Parabola

1. Quadratic
Equations and
Parabola

2. Parabola
A parabola is a curve
where any point is at
an equal distance from
a fixed point (the focus
and a fixed straight line
(the directrix).
Source:
http://www.mathsisfun.com/geometry/parabola.html

3. • the directrix and focus
• the axis of symmetry (goes
through the focus, at right
angles to the directrix)
• the vertex (where the
parabola makes its sharpest
turn) is halfway between
the focus and directrix.
Vocabulary
Source:
http://www.mathsisfun.com/geometry/parabola.html

4. Equations
Standard form:
𝑦 = 𝑎𝑥2
+ 𝑏𝑥 + 𝑐
Vertex form
𝑦 = 𝑎 𝑥 − ℎ 2
+ 𝑘

5. Graphing
Steps:
a. Identify the orientation
If a is positive, then the parabola opens
upward.
If a is negative, then the parabola opens
downward.
Example:
𝑦 = 2𝑥2
+ 𝑥 + 1 opens upward
𝑦 = −𝑥2
+ 1 opens downward

6. Graphing
Steps:
b. Find the vertex
1. 𝑦 = 𝑎𝑥2
, vertex: 0,0
2. 𝑦 = 𝑎𝑥2
+ 𝑘, vertex: 0, 𝑘
3. 𝑦 = 𝑎 𝑥 − ℎ 2
, vertex: ℎ, 0
4. 𝑦 = 𝑎 𝑥 − ℎ 2
+ 𝑘, vertex: ℎ, 𝑘

7. Graphing
Steps:
c. Find the 𝑥-intercept
d. Find the 𝑦-intercept
e. Find other coordinates
f. Find the axis of symmetry
g. Plot the points

8. Graphing
a. Since 𝑎 = 1 is positive, the parabola
opens upward.
b. vertex: 0,0
c. 𝑥-intercept
set 𝑦 = 0: 𝑥2
= 0
𝑥 = 0
𝑥-intercept: 0,0
1. 𝑦 = 𝑎𝑥2
Example: 𝑦 = 𝑥2

9. Graphing
1. 𝑦 = 𝑎𝑥2
Example: 𝑦 = 𝑥2
d. 𝑦-intercept
set 𝑥 = 0: y = 𝑥2
y = 0
𝑦-intercept: 0,0
e. Other coordinates
𝑥 −3 −2 −1 0 1 2 3
𝑦 9 4 1 0 1 4 9

10. Graphing
1. 𝑦 = 𝑎𝑥2
Example: 𝑦 = 𝑥2
f. axis of symmetry: 𝑥 = 0
g. plot the points

11. Graphing
1. 𝑦 = 𝑎𝑥2
Example: 𝑦 = 𝑥2
(0,0)
(1,1)(-1,1)
(2,4)(-2,4)
(3,9)(-3,9)

12. Graphing
a. Since 𝑎 = −2 is negative, the parabola
opens downward.
b. vertex: 0,0
c. 𝑥-intercept
set 𝑦 = 0: −2𝑥2
= 0
𝑥 = 0
𝑥-intercept: 0,0
1. 𝑦 = 𝑎𝑥2
Example: 𝑦 = −2𝑥2

13. Graphing
1. 𝑦 = 𝑎𝑥2
Example: 𝑦 = −2𝑥2
d. 𝑦-intercept
set 𝑥 = 0: y = −2𝑥2
y = 0
𝑦-intercept: 0,0
e. Other coordinates
𝑥 −3 −2 −1 0 1 2 3
𝑦 −18 −8 −2 0 −2 −8 −18

14. Graphing
1. 𝑦 = 𝑎𝑥2
Example: 𝑦 = −2𝑥2
f. axis of symmetry: 𝑥 = 0
g. plot the points

15. Graphing
1. 𝑦 = 𝑎𝑥2
Example: 𝑦 = −2𝑥2
(0,0)
(1,-2)
(-3,-18)
(2,-8)(-2,-8)
(3,-18)
(-3,-2)

16. Graphing
a. Since 𝑎 = 2 is positive, the parabola opens
upward.
b. vertex: 0,1
c. 𝑥-intercept
set 𝑦 = 0: 𝑥2
= −
1
2
𝑥 =
1
2
𝑖
𝑥-intercept: none
1. 𝑦 = 𝑎𝑥2
+ 𝑘
Example: 𝑦 = 2𝑥2
+ 1

17. Graphing
1. 𝑦 = 𝑎𝑥2
+k
Example: 𝑦 = 2𝑥2
+ 1
d. 𝑦-intercept
set 𝑥 = 0: y = 2 0 2
+ 1
y = 1
𝑦-intercept: 0,1
e. Other coordinates
𝑥 −3 −2 −1 0 1 2 3
𝑦 19 9 3 1 3 9 19

18. Graphing
1. 𝑦 = 𝑎𝑥2
+ 𝑘
Example: 𝑦 = 2𝑥2
+ 1
f. axis of symmetry: 𝑥 = 0
g. plot the points

19. Graphing
1. 𝑦 = 𝑎𝑥2
+ 𝑘
Example: 𝑦 = 2𝑥2
+ 1
(0, 1)
(2, 9)(-2, 9)
(1, 3)(-1, 3)

20. Graphing
a. Since 𝑎 = 1 is positive, the parabola opens
upward.
b. vertex: 1,0
c. 𝑥-intercept
set 𝑦 = 0: 𝑥 − 1 2
= 0
𝑥 = 1
𝑥-intercept: 1,0
1. 𝑦 = 𝑎 𝑥 − ℎ 2
Example: 𝑦 = 𝑥 − 1 2

21. Graphing
d. 𝑦-intercept
set 𝑥 = 0: y == 0 − 1 2
y = 1
𝑦-intercept: 0,1
e. Other coordinates
𝑥 −2 −1 0 1 2 3 4
𝑦 9 4 1 0 1 4 9
1. 𝑦 = 𝑎 𝑥 − ℎ 2
Example: 𝑦 = 𝑥 − 1 2

22. Graphing
f. axis of symmetry: 𝑥 = 1
g. plot the points
1. 𝑦 = 𝑎 𝑥 − ℎ 2
Example: 𝑦 = 𝑥 − 1 2

23. Graphing
(1, 0)
(3, 4)(-1, 4)
(2, 1)(0, 1)
(4, 9)(-2, 9)
1. 𝑦 = 𝑎 𝑥 − ℎ 2
Example: 𝑦 = 𝑥 − 1 2