Pre-Calculus Quarter 1 - Module 3 PDF

SENIOR HIGH SCHOOL Pre-Calculus Quarter 1 - Module 3 Definition and Standard Equation of an Ellipse i About the Module This module was designed and written with you in mind. It is here to help you master about Ellipse. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged based on the Most Essential Learning Competencies (MELCs) released by the Department of Education (DepEd) for this school year 2020 – 2021. This module is divided into two lessons, namely: Lesson 1 – Definition of ellipse Lesson 2 – Derivation of the standard equation of ellipse After going through this module, you are expected to: deﬁne an ellipse; and determine the standard form of equation of an ellipse. ii What I Know (Pretest) Instruction: Choose the letter of the correct answers to the following items. Write them on a separate sheet of paper. 1. It is formed when the tilted plane intersects only one cone to form a bounded curve. A. Parabola B. Ellipse C. Circle D. Hyperbola 𝑥2 𝑦2 2. Given the equation of an ellipse 25 + 16=1. Determine the foci of the ellipse. A. 𝐹1 (-3,0) and 𝐹2 (3,0) C. 𝐹1 (4,0) and 𝐹2 (-4,0) B. 𝐹1 (0, -3) and 𝐹2 (0,3) D. 𝐹1 (0, -4) and 𝐹2 (0,4) 3. What is the equation in standard form of the ellipse whose foci are 𝐹1 (−8, 0) and 𝐹2 (8, 0), such that for any point on it, the sum of its distances from the foci is 20? 𝑥2 𝑦2 𝑥2 𝑦2 𝑦2 𝑥2 𝑦2 𝑥2 A. 100 + 36 =1 B. 36 + 100 = 1 C. 36 + 100 = 1 D. 100 + 36 =1 4. What do you call the segment through the center of an ellipse perpendicular to the major axis? A. directrix C. minor axis B. axis of symmetry D. asymptotes 5. If 𝐹1 and 𝐹2 are two disinct points. What do you call the set of all points P, whose distances from 𝐹1 and 𝐹2 add up to a certain constant? A. Parabola B. Ellipse C. Circle D. Hyperbola 6. What do you call the midpoint of both major axis and minor axis of an ellipse? A. Vertex B. Focus C. Center D. Co-vertex 7. Which of the following does NOT have the shape of an ellipse? A. a football C. a whispering gallery B. a satellite and planet orbits D. an olympic swimming pool 8. What is the standard equation of a horizontal ellipse with center (ℎ, 𝑘)? (𝑥−ℎ)2 (𝑦−𝑘)2 (𝑥−𝑘)2 (𝑦−ℎ)2 A. + =1 C. + =1 𝑎2 𝑏2 𝑏 2 𝑎2 (𝑥−ℎ)2 (𝑦−𝑘)2 (𝑥−𝑘)2 (𝑦−ℎ)2 B. + 2 =1 D. + 2 =1 𝑏2 𝑎 𝑎2 𝑏 9. The vertices of an ellipse are (±4,0) and the foci are (±2,0). What is the standard equation? (𝑥−2)2 𝑦+4)2 𝑥2 𝑦2 A. 16 + ( 12 =1 C. 12 + 16 = 1 𝑥2 𝑦2 (𝑥+4)2 (𝑦−2)2 B. 16 + 12 =1 D. 16 + 12 =1 10. What is the center of an ellipse with general equation: 9𝑥 2 +16𝑦 2 + 72𝑥 − 96𝑦 + 144 = 0? A. (3,5) B. (5, -4) C. (-4,3) D. (2, -4) 1 𝑥2 𝑦2 11. What are the vertices of an ellipse with the equation + = 1? 169 25 A. (±13,0) B. (0, ±13) C. (±5,0) D. (0, ±5) 𝑥2 𝑦2 12. What are the foci of an ellipse with the equation 144 + 169 = 1? A. (±13,0) B. (0, ±13) C. (±5,0) D. (0, ±5) 13. What are the co-vertices of an ellipse with standard equation: (𝑥+7)2 (𝑦−4)2 16 + 25 = 1? A. 𝑊1 (−11, 4) and 𝑊2 (−3, 4) C. 𝑊1 (−11,4) and 𝑊2 (−4,0) B. 𝑊1 (−4, 0) and 𝑊2 (−4, 6) D. 𝑊1 (−4, 0) and 𝑊2 (−3,4 14. The arch of a bridge is in the shape of a semi-ellipse, with its major axis at the water level. Suppose the arch is 20 ft high in the middle, and 120 ft across its major axis. How high above the water level is the arch, at a point 20 ft from the center (horizontally)? Round off to 2 decimal places. A. 18.90 ft B. 17.78 ft C. 18.86 ft D. 19.89 ft 15. Which of the following is the graph of an ellipse with equation: (𝑥−1)2 (𝑦−3)2 10 + 11 = 1? A. C. B. D. 2 Lesson Definition of Ellipse 1 What I Need to Know At the end of this lesson, you are expected to: o define ellipse; and o identify various terms that will later be used to describe an ellipse. What’s In In geometry, when the plane intersects only one cone to form a bounded curve it is called an ellipse. Figure 1: Ellipse taken from the cone What’s New Activity 1.1: Drawing an Ellipse-The String Method (20 points) Instruction: Follow the steps below to create an ellipse using the String Method. Materials needed: 1 whole sheet of bond paper (or any paper), string (or thread), pen, pins (thumbtacks), a straight edge and a piece of cardboard (or carton). STEP 1: Using a straight edge, draw the desired length of the ellipse’s Major Axis. STEP 2: Using a straight edge to draw the Minor Axis. It should be perpendicular to the Major Axis and pass through the center point again. STEP 3: Locate focal points 1 and 2 by using half the length of the Major Axis to measure from point C back to the Major Axis. 3 STEP 4: Set pins (thumbtacks, nails, or screws) at points1,2 and C. Then tightly stretch a string around three points and tie the ends together. Observe that the string has 2 lengths, C to point 1 and C to point 2. STEP 5: Keeping the string’s length set, remove the pin from point C and use the string as a guide and pen to draw the ellipse. After following the 5 steps above, answer the question below. 1. What have you observed from the sum of the length of the string as it moves along and form the shape of an ellipse? Rubrics for scoring: Points Description 16-20 Followed the 5 steps and answered the question 11-15 Followed the 5 steps but did not answer the question 6-10 Followed the 3 steps only 1-5 Followed 1 steps 0 Followed NONE of the steps NOTE: For modular class: Insert your output in your portfolio and submit on the scheduled date of collection. For online class: Take a picture of your output and send it to our Google Classroom or messenger for checking. What Is It What is an ELLIPSE? An ellipse is one of the conic sections that most students have not encountered formally before, unlike circles and parabolas. In the deﬁnition of a circle from Module 1, we ﬁxed a point called the center and considered all the points which were a ﬁxed distance from that one point. A circle is an ellipse where both foci are the same point. For our next conic section, the ellipse, we ﬁx two distinct points and a distance to use in our deﬁnition (Stewart, 2012). An ellipse is an oval curve that looks like an elongated circle. More precisely, we have the following deﬁnition. 4 Geometric definition of an Ellipse An ellipse is the set of all points(𝑥, 𝑦) in the plane, the sum of whose distances 𝑑 from two ﬁxed points F1 and F2 is constant. (See Figure 2.) These two ﬁxed points are the foci (plural of focus) of the ellipse (Zeager, 2010). Figure 2: Distance of one point to the Focal points of an ellipse Your previous activity traces out an ellipse because the sum of the distances from the point of the pencil to the foci will always equal to the length of the string, which is constant. The center of the ellipse is the midpoint of the line segment connecting the two foci. The major axis of the ellipse is the line segment connecting two opposite ends of the ellipse which also contains the center and foci. The minor axis of the ellipse is the line segment connecting two opposite ends of the ellipse which contains the center but is perpendicular to the major axis. The vertices of an ellipse are the points of the ellipse which lie on the major axis. Notice that the center is also the midpoint of the major axis, hence it is the midpoint of the vertices. In figure 3 below we have, an ellipse with center 𝐶, foci 𝐹1 and 𝐹2 and vertices 𝑉1 and 𝑉2. Figure 3: Coordinates of Ellipse Ellipses are common in physics, astronomy and engineering. For example, the orbits of the planets in our solar system around the sun happen to be elliptical in shape. The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. Also, just like parabolas, ellipses have reﬂective properties that have been used in the construction of certain structures. 5 What’s More Activity 1.2: NOW IT’S YOUR TURN! Instruction: Based on your own understanding, draw an example of ellipse (vertical or horizontal) and identify the coordinates of your vertices, co-vertices, foci, and center. An example is shown below. Example: Vertices: 𝑉1 (2, −5); 𝑉2 (2,6) 𝑉2 𝑊1 (−2,0.5); 𝑊2 (6, 0.5) 𝐹2 Foci: 𝐹1 (2, −3); 𝐹2 (2,4) 𝑊1 𝐶 𝑊2 Center: (2, 0.5) 𝐹1 𝑉1 What I Need to Remember An ellipse is formed by a plane intersecting a cone at an angle to its base. All ellipses have two focal points, or foci. The sum of the distances from every point on the ellipse to the two foci is constant. All ellipses have a center and a major and minor axis. 6 Lesson Standard Equation of an Ellipse 2 What I Need to Know At the end of