1. Parabola and its Vertex
Remember, parabolas have equations :
The Standard Form for a parabola :

2. Parabola and its Vertex
Remember, parabolas have equations :
The Standard Form for a parabola : OR

3. 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

4. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square ( similar to what we did w/ circles )
b) formula

5. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Get your x2 and x together, leave a blank, and place the constant outside parens…
Completing the square :

6. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
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…
Completing the square :

7. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Divide the linear coefficient by two and place the result where h would go…

8. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Square that result and place it in the blank INSIDE parens…

9. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
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…

10. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Combine the constants…

11. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
This parabola has a vertex of ( 4 , - 9 ) and because a =1, opens upward

12. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Formula :
This equation finds the x – coordinate of the vertex or ( h )…

13. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Formula :
From the equation above: a = 1 , b = - 8 , c = 7

14. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Formula :
From the equation above: a = 1 , b = - 8 , c = 7
So h = 4

15. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Formula :
h = 4
Substitute x = 4 into original equation to find the y coordinate of the vertex or k…

16. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Formula :
h = 4
k = - 9
Substitute x = 4 into original equation to find the y coordinate of the vertex or k…

17. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Get your x2 and x together, leave a blank, and place the constant outside parens…
Completing the square :

18. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Factor out the coefficient of x and x2…

19. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Divide the linear coefficient by 2 and fill in “h”…

20. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Divide the linear coefficient by 2 and fill in “h”…

21. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Square that result and place in blank inside parens…

22. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Since there is a 2 outside parens, multiply 2 by 1 and place the opposite result in last blank…

23. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Since there is a 2 outside parens, multiply 2 by 1 and place the opposite result in last blank…

24. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Combine outside integers…

25. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
Completing the square :
Vertex = ( 1 , 3 )
a = 2
Opens upward

26. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
EXAMPLE : Find the vertex of
FORMULA :

27. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
a = 2
b = - 4
c = 5
EXAMPLE : Find the vertex of
FORMULA :
Vertex = ( 1 , )

28. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
a = 2
b = - 4
c = 5
EXAMPLE : Find the vertex of
FORMULA :
Vertex = ( 1 , 3 )

29. Parabola and its Vertex
Finding the vertex
Two ways a) completing the square
b) formula
a = 2
b = - 4
c = 5
EXAMPLE : Find the vertex of
FORMULA :
Vertex = ( 1 , 3 )
a = 2

30. 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.

31. 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.
We need to find “ a “

32. 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.
We need to find “ a “
Substitute vertex ( 3 , - 2 ) for ( h , k ) and ( 0 , 7 ) for y – intercept ( x , y ) into standard form…

33. 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.
We need to find “ a “
Add 2 to both sides…

34. 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.
We need to find “ a “
( - 3 )2 = 9…

35. 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.
We need to find “ a “
Divide both sides by 9 …

36. 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.
Substitute a = 1 into standard form along with the vertex ( 3 , - 2 ) …

37. 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.
x – intercepts occur where y = 0.
Set y = 0 and solve for “x”

38. 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.
x – intercepts occur where y = 0.
Set y = 0 and solve for “x”

39. 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.
Add 2 to both sides…

40. 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.
Divide both sides by 1…

41. 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.
Divide both sides by 1…

42. 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.
Square root both sides…

43. 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.
Square root both sides…

44. 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.
Add 3 to both sides…

45. 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.
x – intercepts :

46. 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.
x – intercepts :
If you get a negative under the square root on this step, there ARE NO x – intercepts !!!