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Pre-Calculus

Quarter 1

DO_Q1_ Pre-Calculus_Module 1

ii

Pre-Calculus

Alternative Delivery Mode

Quarter 1 – Module 1

First Edition, 2020

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Published by the Department of Education

Secretary: Leonor Magtolis Briones

Undersecretary: Diosdado M. San Antonio

Printed in the Philippines by ________________________

Department of Education – National Capital Region

Office Address: Pio Valenzuela St., Marulas, Valenzuela City

Telefax: (02) 292 – 3247

E-mail Address: [email protected]

Development Team of the Module

Writers: CHARMAINE F. ALIDO, Gen. T. De Leon NHS, SDO – Valenzuela

ALEJANDRO B. FAJARDO, Vicente P. Trinidad NHS, SDO – Valenzuela GRETCHEN MAE G.

GEMENTIZA, Valenzuela NHS, SDO - Valenzuela

RODEL D. ROJAS, Bignay National High School, SDO – Valenzuela

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Illustrator:

Layout Artist: FRANCISCO P. FRONDA III

Management Team:

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FILMORE A. CABALLERO, CID Chief

JEAN A. TROPEL, Division EPS In-Charge of LRMS

MARILYN B. SORIANO, Division Mathematics Coordinator

Choose the letter of the best answer.

____1. What do you call to the locus of a point which moves at a constant distance

from a fixed point called its center?

A. Circle B. Ellipse C. Hyperbola D. Parabola

____2. It is the intersection of a plane and a double-napped cone.

A. circle B. parabola C. ellipse D. conic

____3 Which of the following is the standard form of the circle?

A. (x – h)

2

• (y – k)

2

= r

2

C.

𝑥

2

𝑎

2

𝑦

2

𝑏

2

B. (x – h)

2

= 4c(y – k) D.

𝑥

2

𝑎

2

𝑦

2

𝑏

2

____4. In the standard form of equation of a circle, (x - h)

2

• (y - k)

2

= r

2

, (h, k) is

known as __________.

A. center B. vertex C. directrix D. focus

For nos. 5-6. Given: Point (1, 4) is on a circle whose center is at (-2, - 3).

____5. What is the radius of the circle?

A. √ 58 B. 58 C. 7.61 D. 2√ 39

____6. What is the standard form of the equation of the circle?

A. (x - 2)

2

• (y - 3)

2

= 58 C. (x - 3)

2

• (y - 2)

2

= √

B. (x + 2)

2

• (y + 3)

2

= √58 D. (x + 2)

2

• (y + 3)

2

= 58

____7. Find the standard form of a circle having a center at (0,0) and passing

through the point ( 3, - 4)

A. 𝑥

2

2

= 1 C. 𝑥

2

2

B. 𝑥

2

2

= 9 D. 𝑥

2

2

____8. What is the radius of the equation in number 18?

A. 1 B. 2 C. 4 D. 5

____9. What is the center of this equation of a circle (x – 3)

2

• y

2

A. (0, 3) B. (3,0) C. (0,-3) D. (-3,0)

____10. Find the radius of the equation given in number 9.

A. 1 B. 3 C. 4 D. 16

What I Know?

DO_Q1_ Pre-Calculus_Module 1

Lesson

Circles

Expected Learning Outcome/s: (LC Codes: STEM_PC11AG-Ia- 1 - 3 )

1. Illustrate the different types of conic sections.
2. Define a circle.
3. Graph a circle given an equation in center – radius form.

The different types of conic sections are circle, parabola, ellipse, hyperbola,

and degenerate cases. If a plane is made to cut a right circular cone, the section

obtained is called a conic section , or simply a conic.

The shape of the conic will depend on the position of the cutting plane. If the

cutting plane is parallel to the base of the cone, we have a circle. If the cutting plane

intersects only one cone to form an unbounded curve, the section is called parabola.

If the cutting plane intersects only one cone to form a bounded curve, the section is

called an ellipse and if the cutting plane intersects both nappes or sheets, the conic

section is called a hyperbola (Figure 1.1).

Other exceptional types of conic sections called degenerate conics are the

point, two coincident lines (one line) and two intersecting lines (two lines). These are

obtained when the cutting plane passes through the vertex (intersection of two cones).

See Figures 1.2, 1.3 and 1.4.

What I Need to Know?

What’s In?

DO_Q1_ Pre-Calculus_Module 1

Finding the Equation of a Circle

Give the standard form of the circle satisfying the following given conditions.

