1. The Beginning of Chapter 8:
Conic Sections (8.1a)
Parabolas!!!

2. V Imagine two non-perpendicular lines intersecting at point V.
Rotating one of the lines (the
generator) around the other (the
axis) yields a pair of right
circular cones…
A conic section is formed by the
intersection of a plane and these
cones. V
Basics: Parabola, Ellipse, Hyperbola
Degenerates: Point, Line, Intersecting
Lines
Check out the diagram on p.631…

3. Conic Sections All conic sections can be defined algebraically as the
graphs of second-degree (quadratic) equations
in two variables…in the form:
where A, B, and C are not all zero.
Today, we focus on parabolas!!!

4. Definition: Parabola
A parabola is the set of all points in a plane equidistant from a
particular line (the directrix) and a particular point (the focus)
in the plane.
Point on the parabola
Axis
Dist. to
focus
Dist. to
directrix
Focus
Vertex
Directrix

5. Let’s equate these two distances:
Deriving the equation of a parabola
Focus F(0, p) p
P(x, y) p
Let’s equate these
two distances:
Directrix: y = –p
D(x, –p)

6. Deriving the equation of a parabola
Standard form of the equation of an up- or down-opening
parabola. If p > 0, it opens up, if p < 0, it opens down.

7. Deriving the equation of a parabola
Focus F(0, p) p
P(x, y) p
Directrix: y = –p
D(x, –p)
The value p is the focal length of the parabola.
A segment with endpoints on a parabola is a chord.
The value |4p| is the focal width.

8. Deriving the equation of a parabola
Focus F(0, p) p
P(x, y) p
Directrix: y = –p
D(x, –p)
Parabolas that open right or left are inverse relations of the
upward or downward opening parabolas…standard form:

9. Parabolas with Vertex (0, 0)
Standard Equation
Opens
Upward or
downward
To the right or
to the left
Focus
Directrix
Axis
y-axis
x-axis
Focal Length
Focal Width

10. Guided Practice Focus: (0, –1/2), Directrix: y = 1/2, Focal Width: 2
Find the focus, the directrix, and the focal width of the given
parabola. Then, graph the parabola by hand.
Focus: (0, –1/2),
Directrix: y = 1/2,
Focal Width: 2

11. Guided Practice
Find an equation in standard form for the parabola whose
directrix is the line x = 2 and whose focus is the point (–2, 0).
Would a graph help???
y = –8x 2

12. Guided Practice
Find an equation in standard form for the parabola whose
vertex is (0, 0), opens downward, and has a focal width of 4.
Would a graph help??? 2
x = – 4y

13. Whiteboard Practice … Standard Form:
Find an equation in standard form for the parabola with vertex
(0, 0), opening upward, with focal width = 3.
(since parabola
opens upward)
Standard Form:

14. Translations of Parabolas

15. We have only considered parabolas with the vertex on the
origin………………… what happens when it’s not??? V
F (h, k + p)
(h, k)
F (h + p, k)
V (h, k)
Such translations do not change the focal length, the focal
width, or the direction the parabola opens!!!

16. Parabolas with Vertex (h, k)
Standard Equation
Opens
Upward or downward
Focus
Directrix
Axis
Focal Length
Focal Width

17. Parabolas with Vertex (h, k)
Standard Equation
Opens
To the right or to the left
Focus
Directrix
Axis
Focal Length
Focal Width

18. Practice Problems
Find the standard form of the equation for the parabola with
vertex (3, 4) and focus (5, 4).
Which general equation do we use?
What are the values of h and k?
Now, how do we find p?

19. Practice Problems Use a function grapher to graph the given parabola.
First, we must solve for y!!!
Now, plug these two equations
into your calculator!!!

20. Practice Problems
Prove that the graph of the given equation is a parabola, then
find its vertex, focus, and directrix.
We need to complete the square…
The CTS step!!!
We have h = 2, k = –1,
and p = 6/4 = 1.5
Vertex: (2, –1), Focus: (3.5, –1), Directrix: x = 0.5

21. Practice Problems Standard Form:
Find an equation in standard form for the parabola that satisfies
the given conditions.
Vertex (–3, 3), opens downward, focal width = 20
(since parabola
opens downward)
Standard Form:

22. Practice Problems Standard Form:
Find an equation in standard form for the parabola that satisfies
the given conditions.
Vertex (2, 3), opens to the right, focal width = 5
(since parabola
opens to the right)
Standard Form: