2. Properties of Parabolas that open
Up or Down

3. Properties of Parabolas that open
Up or Down
 General Form: y = ax2+bx+c

4. Properties of Parabolas that open
Up or Down
 General Form: y = ax2+bx+c

5. Properties of Parabolas that open
Up or Down
 General Form: y = ax2+bx+c
 Standard form (also called vertex form) is y = a(x - h)2 + k,
where the vertex is (h,k)
 If a is positive it opens up
 If a is negative it opens down
 Axis of Symmetry (aos) is x = -b/2a

6. Properties of Parabolas that open
Right or Left

7. Properties of Parabolas that open
Right or Left
 General Form: x = ay2+by+c

8. Properties of Parabolas that open
Right or Left
 General Form: x = ay2+by+c

9. Properties of Parabolas that open
Right or Left
 General Form: x = ay2+by+c
 Standard Form: x = a(y – k)2+ h, where the vertex is (h,k)
 If a is positive it opens right
 If a is negative it opens left

10. Properties of Parabolas that open
Right or Left
 General Form: x = ay2+by+c
 Standard Form: x = a(y – k)2+ h, where the vertex is (h,k)
 If a is positive it opens right
 If a is negative it opens left
 Axis of Symmetry (aos) is y = -b/2a

11. Differences

12. Differences
 Properties of Parabolas
that open Left or Right

13. Differences
 Properties of Parabolas
that open Left or Right
 x=

14. Differences
 Properties of Parabolas
that open Left or Right
 x=
 aos: y =

15. Differences
 Properties of Parabolas
that open Left or Right
 x=
 aos: y =
 vertex is (h,k)

16. Differences
 Properties of Parabolas  Properties of Parabolas
that open Up or Down: that open Left or Right
 y=  x=
 aos: x =  aos: y =
 Vertex is (h,k)  vertex is (h,k)

17. Examples :
Tell the direction that each parabola opens, the vertex and the axis
of symmetry (aos).
2 2
y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1

18. Examples :
Tell the direction that each parabola opens, the vertex and the axis
of symmetry (aos).
2 2
y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1
 Opens up

19. Examples :
Tell the direction that each parabola opens, the vertex and the axis
of symmetry (aos).
2 2
y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1
 Opens up
 Vertex is (-2,-6)
 make sure to change the
sign of the value in the
parenthesis

20. Examples :
Tell the direction that each parabola opens, the vertex and the axis
of symmetry (aos).
2 2
y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1
 Opens up
 Vertex is (-2,-6)
 make sure to change the
sign of the value in the
parenthesis
 aos is x = -2
 once you have found the
vertex you can just take
the x coordinate and that
is your aos

21. Examples :
Tell the direction that each parabola opens, the vertex and the axis
of symmetry (aos).
2 2
y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1
 Opens up
 Opens left
 Vertex is (-2,-6)
 make sure to change the
sign of the value in the
parenthesis
 aos is x = -2
 once you have found the
vertex you can just take
the x coordinate and that
is your aos

22. Examples :
Tell the direction that each parabola opens, the vertex and the axis
of symmetry (aos).
2 2
y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1
 Opens up
 Opens left
 Vertex is (-2,-6)
 vertex: (1,-1)
 make sure to change the
sign of the value in the
parenthesis
 aos is x = -2
 once you have found the
vertex you can just take
the x coordinate and that
is your aos

23. Examples :
Tell the direction that each parabola opens, the vertex and the axis
of symmetry (aos).
2 2
y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1
 Opens up
 Opens left
 Vertex is (-2,-6)
 vertex: (1,-1)
 make sure to change the
sign of the value in the
 The thing to remember
parenthesis here is that the y-
coordinate is now in the
 aos is x = -2 parenthesis and the x –
 once you have found the coordinate is in the back.
vertex you can just take
the x coordinate and that
is your aos

24. Examples :
Tell the direction that each parabola opens, the vertex and the axis
of symmetry (aos).
2 2
y = 5 ( x + 2) − 6 x = −2 ( y + 1) + 1
 Opens up
 Opens left
 Vertex is (-2,-6)
 vertex: (1,-1)
 make sure to change the
sign of the value in the
 The thing to remember
parenthesis here is that the y-
coordinate is now in the
 aos is x = -2 parenthesis and the x –
 once you have found the coordinate is in the back.
vertex you can just take  aos: y = -1
the x coordinate and that
is your aos

