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Parabola and its Vertex Remember, parabolas have equations : The Standard Form for a parabola : 0 2 c bx ax 2 h x a k y
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Parabola and its Vertex
Remember, parabolas have equations :
The Standard Form for a parabola :
02 cbxax
2hxaky
Parabola and its Vertex
Remember, parabolas have equations :
The Standard Form for a parabola :
OR
02 cbxax
2hxaky
khxay 2
Parabola and its Vertex
Remember, parabolas have equations :
The Standard Form for a parabola :
OR
Where : ( h , k ) is the vertex
If a > 0 the parabola opens up
If a < 0 the parabola opens down
02 cbxax
2hxaky
khxay 2
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square ( similar to what we did w/ circles )
b) formula
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Completing the square :
____7____
7___8
78
2
2
2
xy
xxy
xxy Get your x2 and x together, leave a blank, and place the constant outside parens…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Completing the square :
____7____
7___8
78
2
2
2
xy
xxy
xxy Get your x2 and x together, leave a blank, and place the constant outside parens…
Write the standard form with h missing and a blank behind the constant…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Completing the square :
____74
7___2
8
78
2
2
2
xy
xxy
xxy
Divide the linear coefficient by two and place the result where h would go…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Completing the square :
____74
7162
8
78
2
2
2
xy
xxy
xxy
Square that result and place it in the blank INSIDE parens…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Completing the square :
1674
7162
8
78
2
2
2
xy
xxy
xxy
There is a 1 in front of the set of parens, …multiply “1” by the 16 and the opposite of that result goes in the last blank…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Completing the square :
94
1674
7162
8
78
2
2
2
2
xy
xy
xxy
xxy
Combine the constants…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Completing the square :
94
1674
7162
8
78
2
2
2
2
xy
xy
xxy
xxy
This parabola has a vertex of ( 4 , - 9 ) and because a =1, opens upward
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Formula :a
bx
2
This equation finds the x – coordinate of the
vertex or ( h )…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Formula :a
bx
2
From the equation above: a = 1 , b = - 8 , c = 7
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Formula :a
bx
2
From the equation above: a = 1 , b = - 8 , c = 7
4
2
8
)1(2
8
x So h = 4
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Formula :a
bx
2
Substitute x = 4 into original equation to find the y coordinate of the vertex or k…
4
2
8
)1(2
8
x
h = 4
9
73216
7484 2
y
y
y
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 782 xxy
Formula :a
bx
2
Substitute x = 4 into original equation to find the y coordinate of the vertex or k…
4
2
8
)1(2
8
x
h = 4
k = - 9
9
73216
7484 2
y
y
y
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
Completing the square :
____5____2
5___42
542
2
2
2
xy
xxy
xxy Get your x2 and x together, leave a blank, and place the constant outside parens…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
Completing the square :
____5____2
5___22
5___2
4
2
2
542
2
2
2
2
xy
xxy
xxy
xxy
Factor out the coefficient of x and x2…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
Completing the square :
____5____2
5___2
22
5___2
4
2
2
542
2
2
2
2
xy
xxy
xxy
xxy
Divide the linear coefficient by 2 and fill in “h”…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
Completing the square :
____512
5___2
22
5___2
4
2
2
542
2
2
2
2
xy
xxy
xxy
xxy
Divide the linear coefficient by 2 and fill in “h”…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
Completing the square :
____512
512
22
5___2
4
2
2
542
2
2
2
2
xy
xxy
xxy
xxy
Square that result and place in blank inside parens…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
Completing the square :
____512
512
22
5___2
4
2
2
542
2
2
2
2
xy
xxy
xxy
xxy
Since there is a 2 outside parens, multiply 2 by 1 and place the opposite result in last blank…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
Completing the square :
2512
512
22
5___2
4
2
2
542
2
2
2
2
xy
xxy
xxy
xxy
Since there is a 2 outside parens, multiply 2 by 1 and place the opposite result in last blank…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
Completing the square :
312
512
22
5___2
4
2
2
542
2
2
2
2
xy
xxy
xxy
xxy
Combine outside integers…
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
Completing the square :
312
542
2
2
xy
xxy
Vertex = ( 1 , 3 )
a = 2
Opens upward
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
FORMULA :
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
a = 2
b = - 4
c = 5
FORMULA :
1
4
4
)2(2
42
x
x
a
bx
Vertex = ( 1 , )
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
a = 2
b = - 4
c = 5
FORMULA :
1
4
4
)2(2
42
x
x
a
bx
3
542
51412 2
y
y
y
Vertex = ( 1 , 3 )
Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of 542 2 xxy
a = 2
b = - 4
c = 5
FORMULA :
1
4
4
)2(2
42
x
x
a
bx
3
542
51412 2
y
y
y
Vertex = ( 1 , 3 )
a = 2
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
5
15
15
14 14
14109
2
2
2
a
a
a
a
khxay We need to find “ a “
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
1
99
39
2 2
2307
2
2
2
a
a
a
a
khxay
Substitute vertex ( 3 , - 2 ) for ( h , k ) and ( 0 , 7 ) for y – intercept ( x , y ) into standard form…
We need to find “ a “
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
1
99
39
2 2
2307
2
2
2
a
a
a
a
khxay We need to find “ a “
Add 2 to both sides…
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
1
99
39
2 2
2307
2
2
2
a
a
a
a
khxay We need to find “ a “
( - 3 )2 = 9…
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
1
99
39
2 2
2307
2
2
2
a
a
a
a
khxay We need to find “ a “
Divide both sides by 9 …
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
1
99
39
2 2
2307
2
2
2
a
a
a
a
khxay
Substitute a = 1 into standard form along with the vertex ( 3 , - 2 ) …
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
1
99
39
2 2
231
2
2
2
a
a
a
xy
khxay
x – intercepts occur where y = 0.
Set y = 0 and solve for “x”
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
1
99
39
2 2
2310
231
2
2
2
2
a
a
a
x
xy
khxay
x – intercepts occur where y = 0.
Set y = 0 and solve for “x”
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
1
99
312
2 2
2310
231
2
2
2
2
a
a
x
x
xy
khxay
Add 2 to both sides…
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
1
99
31
1
1
2
2 2
2310
231
2
2
2
2
a
a
x
x
xy
khxay
Divide both sides by 1…
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
32
31
1
1
2
2 2
2310
231
2
2
2
2
2
x
x
x
xy
khxay
Divide both sides by 1…
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
32
31
1
1
2
2 2
2310
231
2
2
2
2
2
x
x
x
xy
khxay
Square root both sides…
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
32
32
31
1
1
2
2 2
2310
231
2
2
2
2
2
x
x
x
x
xy
khxay
Square root both sides…
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
23
32
32
31
1
1
2
2 2
2310
231
2
2
2
2
2
x
x
x
x
x
xy
khxay
Add 3 to both sides…
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
23
32
32
31
1
1
2
2 2
2310
231
2
2
2
2
2
x
x
x
x
x
xy
khxay x – intercepts :
0,59.1 , 0,41.4
OR
0,23 , 0,23
EXAMPLE : Find the x – intercepts of the parabola with vertex ( 3 , - 2 ) and y – intercept ( 0 , 7 ). Write your answer in ( x1 , y1 ), ( x2 , y2 ) form. If necessary, round to 2 decimal places.
Parabola and its Vertex
23
32
32
31
1
1
2
2 2
2310
231
2
2
2
2
2
x
x
x
x
x
xy
khxay x – intercepts :
0,59.1 , 0,41.4
OR
0,23 , 0,23
If you get a negative under the square root on this step, there ARE NO x – intercepts !!!