1. Center at the origin, radius 3

units

a. Find the center

b. Find the radius

c. Find the equation of the

circle

Reasons

a. C(0,0) center at the origin

b. r = 3 units given

c. x

2

• y

2

= r

2

formula

x

2

• y

2

2

substitution

x

2

+ y

2

= 9 simplify

2. Center (-3,6), radius 6 units

a. Find the center

b. Find the radius

c. Find the equation of the

circle

a. C(-3,6) given

b. r = 6 units given

c. (x - h)

2

+(y - k)

2

= r

2

formula

[x-(-3)]

2

+(y - 6)

2

2

substitution

(x+3)

2

+ (y-6)

2

= 36 simplify

3. Center (4,5), tangent to y-axis

a. Find the center

b. Find the radius

c. Find the equation of the

circle

a. C(4,5) given

b. r = 4 units using the distance

formula

D=√(𝑥

2

1

2

2

1

2

Distance from A(4,5) to B(0,5)

c. (x-h)

2

+(y-k)

2

= r

2

formula

(x – 4)

2

+(y - 5)

2

= 4

2

substitution

(x - 4)

2

+(y - 5)

2

= 16 simplify

4. Has a diameter with endpoints

A(4,4) and B(-2,-4)

a. Find the center

b. Find the radius

c. Find the equation of the

circle

a. C(1,0) using Midpoint formula

=

( 𝑥

1

+𝑥

2

) ,

2

( 𝑦

1

+𝑦

2

)

2

b. r = 5 units using the distance

formula

1

2

2

1

2

2

1

2

(half of the diameter)

c. (x-h)

2

+(y-k)

2

= r

2

formula

(x – 1)

2

+(y-0)

2

2

substitution

(x - 1)

2

+ y

2

= 25 simplify

What’s More?

DO_Q1_ Pre-Calculus_Module 1

❖ Circle - is the locus of a point which moves at a constant distance from a

fixed point called its center.

❖ The constant distance of any point from the center is called the radius.

❖ The equation of the circle centered at the origin and r >0 is x

2

+ y

2

= r

2

.

❖ The equation of a circle centered at (h,k) and radius is r >0 is

(x-h)

2

+(y-k)

2

= r

2

Give the standard form of the circle satisfying the following given

conditions.

1. Center at the origin, radius 7 units

a. Find the center

b. Find the radius

c. Find the equation of the circle

2. Center (2,-3), radius 4 units

a. Find the center

b. Find the radius

c. Find the equation of the circle

3. Center (- 3,- 4), tangent to x-axis

a. Find the center

b. Find the radius

c. Find the equation of the circle

4. Has a diameter with endpoints

A(-1,2) and B(5,6)

a. Find the center

b. Find the radius

c. Find the equation of the circle

5. Given the illustration below

a. ______________

b. ______________

c. ______________

a. ______________

b. ______________

c. _____________

(Hint: Use distance formula in

finding the radius)

a. ______________

b. ______________

c. ______________

(Hint: Use midpoint formula in

finding the center of the circle and

distance formula in finding the

length of the diameter. Radius is

1

2

a. ______________

b. ______________

c. ______________

a. ______________

b. ______________

c. ______________

Short Answer. Supply the correct answer.

What I Have Learned?

What I Can Do?

DO_Q1_ Pre-Calculus_Module 1

Choose the letter of the best answer.

____1. What do you call a curve that has points which are equidistant from the

fixed point and the given line?

A. Circle B. Ellipse C. Hyperbola D. Parabola

____2. This is the length of space from the focus to the vertex. It is also the distance

between the directrix line and the vertex.

A. Latus Rectum C. Focal Distance

B. Directrix line D. Axis of Symmetry

____3. Which of the following is the vertex form of parabola that opens to the left?

A. (y – k)

2

= - 4c(x – h) C. (x – h)

2

= - 4c(y – k)

B. (y – k)

2

= 4c(x – h) D (x – h)

2

= 4c(y – k)

For numbers 4 - 10. Given the equation of parabola : 𝑥

2

____4. What is the standard equation of the parabola?

A. (x – 2)

2

= - 4 (y – 3) C. (x – 2)

2

= 4 (y – 3)

B. (x + 2)

2

= - 4 (y + 3) D. (x + 2)

2

= 4 (y + 3)

____5. What is the graph of this parabola?

A. opens upward C. opens downward

B. opens to the right D. opens to the left

____6. Solve the measure of the focal distance.

A.

