Quadratics Parts of a Parabola and Vertex Form. Quadratic Functions Function: Standard Form (Vertex...
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Quadratics Parts of a Parabola and Vertex Form
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Transcript of Quadratics Parts of a Parabola and Vertex Form. Quadratic Functions Function: Standard Form (Vertex...
Quadratics
Parts of a Parabola and Vertex Form
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
Show that g represents a quadratic function. Identify a, b, and c.
723)( xxxg
14193
142213
723)(
2
2
xx
xxx
xxxg
14
19
3
c
b
a
Show that g represents a quadratic function. Identify a, b, and c.
2582)( xxxg
327610
3280410
25162
2582)(
2
2
xx
xxx
xx
xxxg
32
76
10
c
b
a
Show that g represents a quadratic function. Identify a, b, and c.
46)( 2 xxg
3212
43666
466
46)(
2
2
2
xx
xxx
xx
xxg
32
12
1
c
b
a
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
2)( xxf
UpConcave
Parent
23)( 2 xxf
UpConcave
FormVertex
3.Rt
2Up
What is the vertex, max or min, and los?
23)( 2 xxfVertex Form:
)2,3(:Vertex
2:min y
3: xlos
Let’s look at a transformation
2)( xxf
UpConcave
Parent
24)( xxf
UpConcave
FormVertex
4LeftWhat is the vertex, max or min, and los?
24)( xxfVertex Form:
)0,4(: Vertex
0:min y
4: xlos
Let’s look at a transformation
2)( xxf
UpConcave
Parent
2)0()( 2 xxf
DownConcave
FormVertex
Flips
2Down
What is the vertex, max or min, and los?
2)( 2 xxf
Vertex Form:
)2,0(: Vertex
2:min y
0: xlos
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.
473)( 2 xxxf
17.1:
08.8:min
08.8,17.1:
xlos
y
Vertex
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.
473)( 2 xxxf
47.0
8.2
x
x
Example:
Find the vertex, los, max/min, zeros, and tell whether concave up or concave down.
832)( 2 xxxf
DownConcave
xxzeros
y
xlos
Vertex
89.2,39.1:
13.9:max
75.0:
13.9,75.0:
Example:
Find the vertex, los, max/min, zeros, and tell whether concave up or concave down.
523)( 2 xxxf
UpConcave
xxzeros
y
xlos
Vertex
1,70.1:
3.5:min
3.0:
3.5,3.0:
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.
762)( 2 xxxf
DownConcave
SolutionsREALNoSolutions
NONEx
y
xlos
Vertex
:
:int
5.2:max
5.1:
5.2,5.1:
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.
235)( 2 xxxf
UpConcave
Nonezeros
y
xlos
Vertex
:
55.1:min
30.0:
55.1,30.0:
Example:
Find the vertex, los, max/min, zeros, and tell whether concave up or concave down.
xxxf 7)( 2
DownConcave
xxzeros
y
xlos
Vertex
7,0:
25.12:max
5.3:
25.12,5.3:
Example:
Find the vertex, los, max/min, zeros, and tell whether concave up or concave down.
2023)( 2 xxxf
UpConcave
xxzeros
y
xlos
Vertex
3.2,9.2:
3.20:min
3.0:
3.20,3.0:
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.
max,DownConcave
178)( 2 xxxf
147)( 2 xxxf
xxxf 238)(
262)( xxxf
min,UpConcave
max,DownConcave
min,UpConcave
Write the equation in standard form of the parabola whose vertex is (1, 2) and passes through the point (3, -6).
khxay 2)(
(h, k)
(x, y)
2
48
2136 2
a
a
a
212
:2 xy
FORMSTANDARD
Write the equation in standard form of the parabola whose vertex is (-2, -1) and passes through the point (0, 3).
khxay 2)(
(h, k)(x, y)
1
44
1203 2
a
a
a
12
:2 xy
FORMSTANDARD
Write the equation in standard form of the parabola whose vertex is (4, -1) and passes through the point (2, 3).
khxay 2)(
(h, k)
(x, y)
1
44
1423 2
a
a
a 14
:2 xy
FORMSTANDARD
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?
,18016 2 ttth
Graph on GDC.Find the zeros. Answer: 11.25 seconds
Graph on GDC.Find the maximum y-value.
Answer: 506.25 feet
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 21and find the zeros. Answer: 1 second
Set equation = to 16and find the zeros. Answer: 1.56 seconds and
0.44 seconds.
. 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 50and y max to 8000.
Answer: $25
Answer: $6,000
Set equation = to 5424and 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 50and y max to 8000.