25. Putting an Equation in Standard
Form

26. Putting an Equation in Standard
Form
 Complete the square.

27. Putting an Equation in Standard
Form
 Complete the square.
 Example: y = x2 - 2x + 4
 y = (x2 - 2x + ___ ) + 4 – (a) ___ put (-b/2)2 in the
blanks and the value for a in the parenthesis before the
last blank
 y = (x2 - 2x + (2/2)2 ) + 4 – (1)(2/2)2
 y = (x2 -2x + 1) + 4 – (1)(1) now factor the first set of ()
 y = (x - 1)2 + 4 – 1
 y = (x - 1)2 + 3 now it easy to find the vertex and aos.

28. Determine the direction in which the following open.
Solve for either x or y whichever one is only in the problem once or is not
squared.
2 2
6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0

29. Determine the direction in which the following open.
Solve for either x or y whichever one is only in the problem once or is not
squared.
2 2
6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0
 Solve for y
 2y = -6x2 – 4x +10
 y = -3x2 -2x + 5

30. Determine the direction in which the following open.
Solve for either x or y whichever one is only in the problem once or is not
squared.
2 2
6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0
 Solve for y
2y = -6x2 – 4x +10

 y = -3x2 -2x + 5
 Opens down

31. Determine the direction in which the following open.
Solve for either x or y whichever one is only in the problem once or is not
squared.
2 2
6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0
 Solve for y  Solve for x
2y = -6x2 – 4x +10

 y = -3x2 -2x + 5
 Opens down

32. Determine the direction in which the following open.
Solve for either x or y whichever one is only in the problem once or is not
squared.
2 2
6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0
 Solve for y  Solve for x
2y = -6x2 – 4x +10
  -5x = 5y2 – 10y -10
 y = -3x2 -2x + 5
 Opens down

33. Determine the direction in which the following open.
Solve for either x or y whichever one is only in the problem once or is not
squared.
2 2
6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0
 Solve for y  Solve for x
2y = -6x2 – 4x +10
  -5x = 5y2 – 10y -10
 y = -3x2 -2x + 5  x = - y2 + 2y + 2
 Opens down

34. Determine the direction in which the following open.
Solve for either x or y whichever one is only in the problem once or is not
squared.
2 2
6x + 2y + 4x = 10 10y − 5y − 5x + 10 = 0
 Solve for y  Solve for x
2y = -6x2 – 4x +10
 -5x = 5y2 – 10y -10

 y = -3x2 -2x + 5  x = - y2 + 2y + 2
 Opens down  Opens left

35. Write the Standard Form of the equation
with a Vertex at (-1,2) and goes
through the point (2, 8).

36. Write the Standard Form of the equation
with a Vertex at (-1,2) and goes
through the point (2, 8).
 Identify h, k , x and y
 h = -1, k = 2 these are from the vertex
 x = 2, y = 8 these are from the other point

37. Write the Standard Form of the equation
with a Vertex at (-1,2) and goes
through the point (2, 8).
 Identify h, k , x and y
 h = -1, k = 2 these are from the vertex
 x = 2, y = 8 these are from the other point
 Plug in what you know
 y = a(x - h)2 + k
 8 = a(2 – (-1))2 + 2

38. Write the Standard Form of the equation
with a Vertex at (-1,2) and goes
through the point (2, 8).
 Identify h, k , x and y
 h = -1, k = 2 these are from the vertex
 x = 2, y = 8 these are from the other point
 Plug in what you know
 y = a(x - h)2 + k
 8 = a(2 – (-1))2 + 2
 Now solve for a
 8 = 9a + 2
 6 = 9a
 6/9 = a or a = ⅔

39. Write the Standard Form of the equation
with a Vertex at (-1,2) and goes
through the point (2, 8).
 Identify h, k , x and y
 h = -1, k = 2 these are from the vertex
 x = 2, y = 8 these are from the other point
 Plug in what you know
 y = a(x - h)2 + k
 8 = a(2 – (-1))2 + 2
 Now solve for a
 8 = 9a + 2
 6 = 9a
 6/9 = a or a = ⅔
 Write the answer in Standard Form, plugging in a h and k
 y = ⅔(x + 1)2 + 2