1

2

B.

3

4

C. 1 D. 8

____7. Find the length of latus rectum?

A. 2 B. 4 C. 6 D. 8

____8. What is the coordinate of the vertex?

A. (2, 3) B. (-2, 3) C. (-2, - 3) D. (2, - 3)

____9. What is the coordinate of the focus?

A. (2, 2) B. (-2, 2) C. (-2, - 2) D. (2, - 2)

____10. Which are the endpoints of the latus rectum?

A. (2,0) , (2, 4) B. (0,2), (4,2) C. (-2,0) , (-2, 4) D. (0,-2), (4,-2)

What I Know?

DO_Q1_ Pre-Calculus_Module 1

Lesson

Parabolas

Expected Learning Outcomes:

(LC Codes: STEM_PC11AG-Ia-5, STEM_PC11AG-Ib- 1 )

1. Define a parabola.

2. Determine the standard form of equation of a parabola.

“FOLLOWING DIRECTIONS”

1. Graph the following ordered pairs in a

rectangular coordinate plane:

A(0, 0), B(0, 2), C(4, 2), D(-4,2)

2. Draw a curve that passes through the

points of A, C, and D

3. Draw a line that passes through the

points of A and B

4. Draw a horizontal line which passes through

the equation 𝑦 = − 2. Name it as 𝑙𝑖𝑛𝑒 𝑚

5. Draw a line segment connecting the points of B and C.
6. From point C, draw a line segment perpendicular to a line in number 3. Name

the line segment as 𝑙.

Guide Questions:

1. Describe the graph drawn in number 2?
2. What is the distance of point A from point B?
3. Describe the distance from point B to C and the distance from point B to D.
4. What is the length of line segment BC(𝐵𝐶)?
5. What is the length of line segment 𝑙?
6. Describe the length of 𝐵𝐶 and line segment 𝑙?

From the activity above, Let B be a given

point, and 𝑙𝑖𝑛𝑒 𝑚, a given line not containing B.

The set of all points P such that its distances from B

What I Need to Know?

What’s In?

What’s New?

DO_Q1_ Pre-Calculus_Module 1

2. Directrix line equation 𝑦– 6 = 0 ,

vertex at ( 0 , 5 )

a. List down all the given

b. Identify the graph of parabola

c. Substitute the needed values to

the equation of parabola

d. simplify

3. Vertex (1, - 4) and 𝐿𝑅

1

a. List down all the given

b. Identify the graph of parabola

c. Substitute the needed values to

the equation of parabola

d. simplify

a. 𝑉( 0 , 5 ) 𝐷𝐿: 𝑦 = 6

b. (𝑥ℎ)

2

= − 4 𝑐(𝑦 − 𝑘) → opens down

c. (𝑥 − 0 )

2

d. 𝑥

2

a. 𝑉( 1 , − 4 ) 𝐿𝑅

1

b. (𝑦 − ℎ)

2

= 4 𝑐(𝑥 − ℎ) → opens right

c. (𝑦 − (− 4 ))

2

d. (𝑦 + 4 )

2

To graph a parabola in a rectangular coordinate system given its standard form

equation,

1. Determine which of the standard forms applies to the given equation:

2

= ± 4 𝑐(𝑥 − ℎ) or (𝑥 − ℎ)

2

2. Use the standard form identified in Step 1 to determine the axis of symmetry,

focus, equation of the directrix, and endpoints of the latus rectum.

a. If the equation is in the form (𝑦 − ℎ)

2

= ± 4 𝑐(𝑥 − ℎ), then

• the axis of symmetry is 𝑦 = 𝑘
• set 4 c equal to the coefficient of x in the given equation to solve for c. If

𝑐 > 0 , the parabola opens right. If 𝑐 < 0 , the parabola opens left.

• use c to find the coordinates of the focus,

• use c to find the equation of the directrix, 𝑥 = ℎ ± 𝑐
• use c to find the endpoints of the latus rectum. For a graph that opens

left,use 𝐿𝑅

1

(h - c, k - 2c) and 𝐿𝑅

2

(h - c, k + 2c). For a graph that opens

right, 𝐿𝑅

1

(h +c, k - 2c) and 𝐿𝑅

2

(h +c, k + 2c)

b. If the equation is in the form (𝑥 − ℎ)

2

= ± 4 𝑐(𝑦 − 𝑘), then

• the axis of symmetry is 𝑥 = ℎ
• set 4 c equal to the coefficient of y in the given equation to solve for c. If

𝑐 > 0 , the parabola opens up. If 𝑐 < 0 , the parabola opens down.

