
1
3
6
2








a
a
a
  2
1
2
:
2



 x
y
FORM
STANDARD

35. Write the equation in standard form of the
parabola whose vertex is (-2, -1) and
passes through the point (0, 3).
k
h
x
a
y 

 2
)
(
(h, k)
(x, y)
 
1
4
4
1
2
0
3
2





a
a
a
  1
2
:
2


 x
y
FORM
STANDARD

36. Write the equation in standard form of the
parabola whose vertex is (4, -1) and passes
through the point (2, 3).
k
h
x
a
y 

 2
)
(
(h, k)
(x, y)
 
1
4
4
1
4
2
3
2





a
a
a
  1
4
:
2


 x
y
FORM
STANDARD

37. A golf ball is hit from the ground. Its height in feet above the
ground is modeled by the function
where t represents the time in seconds after the ball is hit.
How long is the ball in the air?
What is the maximum height of the ball?
  ,
180
16 2
t
t
t
h 


Graph on GDC.
Find the zeros. Answer: 11.25 seconds
Graph on GDC.
Find the maximum y-value.
Answer: 506.25 feet

38.  A. What is the maximum height of the ball?
B. At what time does the ball reach its maximum height?
C. At what time(s) is the ball 16 feet high in the air?
Graph and find the
maximum y-value.
Answer: 21 feet
Set equation = to 21
and find the zeros. Answer: 1 second
Set equation = to 16
and find the zeros. Answer: 1.56 seconds and
0.44 seconds.

39.  .
A. What ticket price gives the maximum profit?
B. What is the maximum profit?
C. What ticket price would generate a profit of $5424?
Graph and find the maximum x-value.
Hint: Press Zoom 0 and change x-max to 50
and y max to 8000.
Answer: $25
Answer: $6,000
Set equation = to 5424
and find the zeros.
Hint: Press zoom 0.
Answer: $19 or $31
Graph and find the maximum y-value.
Hint: Press Zoom 0 and change x-max to 50
and y max to 8000.

40. Classwork: Quadratics Review
Homework: Worksheet 4B