earth science neb

2. Conic Sections:
The parabola

3. Parabola
• Review: The geometric definition relies on a cone and a plane
intersecting it.
• Algebraic definition: All points that are equidistant from a given line
(the directrix)and a fixed point not on the directrix (the focus)

4. Parabola
Focus
Directrix
x
y
Any point on
the parabola is
equidistant to
the focus and
the directrix.
Example:
Point A: d1=d2
Point B: d1=d2
A
d1
d2
d1
d2
B

5. Items referenced on the graph of a
parabola:
Vertex
Focus
Directrix
Axis of Symmetry
x
y

6. Facts: Parabola Equations
• One variable is squared and one is not. (How does this differ from
linear equations?)
• There are many ways the equation of a parabola can be written. We
will get the quadratic part (variable that is squared) on the left of the
equal sign and the linear part (variable is to the first power) on the
right of the equal sign.
• Equation:
(x - h)2 = c(y – k) OR (y - k)2 = c(x – h)

7. (x - h)2 = c(y – k) OR (y - k)2 = c(x – h)
where the vertex is at (h,k) and |c| is the width at the
focus
To graph:
1. Put in standard form (above) – squared term on left
2. Decide which way the parabola opens.
Look at the right side. If y: + c → opens up
If y: - c → opens down
If x: + c → opens right
If x: - c → opens left

8. (x - h)2 = c(y – k) OR (y - k)2 = c(x – h)
where the vertex is at (h,k) and |c| is the width at the
focus
To graph:
3. Plot the vertex (h,k) Note what happens to the signs.
4. Plot the focus: move │¼ c │ from the vertex in the
direction that the parabola opens. Mark with an f.
5. Draw the directrix: │¼ c │ from the vertex in the
opposite direction of the focus (Remember that the
directrix is a line.)

9. (x - h)2 = c(y – k) OR (y - k)2 = c(x – h)
where the vertex is at (h,k) and |c| is the width at the
focus
To graph:
6. Plot the endpoints of the latus rectum/focal chord
(width at the focus). The width is the │c│ at the
focus.
7. Sketch the parabola by going through the vertex
and the endpoints of the latus rectum. (Be sure to
extend the curve and put arrows.)
8. Identify the axis of symmetry. (The line that goes
through the vertex dividing the parabola in half.)

10. Exp. 1: Graph (x - 5)2 = 12(y – 6)
To graph:
1. Put in standard form– squared term on left
Done
2. Decide which way the parabola opens.
Look at the right side. If y: + c → opens up
If y: - c → opens down
If x: + c → opens right
If x: - c → opens left
Up because y is on the right and 12 is positive

11. Exp. 1: Graph (x - 5)2 = 12(y – 6)
To graph:
3. Plot the vertex (h,k) Note what happens to the signs.
(5,6)
4. Plot the focus: move │¼ c │ from the vertex in the
direction that the parabola opens. Mark with an f.
(5,9): found by moving up 3 from the vertex
5. Draw the directrix: │¼ c │ from the vertex in the
opposite direction of the focus (Remember that the
directrix is a line.)
y = 3: found by moving down 3 from the vertex

12. Exp. 1: Graph (x - 5)2 = 12(y – 6)
To graph:
6. Plot the endpoints of the latus rectum/focal chord
(width at the focus). The width is the │c│ at the
focus.
L.R. = 12 with endpoints at (-1,9) & (11,9)
7. Sketch the parabola by going through the vertex
and the endpoints of the latus rectum. (Be sure to
extend the curve and put arrows.)
8. Identify the axis of symmetry. (The line that goes
through the vertex dividing the parabola in half.)
x = 5

13. Exp. 1: Graph (x - 5)2 = 12(y – 6)
Vertex: (5,6)
Focus: (5,9)
Directrix: y = 3
L.R.: 12
Axis: x = 5
f