Quadratics
Parts of a Parabola and Vertex
Form
Quadratic Functions
f ( x )  ax  bx  c
2

Function:

Standard Form (Vertex Form):
f ( x )  a ( x  h) 2  k

Graphs a parabola

All are symmetric to a line called the axis of
symmetry or line of symmetry (los)
Show that g represents a quadratic
function. Identify a, b, and c.
g ( x )  3 x  2  x  7 
g ( x)  3 x  2  x  7 
 3 x  21x  2 x  14
a3
b  19
 3 x  19 x  14
c  14
2
2
Show that g represents a quadratic
function. Identify a, b, and c.
g ( x )  2 x  85 x  2 
g ( x)  2 x  85 x  2 
  2 x  16 5 x  2 
 10 x  4 x  80 x  32
a  10
b  76
 10 x  76 x  32
c  32
2
2
Show that g represents a quadratic
function. Identify a, b, and c.
g ( x)  x  6  4
2
g ( x)  x  6  4
2
  x  6  x  6   4
 x  6 x  6 x  36  4
2
 x  12 x  32
2
a 1
b  12
c  32
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.
Let’s look at a transformation
Rt . 3
What is the vertex, max or min, and los?
f ( x)  x
2
f ( x )   x  3  2
2
Up 2
Parent
Concave Up
Vertex Form
Concave Up
f ( x )   x  3  2
2
 Vertex
Form:
Vertex : (3, 2)
min : y  2
los : x  3
Let’s look at a transformation
What is the vertex, max or min, and los?
f ( x)  x
2
Parent
Concave Up
Left 4
f ( x)  x  4
2
Vertex Form
Concave Up
f ( x)  x  4
2
 Vertex
Form:
Vertex : ( 4,0)
min : y  0
los : x  4
Let’s look at a transformation
What is the vertex, max or min, and los?
f ( x)  x
2
Flips
f ( x )   ( x  0) 2  2
Down 2
Parent
Concave Up
Vertex Form
Concave Down
f ( x)   x  2
2
 Vertex
Form:
Vertex : (0,2)
min : y  2
los : x  0
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.
Let’s Try It….

Find the vertex, min, and los.
f ( x)  3x  7 x  4
2
Vertex :  1.17,  8.08
min : y  8.08
los : x  1.17
Use your GDC to find the zeros
(x-intercepts)….



Press 2nd Trace (Calc)
Press zero.
Again, to the left, to the right, ENTER.
Find the zeros for the last
example.
f ( x)  3x  7 x  4
2
x  2 . 8
x  0.47
Example:

Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
f ( x )  2 x  3 x  8
2
Vertex : 0.75, 9.13
los : x  0.75
max : y  9.13
zeros : x  1.39, x  2.89
Concave Down
Example:

Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
f ( x)  3x  2 x  5
2
Vertex :  0.3,  5.3
los : x  0.3
min : y  5.3
zeros : x  1.70, x  1
Concave Up
These are the solutions.
What are the x-intercepts?
Example:

Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
f ( x )  2 x  6 x  7
2
Vertex : 1.5,  2.5
los : x  1.5
max : y  2.5
x  int : NONE
Solutions : No REAL Solutions
Concave Down
These are the solutions.
What are the x-intercepts?
Example

Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
2
f ( x)  5 x  3x  2
Vertex : 0.30,1.55
los : x  0.30
min : y  1.55
zeros : None
Concave Up
Example:

Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
f ( x)   x  7 x
2
Vertex : 3.5,12.25
los : x  3.5
max : y  12.25
zeros : x  0, x  7
Concave Down
Example:

Find the vertex, los, max/min, zeros, and tell
whether concave up or concave down.
f ( x )  3 x  2 x  20
2
Vertex :  0.3,  20.3
los : x  0.3
min : y  20.3
zeros : x  2.9, x  2.3
Concave Up
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.
f ( x)  8 x  7 x  1
Concave Up, min
f ( x )  7 x 2  4 x  1
Concave Down , max
f ( x)  8  3x  x
Concave Down , max
f ( x)  2  6 x  x 2
Concave Up, min
2
2
Write the equation in standard form of the
parabola whose vertex is (1, 2) and passes
through the point (3, -6). (h, k)
(x, y)
y  a ( x  h)  k
2
 6  a3  1  2
STANDARD FORM :
 8  4a
y  2 x  1  2
2
a  2
2
Write the equation in standard form of the
parabola whose vertex is (-2, -1) and
(h, k)
passes through the point (0, 3).
(x, y)
y  a ( x  h)  k
2
3  a0  2   1
2
4  4a
a 1
STANDARD FORM :
y  x  2  1
2
Write the equation in standard form of the
parabola whose vertex is (4, -1) and passes
(h, k)
through the point (2, 3).
(x, y)
y  a ( x  h)  k
2
3  a2  4   1 STANDARD FORM :
2
4  4a
a 1
y  x  4  1
2
A golf ball is hit from the ground. Its height in feet above the
ground is modeled by the function ht   16t 2  180t ,
where t represents the time in seconds after the ball is hit.
How long is the ball in the air?
Graph on GDC.
Find the zeros.
Answer: 11.25 seconds
What is the maximum height of the ball?
Graph on GDC.
Find the maximum y-value.
Answer: 506.25 feet
 A. What is the maximum height of the ball?
Graph and find the
maximum y-value.
Answer: 21 feet
B. At what time does the ball reach its maximum height?
Set equation = to 21
and find the zeros.
Answer: 1 second
C. At what time(s) is the ball 16 feet high in the air?
Set equation = to 16
and find the zeros.
Answer: 1.56 seconds and
0.44 seconds.

.
A. What ticket price gives the maximum profit?
Graph and find the maximum x-value.
Hint: Press Zoom 0 and change x-max to 50
and y max to 8000.
Answer: $25
B. What is the maximum profit?
Graph and find the maximum y-value.
Hint: Press Zoom 0 and change x-max to 50
and y max to 8000.
Answer: $6,000
C. What ticket price would generate a profit of $5424?
Set equation = to 5424
and find the zeros.
Hint: Press zoom 0.
Answer: $19 or $31