1. Worksheet Key
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9.2: Parabolas

2. Worksheet Key
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9.2: Parabolas

3. CONIC SECTIONS: PARABOLAS
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9.2: Parabolas

4. Shape of a parabola from a cone
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9.2: Parabolas

5. Real-Life Examples
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9.2: Parabolas

6. Definitions
Parabola: Graph of a quadratic equation to which a set of points in a plane that are the same distance from a given point
Vertex: Midpoint of the graph; the turning point
Focus: Distance from the vertex; located inside the parabola
Directrix: A fixed line used to define its shape; located outside of the parabola
Axis of Symmetry: A line that divides a plane figure or a graph into congruent reflected halves.
Latus Rectum: A line segment through the foci of the shape in which it is perpendicular through the major axis and endpoints of the ellipse
Eccentricity: Ratio to describe the shape of the conic, e = 1
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9.2: Parabolas

7. Horizontal Axis Standard Form:
Formulas to know:
Horizontal Axis Standard Form:
If the ‘x’ is not being squared:
P is placement point; distance from the focus to the vertex and vertex to the directrix
If p is positive, the parabola opens right
If p is negative, the parabola opens left
Focus Point:
Directrix:
Axis of Symmetry:
Points of Latus Rectum
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9.2: Parabolas

8. Vertical Axis Standard Form:
Formulas to know:
Vertical Axis Standard Form:
If the ‘y’ is not being squared:
P is placement point; distance from the focus to the vertex and vertex to the directrix
If p is positive, the parabola opens up
If p is negative, the parabola opens down
Focus Point:
Directrix:
Axis of Symmetry:
Points of Latus Rectum
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9.2: Parabolas

9. All Standard Form Equations
Formulas to know:
All Standard Form Equations
Center
Length of Latus Rectum
Eccentricity e = 1
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9.2: Parabolas

10. Review of Parent Function Parabolas
y = x2 or x2 = y
x = y2 or y2 = x
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9.2: Parabolas

11. The Parabola Brief Clip
Where do we see the focus point used in real-life?
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9.2: Parabolas

12. Horizontal parabola due to its ‘Axis of Symmetry’
Vertex: (h, k)
Focus point: (p, 0)
Directrix: x = –p
Axis of Symmetry: y = k
Length of Latus Rectum: |4p|
Latus Rectum: (h + p, k + 2p)
Horizontal parabola due to its ‘Axis of Symmetry’
(h, k) F
(p, 0)
y = k
x = –p
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9.2: Parabolas

13. Vertical parabola due to its ‘Axis of Symmetry’
Vertex: (h, k)
Focus point: (0, p)
Directrix: y = –p
Axis of Symmetry: x = k
Length of Latus Rectum: |4p|
Latus Rectum: (h + 2p, k + p)
(p, 0) F
(h, k)
y = –p
Vertical parabola due to its ‘Axis of Symmetry’
x = k
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9.2: Parabolas

14. Review of Parent Function Parabolas
y = x2 or x2 = y
x = y2 or y2 = x
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9.2: Parabolas

15. Steps in Writing Equations of Parabolas
Identify whether the equation opens Up/Down or Left/Right
Divide the coefficient (if necessary) to keep the variable by itself
On the side without the squared into the equation (which usually is a fraction), drop off all the variables
Multiply the coefficient (not involved with squared) with ¼ to solve for p
Put it in standard form and graph
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9.2: Parabolas

16. Example 1
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2.
First, figure out what variable is squared? Put into the suitable equation.
What is the vertex? (0, 0)
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9.2: Parabolas

17. Example 1
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2.
To find p:
Isolate the equation to where the variable squared has a coefficient of 1.
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9.2: Parabolas

18. Example 1
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2.
To find p:
Take the coefficient in front of the isolated un-squared variable and multiply it by ¼
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9.2: Parabolas

19. Example 1
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2.
To determine the Latus Rectum:
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9.2: Parabolas

20. Example 1
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2.
(0, 0)
1/16 Vertical
Vertex: P:
Focus Point:
Directrix:
Axis of Symmetry:
Latus Rectum:
(0, 1/16) F
y = –1/16
x = 0
(+1/8, 1/16)
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9.2: Parabolas

21. Your Turn
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for x = (1/20)y2.
(0, 0) 5
Vertex: P:
Focus Point:
Directrix:
Axis of Symmetry:
Latus Rectum:
(5, 0) F
x = –5
y = 0
(5, +10)
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9.2: Parabolas

22. Example 2
Write in standard form equation of a parabola with the vertex is at the origin and the focus is at (2, 0). F
P = 2
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9.2: Parabolas

23. Your Turn
Write a standard form equation of a parabola where the directrix is y = 6 and focus point (0, –6).
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9.2: Parabolas

24. Example 3
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for (y + 1)2 = 8(x + 1).
(–1, –1) 2
Vertex: P:
Focus Point:
Directrix:
Axis of Symmetry:
Latus Rectum:
(1, –1)
x = –3 F
y = -1
(1, 3), (1, –5)
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9.2: Parabolas

25. Example 4
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for (x – 7)2 = –8(y – 2).
(7, 2) –2
Vertex: P:
Focus Point:
Directrix:
Axis of Symmetry:
Latus Rectum:
(7, –2)
y = 4 F
x = 7
(7, 5), (7, 9)
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9.2: Parabolas

26. Your Turn
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for (y – 1)2 = –4(x – 1).
(1, 1) –1
Vertex: P:
Focus Point:
Directrix:
Axis of Symmetry:
Latus Rectum:
(0, 1) F
x = 2
y = 1
(0, 3), (0, –1)
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9.2: Parabolas

27. Example 5
Write a standard form equation of a parabola where the vertex is (–7, –3) and focus point (2, –3). F
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9.2: Parabolas

28. Example 6
Write a standard form equation of a parabola where the vertex is (–2, 1) and the directrix is at x = 1.
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9.2: Parabolas

29. Your Turn
Write a standard form equation of a parabola where the axis of symmetry is at y = –1, directrix is at x = 2 and the focus point (4, –1).
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9.2: Parabolas

30. Assignment
Worksheet
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9.2: Parabolas