1. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
parabolas
Learning Outcomes of the Lesson:
At the end of the lesson, the learners are able to:
1. define a parabola;
2. determine the standard equation of a parabola; and
3. solve situational problems involving parabolas.
2. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Definition and Equation
of a Parabola
Definition and Equation
of a Parabola
Definition and Equation
of a Parabola
3. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Definition
Let 𝐹 be a given point, and 𝑙 be a given line not containing 𝐹. The set
of all points 𝑃 such that its distances from 𝐹 and from 𝑙 are the same,
is called a parabola. The point 𝐹 is its focus and the line 𝑙 its
directrix.
𝑥
𝑦
1 2 3 4 5 6
−6 −5 −4 −3 −2 −1
−1
−2
1
2
3
4
5
6
𝐹 0,2 𝐴 4,2
𝐵 −8,8
𝑃
𝑃𝑙
𝐴𝑙 4, −2
𝐵𝑙 −8, −2
𝑙: 𝑦 = −2
4. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
focus
directrix
axis of symmetry
latus rectum
vertex
5. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
opens upward
opens downward
opens to the roght
opens to the left
6. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Opens Upward
Opens Downward
Opens to the Right
Opens to the Left
𝑥 − ℎ 2 = 4𝑐 𝑦 − 𝑘
𝑥 − ℎ 2 = −4𝑐 𝑦 − 𝑘
𝑦 − 𝑘 2
= 4𝑐 𝑥 − ℎ
𝑦 − 𝑘 2 = −4𝑐 𝑥 − ℎ
ℎ, 𝑘
ℎ, 𝑘
ℎ, 𝑘
ℎ, 𝑘
ℎ, 𝑘 + 𝑐
ℎ, 𝑘 − 𝑐
ℎ + 𝑐, 𝑘
ℎ − 𝑐, 𝑘
𝑦 = 𝑘 − 𝑐
𝑦 = 𝑘 + 𝑐
𝑥 = ℎ − 𝑐
𝑥 = ℎ + 𝑐
𝑥 = ℎ
𝑥 = ℎ
𝑦 = 𝑘
𝑦 = 𝑘
ℎ − 2𝑐, 𝑘 + 𝑐 ,
ℎ + 2𝑐, 𝑘 + 𝑐
ℎ − 2𝑐, 𝑘 − 𝑐 ,
ℎ + 2𝑐, 𝑘 − 𝑐
ℎ + 𝑐, 𝑘 − 2𝑐 ,
ℎ + 𝑐, 𝑘 + 2𝑐
ℎ − 𝑐, 𝑘 − 2𝑐 ,
ℎ − 𝑐, 𝑘 + 2𝑐
ORIENTATION EQUATION VERTEX FOCUS DIRECTRIX AOS
ENDPOINTS OF
LATUS RECTUM
𝑉 ℎ, 𝑘
𝐹 ℎ, 𝑘 + 𝑐
𝑃 𝑥, 𝑦
𝑙: 𝑦 = 𝑘 − 𝑐
𝑃𝑙 𝑥, 𝑘 − 𝑐
𝑐
𝑐
Consider a parabola opening upward with vertex
𝑉 ℎ, 𝑘 . Let the focus 𝐹 and directrix 𝑙 be 𝑐 units
away from the vertex. Hence, 𝐹 ℎ, 𝑘 + 𝑐 and
directrix 𝑙 has an equation 𝑦 = 𝑘 − 𝑐. Let 𝑃 𝑥, 𝑦 be
any point on the parabola. Then
𝑥 − ℎ 2
= 4𝑐 𝑦 − 𝑘
7. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Opens Upward
Opens Downward
Opens to the Right
Opens to the Left
𝑥 − ℎ 2 = 4𝑐 𝑦 − 𝑘
𝑥 − ℎ 2
= −4𝑐 𝑦 − 𝑘
𝑦 − 𝑘 2
= 4𝑐 𝑥 − ℎ
𝑦 − 𝑘 2 = −4𝑐 𝑥 − ℎ
ℎ, 𝑘
ℎ, 𝑘
ℎ, 𝑘
ℎ, 𝑘
ℎ, 𝑘 + 𝑐
ℎ, 𝑘 − 𝑐
ℎ + 𝑐, 𝑘
ℎ − 𝑐, 𝑘
𝑦 = 𝑘 − 𝑐
𝑦 = 𝑘 + 𝑐
𝑥 = ℎ − 𝑐
𝑥 = ℎ + 𝑐
𝑥 = ℎ
𝑥 = ℎ
𝑦 = 𝑘
𝑦 = 𝑘
ℎ − 2𝑐, 𝑘 + 𝑐 ,
ℎ + 2𝑐, 𝑘 + 𝑐
ℎ − 2𝑐, 𝑘 − 𝑐 ,
ℎ + 2𝑐, 𝑘 − 𝑐
ℎ + 𝑐, 𝑘 − 2𝑐 ,
ℎ + 𝑐, 𝑘 + 2𝑐
ℎ − 𝑐, 𝑘 − 2𝑐 ,
ℎ − 𝑐, 𝑘 + 2𝑐
ORIENTATION EQUATION VERTEX FOCUS DIRECTRIX AOS
ENDPOINTS OF
LATUS RECTUM
VERTEX
the point midway the
focus and directrix
FOCUS AND DIRECTRIX
𝑐 units away from the
vertex
AXIS OF SYMMETRY
line passing through the
vertex and focus
LATUS RECTUM
chord perpendicular to the axis of
symmetry passing through the focus
with its endpoints on the parabola. Its
length is 4𝑐.
8. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Let’s Practice!
Find the standard equation of the parabola which satisfies the given
conditions.
1. Vertex (1, −9), focus −3, −9
2. Focus (7,11), directrix 𝑥 = 1
3. Vertex (−5, −7), vertical axis of symmetry, through point 𝑃(7,11)
9. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Let’s Practice!
1. Vertex (1, −9), focus −3, −9
Solution
2 4 6 8
−8 −6 −4 −2
2
4
−4
−2
−8
−6
−12
−10
𝑥
𝑦
𝑉 1, −9
𝐹 −3, −9
10. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Let’s Practice!
2. Focus (7,11), directrix 𝑥 = 1
Solution
2 4 6 8
−8 −6 −4 −2
2
8
10
4
6
𝑥
𝑦
12
𝐹 7,11
1,11
11. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Let’s Practice!
3. Vertex (−5, −7), vertical axis of symmetry, through point 𝑃(7,11)
Solution
2 4 6 8
−8 −6 −4 −2
2
4
−2
𝑥
𝑦
6
8
10
−4
−6
−8
7,11
𝑉 −5,−7
12. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
General Equation of
a Parabola
General Equation of
a Parabola
General Equation of
a Parabola
13. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
𝐴𝑥2
+ 𝐶𝑥 + 𝐷𝑦 + 𝐸 = 0, where 𝐴, 𝐷 ≠ 0 or
𝐵𝑦2
+ 𝐶𝑥 + 𝐷𝑦 + 𝐸 = 0, where 𝐵, 𝐶 ≠ 0
General Equation of a Parabola
Let’s Practice!
Identify the vertex, focus, directrix, axis of symmetry, and endpoints of
the latus rectum of the parabola with the given equation in each item.
Sketch its graph and indicate these points and lines.
1. 3𝑦2 = 24𝑥
2. 𝑥2
+ 6𝑥 + 8𝑦 = 7
3. 16𝑥2 + 72𝑥 − 112𝑦 = −221
14. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Let’s Practice!
1. 3𝑦2 = 24𝑥
20. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Situational Problems
Involving Parabolas
Situational Problems
Involving Parabolas
Situational Problems
Involving Parabolas
21. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Problem 1
A satellite dish has a shape called a paraboloid, where each cross-section is a
parabola. Since radio signals (parallel to the axis) will bounce off the surface of the
dish to the focus, the receiver should be placed at the focus. How far should the
receiver be from the vertex, if the dish is 12 ft across, and 4.5 ft deep at the
vertex?
1 2 3 4 5 6
−6 −5 −4 −3 −2 −1 1
1
2
3
4
5
6
12
4.5
𝑐?
6,4.5
22. JEREMIAH A. ATENTA | Special Science Teacher I | AGUSAN NATIONAL HIGH SCHOOL
Problem 2
The cable of a suspension bridge hangs in the shape of a parabola. The towers
supporting the cable are 400 ft apart and 150 ft high. If the cable, at its lowest, is
30 ft above the bridge at its midpoint, how high is the cable 50 ft away
(horizontally) from either tower?
100 200
−200 −100
100
200
0,30
200,150
400
150
50
50
(150, 𝑦)
?
