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
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
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
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
