parabola. examples give four examples of equations of parabola all having the vertex (2,3), but one...
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Parabola
kysymmetryofAxis
hkyaxEquation
aleftorarightFacingI
hxsymmetryofAxis
khxayEquation
adownoraupwFacingI
khVertex
:
)(:
:)0()0(.
:
)(:
:)0()0(.
),(
2
2
Examples
Give four examples of equations of parabola all having the vertex (2,3), but one is facing upward, one downward, one to the right and one to the left. Identify the axis of symmetry.
Solution
Vertex (2,3)
2:
3)2(5:
:.
2:
3)2(5:
:.
.
2
2
xsymmetryofAxis
xyEquation
downFacingb
xsymmetryofAxis
xyEquation
upFacinga
I
3:
2)3(5:
:.
3:
2)3(5:
:.
.
2
2
ysymmetryofAxis
yxEquation
leftFacinga
ysymmetryofAxis
yxEquation
rightFacinga
II
Examples (1) – (4)
Identify the parabola having the given equation and its axis of symmetry and graph it.
Decide Which parabola have the following Equations & Graph:
542.4
152.3
2)1(3.2
2.1
2
2
2
2
yyx
xxy
yx
yx
Decide Which Parabola have the Given Equations & Graph:
)0,0(
)(0:
0)0(2
:Re
2.1
2
2
isvertxThe
axisxtheylinetheissymmetryofaxistheThe
rightthetofacingparabolaaisThis
yx
formndardtastheinequationthewrite
yx
Finding points of the graphx=2y2
y x (x,y)
-2 8 (8,-2)
-1 2 (2,-1)
0 0 (0,0)
1 2 (2,1)
2 8 (8,2)
3 18 (18,3)
Decide Which Parabola have the Given Equations & Graph:
)1,2(:
1:
2)1(3.2 2
isvertexThw
ylinetheissymmetryofaxistheThe
leftthetofacingparabolaaisThis
yx
Finding points of the graphx=-3(y-1)2+2
y x (x,y)
-2 -25 (-25,-2)
-1 -10 (-10,-1)
0 -1 (-1,0)
1 2 (2,1)
2 1 (-1,2)
3 -10 (-10,3)
What are the x-intercepts and the y-intercepts?
1835.03
211658.1
3
21,
)3
21,0()
3
21,0(
3
21
3
21
3
2)1(
2)1(32)1(30
:0,
1232)10(3
:0,
2)1(3.2
2
22
2
2
andWhere
andataxisythectsserteingraphThe
yyy
yy
xletweterceptsinythefindTo
x
yletweterceptsinxthefindTo
yx
Decide Which Parabola have the Given Equations & Graph:
)16,1(
1:
16)1(
151)1(
15]1)1[(
15)2(
:
152.3
2
2
2
2
2
isvertexThe
xissymmetryofaxisThe
downwardfacingparabolaaofequationanisThis
x
x
x
xxy
squarethecompletingbythatdoWe
formndardstatheinequationthewriteretoneedWe
xxy
Finding points of the graphy=-x2-2x+15 = - (x+1) 2 + 16
x y (x,y)
-2 15 (-2,15)
-1 16 (-1,16)
0 15 (0,15)
1 12 (1,12)
2 7 (2,7)
3 0 (3,0)
-5 0 (-5,00
-3 12 (-3,12)
What are the x-intercepts and the y-intercepts?
)0,5()0,3(
54134141
16)1(
16)1(0:,
)15,0(
1516)10(:,
16)1(
2
2
2
2
andataxisxthecrossgraphThe
xorxx
x
xgetwezerobeyLetting
ataxisythecrossgraphThe
ygetwezerobexLetting
xy
Decide Which Parabola have the Given Equations & Graph:
)1,3(:
1:
3)1(2
52)1(2
5]1)1[(2
5)2(2
542
:
542.4
2
2
2
2
2
2
vertexThe
yissymmetryofaxisThe
rightthetofacingparabolaaofequationanisThis
y
y
y
y
yyx
squarethecompletingbythatdoWe
formndardstathenequationithewriteretoneedWe
yyx
Finding points of the graphx=-2y2+4x+5 = 2 (y+1) 2 + 3
y x (x,y)
-4 21 (21,-4)
-3 11 (11,-3)
-2 5 (5,-2)
-1 3 (3,-1)
0 5 (5,0)
1 11 (11,1)
2 21 (21,2)
What are the x-intercepts and the y-intercepts?
axisyhecrossnotdoesgraphThe
solutionrealnohaswhichy
yequationthegetwexLetting
aaxisxthecrossesgraphThe
xgetweyLetting
yx
,3)1(2
3)1(20:,0
)0,5(
53)10(2:,0
3)1(2
2
2
2
2
More Examples
Example (1)Graph y=x2 - 6x + 7
The graph is facing upward. Why?
y= x2 - 6 x + 7 = (x- 3 )2 – 9 + 7= (x- 3 )2 – 2
The vertex of the parabola is (3 , - 2 )
The intersection with the y-axis is(0,7)y=x2 - 6x + 7 Why?
The intersections with the x-axis: (3+√2 , 0 ) and (3-√2 , 0 ) Why?
7.552.50-2.5
7.5
5
2.5
0
-2.5
-5
x
y
x
y
)2,3(
23 23
7
Example (2)Graph y=2x2 + 2x + 3
The graph is concave upward. Why?y=2[x2 +x] + 3 = 2[(x+½)2–¼] + 3= 2(x+½)2–½ + 3 = 2(x+½)2 + 5/2 The vertex of the parabola is ( - ½ , 5/2 )
The intersection with the y-axis is (0,3) Why?
No intersections with the x-axis: Why?
1.250-1.25-2.5-3.75
8
6
4
2
0
x
y
x
y
3),( 25
21
Example (3)Graph y=-x2 - 4x + 12
The graph is concave downward. Why?y= -(x2 +4x ) + 12=-[(x+ 2 )2 – 4] + 12= -(x+2)2 + 16 The vertex of the parabola is (-2 , 16 )
The intersection with the y-axis is (0,12) Why?
y=-x2 - 4x + 12=-(x2 +4x - 12)=-(x+6)(x-2)The intersections with the x-axis: (2 , 0 ) and (-6 , 0 ) Why?
52.50-2.5-5-7.5
15
10
5
0
-5
-10
x
y
x
y
)16,2(
12
6 2
QuestionDo it now!
Do not see the next slide before you do it!
Graph y=x2 + 2x - 3
Solution y=x2 +2x - 3
The graph is concave upward. Why?y=x2 +2x – 3 = = (x+1)2 – 1 – 3 = (x+1)2 - 4 The vertex of the parabola is (-1 , -4 )
The intersection with the y-axis is (0,-3)y = x2 + 2x - 3=(x+3)(x-1) why?
The intersections with the x-axis: (-3 , 0 ) and (1 , 0 ) Why?
52.50-2.5-5
30
25
20
15
10
5
0
-5x
y
x
y
3)4,1( 31
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