1. 2/24/2019 5:14 AM
11.3: Parabolas

2. Parabolas
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11.2: Hyperbolas

3. Shape of a parabola from a cone
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11.3: Parabolas

4. Real-Life Examples
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11.3: Parabolas

5. 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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11.2: Hyperbolas

6. Horizontal Axis Standard Form: If the ‘x’ is not being squared:
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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11.3: Parabolas

7. Vertical Axis Standard Form: If the ‘y’ is not being squared:
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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11.3: Parabolas

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

9. Review of Parent Function Parabolas
y = x2 or x2 = y x = y2 or y2 = x
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11.3: Parabolas

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

11. 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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11.3: Parabolas

12. 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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11.3: Parabolas

13. Review of Parent Function Parabolas
y = x2 or x2 = y x = y2 or y2 = x
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11.3: Parabolas

14. Steps in Writing Conic Sections 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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11.3: Parabolas

15. 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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11.3: Parabolas

16. 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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11.3: 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:
Take the coefficient in front of the isolated un-squared variable and multiply it by ¼
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11.3: Parabolas

18. 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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11.3: Parabolas

19. Example 1
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for y = 4x2.
(0, 0)
1/16
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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11.3: Parabolas

20. Example 2
Determine the vertex, focus point, directrix, axis of symmetry, latus rectum, and graph the parabola for x = (–1/8)y2.
(0, 0) –2
Vertex: P:
Focus Point:
Directrix:
Axis of Symmetry:
Latus Rectum:
(–2, 0) F
x = 2
y = 0
(–2, +4)
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11.3: 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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11.3: Parabolas

22. Example 3
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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11.3: 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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11.3: Parabolas

24. Example 4
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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11.3: Parabolas

25. Example 5
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, 0) F
y = 4
x = 7
(5, 0), (9, 0)
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11.3: 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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11.3: Parabolas

27. Example 6
Write an standard form equation of a parabola where the vertex is (–7, –3) and focus point (2, –3). F
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11.3: Parabolas

28. Example 7
Write an standard form equation of a parabola where the vertex is (–2, 1) and the directrix is at x = 1.
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11.3: Parabolas

29. Your Turn
Write an 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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11.3: Parabolas

30. Converting to Standard Form
Identify whether it is an ellipse by using the equation, b2 – 4ac where the answer is equal to zero using the equation, Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
Rearrange variables for x’s and y’s through factoring
Take GCF and add everything to other side
Use completing the square; using what’s added to the x’s and y’s is added to the radius
Identify the coefficient which is raised to the first power and divide the term by 2
Take the second term, divide the term by 2, and square that number
Add to both sides to the equation
Put the equation into factored form
Put it in standard form
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11.3: Parabolas

31. Example 8 Change the equation y2 – 9x + 16y + 64 = 0 to standard form.
11.3: Parabolas

32. Example 8 Change the equation y2 – 9x + 16y + 64 = 0 to standard form.
11.3: Parabolas

33. Example 8 Change the equation y2 – 9x + 16y + 64 = 0 to standard form.
11.3: Parabolas

34. Example 9
Change the equation x2 + 14x – 12y + 97 = 0 to standard form.
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11.3: Parabolas

35. Your Turn Change the equation y2 – 16x – 6y + 73 = 0 to standard form.
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11.3: Parabolas

36. Assignment
Worksheet
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11.3: Parabolas