• use c to find the coordinates of the focus, (ℎ , 𝑘 ± 𝑐)
• use c to find equation of the directrix, 𝑦 = 𝑘 ± 𝑐
• use c to find the endpoints of the latus rectum. For a graph that opens

down, use 𝐿𝑅

1

(h - 2c, k - c) and 𝐿𝑅

2

(h + 2c, k - c). For a graph that opens

up, 𝐿𝑅

1

(h - 2c, k + c) and 𝐿𝑅

2

(h + 2c, k + c)

3. Plot the focus, directrix, and latus rectum, and draw a smooth curve to form the

parabola.

What I Have Learned?

DO_Q1_ Pre-Calculus_Module 1

OPENS DOWN

● Standard Form: (x – h)

2

= - 4c(y – k)

● Vertex: V (h, k) ● Focus: F (h, k-c)

● Directrix Line: y = k + c ● Endpoints of LR:

● Axis of Symmetry: x = h 𝐿𝑅 1

(h-2c, k-c)

𝐿𝑅

2

(h+2c, k-c)

OPENS UP

● Standard Form: (x – h)

2

= 4c(y – k)

● Vertex: V (h, k) ● Focus: F (h, k+c)

● Directrix Line: y = k - c ● Endpoints of LR:

● Axis of Symmetry: x = h 𝐿𝑅

1

(h-2c, k+c)

𝐿𝑅

2

(h+2c, k+c)

OPENS LEFT

● Standard Form: (y – k)

2

= - 4c(x – h)

● Vertex: V (h, k) ● Focus: F (h - c, k)

● Directrix Line: x = h + c ● Endpoints of LR:

● Axis of Symmetry: y = k 𝐿𝑅

1

(h - c, k - 2c)

𝐿𝑅 2

(h - c, k + 2c)

OPENS RIGHT

● Standard Form: (y – k)

2

= 4c(x – h)

● Vertex: V (h, k) ● Focus: F (h + c, k)

● Directrix Line: x = h - c ● Endpoints of LR:

● Axis of Symmetry: y = k 𝐿𝑅

1

(h +c, k - 2c)

𝐿𝑅

2

(h +c, k + 2c)

Solving Applied Problem Involving Parabola

A cross-section of a design for a

travel-sized solar fire starter is shown in the

figure. The sun’s rays reflect off the parabolic

mirror toward an object attached to the igniter.

Because the igniter is located at the focus of the

parabola, the reflected rays cause the object to

burn in just seconds.

DO_Q1_ Pre-Calculus_Module 1

____4. Given the condition in number 3, where is the major axis located at?

A. at the origin C. along the x-axis

B. along the y-axis D. (5, - 3)

____5. Find the standard equation of the given equation in 3.

A.

𝑥

2

25

𝑦

2

16

= 1 B.

𝑥

2

25

𝑦

2

16

= 1 C.

𝑥

2

16

𝑦

2

25

= 1 D.

𝑥

2

16

𝑦

2

25

For numbers 6-10. Given the equation of ellipse : 25 𝑥

2

2

____6. Find the standard equation.

A.

𝑥

2

9

𝑦

2

25

= 1 B.

𝑥

2

9

𝑦

2

25

= 1 C.

𝑥

2

25

𝑦

2

9

= 1 D.

𝑥

2

25

𝑦

2

9

____7. Find the coordinates of the vertices.

A. (0, 5), (0, - 5) B. (5, 0),(-5, 0) C. (0, 4), (0, - 4) D. (4, 0), (-4,0)

____8. Find the coordinates of the foci.

A. (0, 5), (0, - 5) B. (5, 0),(-5, 0) C. (0, 4), (0, - 4) D. (4, 0), (-4,0)

____9. Find the endpoints of the covertices.

A. (0, - 3) (0, 3) B. (-3, 0) (3, 0) C. (0, - 6) (0, 6) D. (-6, 0) (6, 0)

____10. Where is the major axis located?

A. at the origin C. along the x-axis

B. along the y-axis D. undefined

Lesson

Ellipse

Expected Learning Outcome/s: (LC Codes: STEM_PC11AG-Ic- 1 - 2 )

1. Define an ellipse.
2. Graph an ellipse given an equation in standard form.

Unlike circle and parabola, an ellipse is one of the conic sections that most

students have not encountered formally before. Its shape is a bounded curve

which looks like a flattened circle. The orbits of the planets in our solar system

around the sun happen to be elliptical in shape ( see Figure 1).

What I Need to Know?

What’s In?

Figure 1 Source: https://courses.lumenlearning.com/suny-earthscience/chapter/introduction-to-the-solar-system/

DO_Q1_ Pre-Calculus_Module 1

Also, just like parabolas, ellipses have reflective properties that have been

used in the construction of certain structures. These applications and more will

be encountered in this lesson.

An ellipse is the set all points in a plane the sum of whose distances from two

fixed points in the plane is constant. The fixed points are called the foci (plural of

focus) of the ellipse, and the line through them is sometimes called the focal axis.

The point on the focal axis midway between the foci is the center , and the points

where the ellipse crosses its focal axis are called the vertices. The line segment

joining the two vertices is called the major axis ; the line segment through the center,

perpendicular to the major axis, and terminating at the ellipse is called the minor

axis. The eccentricity of a curve tells us the amount of roundness of that curve.

Now, the eccentricity of the circle is 0; that of the parabola is 1, while the eccentricity

of the ellipse is between 0 and 1, that is 0 < 𝑒 < 1. A small eccentricity indicates that

the ellipse tends toward being circular, whereas an eccentricity close to 1 indicates

that the ellipse is elongated.

What’s New?

DO_Q1_ Pre-Calculus_Module 1

(𝑥−ℎ )

2

𝑎

2

(𝑦−𝑘 )

2

𝑏

2

(𝑥− 0 )

2

(− 4 )

2

(𝑦− 0 )

2

( 2. 65 )

2

THE ELLIPSE

ELLIPSE - the set all points in a plane the sum of whose distances from two fixed

points in the plane is constant.

FOCI - (plural of focus) fixed points of the ellipse.

FOCAL AXIS - line through the foci.

CENTER - The point on the focal axis midway between the foci.

VERTICES - the points where the ellipse crosses its focal axis.

MAJOR AXIS - The line segment joining the two vertices.

MINOR AXIS - line segment through the center, perpendicular to the major axis, and

terminating at the ellipse.

ECCENTRICITY - tells us the amount of roundness of a curve.

ECCENTRICITY OF THE ELLIPSE - between 0 and 1, ( 0 < 𝑒 < 1 ). Small eccentricity

indicates that the ellipse tends toward being circular. If close to 1 indicates that the

ellipse is elongated.

Summary:

NOTE : c

2

= a

2

- b

2

HORIZONTAL MAJOR AXIS VERTICAL MAJOR

AXIS

Standard Form

(𝑥−ℎ )

2

𝑎

2

(𝑦−𝑘 )

2

𝑏

2

(𝑥−ℎ )

2

𝑏

2

(𝑦−𝑘 )

2

𝑎

2

Foci

1

( h – c, k) 𝐹

1

( h, k - c)

2

( h + c, k) 𝐹

2

( h, k + c)

Vertices

1

( h – a, k) 𝑉

1

( h, k - a)

2

( h + a, k) 𝑉

2

( h, k + a)

Co-vertices

1

( h, k - b) 𝑊

1

( h – b, k)

2

( h, k + b) 𝑊

2

( h + b, k)

Let us now apply the concept of ellipse to some situational

problems.

A tunnel has the shape of a semi ellipse that is 15 ft high at the

center, and 36 ft across at the base. At most how high should a passing

truck be, if it is 12 ft wide, for it to be able to fit through the tunnel?

Round off your answer to two decimal places.

What I Can Do?

DO_Q1_ Pre-Calculus_Module 1

Solution. Refer to the figure above. If we draw the semi ellipse on

rectangular coordinate system, with its center at the origin, an equation

of the ellipse which contains it, is

2

2

2

2

To maximize its height, the corners of the truck, as shown in the

figure, would have to just touch the ellipse. Since the truck is 12 ft wide,

let the point (6, n) be the corner of the truck in the first quadrant, where

n>0, is the (maximum)height of the truck. Since this point is on the

ellipse, it should fit the equation.

Thus, we have

2

2

2

2

2

2

2

2

Give the coordinate of the center, vertices, covertices, and foci of

the ellipse with the given equation.

𝑥

2

169

𝑦

2

25

2

𝑥

2

144

𝑦

2

169

2

2

(𝑥+ 7 )

2

16

(𝑦− 2

) 2

25

2

2

Assessment

DO_Q1_ Pre-Calculus_Module 1