this lesson, you are expected to: o determine standard and general equation of an ellipse; and o identify the vertices, foci, major axis and minor axis of an ellipse through a given equation. What’s In An ellipse is a shape that looks like an oval or a flattened circle. Circles are special cases of ellipses, obtained when the cutting plane is perpendicular to the cone's axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant. A circle has one center, called a focus, but an Figure 4: Properties of ellipse in a cone ellipse has two foci. What’s New Consider the points 𝐹1 (3,0) and 𝐹2 (3,0), as shown in Figure 4 below. Using the distance formula, find the sum of the distances of A (4,2.4) from 𝐹1 and from 𝐹2. How about the sum of the distances of B (and C (0,4)) from 𝐹1 and from 𝐹2 ? Figure 5: Points in an Ellipse 7 What Is It Derivation of the Standard Equation of Ellipse According to Garces (2010), in order to obtain the simplest equation for an ellipse, we place the foci on the x-axis at 𝐹1 (−𝑐, 0) and 𝐹2 (𝑐, 0) so that the origin is halfway between them (see Figure 6 on the next page). For later convenience, we let the sum of the distances from a point on the ellipse to the foci 2𝑎. Then if 𝑃(𝑥, 𝑦) is any point on the ellipse, we have 𝑃𝐹1 + 𝑃𝐹2 = 2𝑎. Figure 6: Derivation of the standard equation of an Ellipse When collect here the features of the graph of an ellipse with standard equation 𝑥2 𝑦2 𝑎2 + 𝑏2 = 1 where 𝑎 > 𝑏. Let 𝑐 = √𝑎2 − 𝑏 2. Figure 7: The minor and major axis of an Ellipse 8 (1) center: origin (0 ,0) (2) foci:𝐹1 (−𝑐, 0) and 𝐹2 (𝑐, 0) Each focus is c units away from the center. For any point on the ellipse, the sum of its distances from the foci is 2𝑎. (3) vertices: 𝑉1 (−𝑎, 0) and 𝑉2 (𝑎, 0) The vertices are points on the ellipse, collinear with the center and foci. If 𝑦 = 0, then 𝑥 = ±𝑎. Each vertex is 𝑎 units away from the center. The segment 𝑉1 𝑉2 is called the major axis. Its length is 2𝑎. It divides the ellipse into two congruent parts. (4) covertices: 𝑊1 (0, −𝑏) and 𝑊2 (0, 𝑏) The segment through the center, perpendicular to the major axis, is the minor axis. It meets the ellipse at the covertices. It divides the ellipse into two congruent parts. If 𝑥 = 0, then 𝑦 = ±𝑏. Each covertex is 𝑏 units away from the center. The minor axis 𝑊1 𝑊2 is 2𝑏 units long. Since 𝑎 > 𝑏, the major axis is longer than the minor axis. More Properties of Ellipses The ellipses we have considered so far are “horizontal” and have the origin as their centers. Some ellipses have their foci aligned vertically, and some have centers not at the origin. Their standard equations and properties are given in the box. The derivations are more involved, but are similar to the one above, and so are not shown anymore. In all four cases below, 𝑎 > 𝑏 and 𝑐 = √𝑎2 − 𝑏 2. The foci 𝐹1 and 𝐹2 are 𝑐 units away from the center. The vertices 𝑉1 and 𝑉2 are 𝑎 units away from the center, the major axis has length 2𝑎, the covertices 𝑊1 and 𝑊2 are 𝑏 units away from the center, and the minor axis has length 2𝑏. Recall that, for any point on the ellipse, the sum of its distances from the foci is 2𝑎. 9 In the standard equation, if the x-part has the bigger denominator, the ellipse is horizontal. If the y-part has the bigger denominator, the ellipse is vertical. Examples: 1. Give the coordinates of the foci, vertices, and covertices of the ellipse with 𝑥2 𝑦2 equation 25 + 9 = 1. Solution: 𝑎2 = 25 𝑏2 = 9 and 𝑐 = √𝑎2 − 𝑏 2 𝑎=5 𝑏=3 𝑐 = √25 − 9 = √16 𝑐=4 x-part has the longest side thus, the major axis of the ellipse is horizontal. Center: (0,0) foci: 𝐹1 (−4,0), 𝐹2 (4,0) 𝑐 units from the center 𝑥2 𝑦2 Figure 8: Graph of 25 + 9 =1 vertices: 𝑉1 (−5,0), 𝑉2 (5,0) 𝑎 units from the center co-vertices: 𝑊1 (0, −3), 𝑊2 (0,3) 𝑏 units from the center 10 2. Give the coordinates of the center, foci, vertices, and covertices of the ellipse (𝑥+3)2 (𝑦−5)2 with the given equation 24 + 49 = 1. Solution: 𝑎2 = 49 𝑏 2 = 24and 𝑐 = √𝑎2 − 𝑏 2 𝑎=7 𝑏 = 4.9 𝑐 = √49 − 24 = 25 𝑐=5 y-part has the longest side thus, the major axis of the ellipse is vertical. Center: (-3,5) foci: 𝐹1 (−3,0), 𝐹2 (−3,10) 𝑐 units from the center vertices: 𝑉1 (−3, −2), 𝑉2 (−3, 12) 𝑎 units from the Figure 9: Graph of (𝑥+3)2 24 + (𝑦−5)2 49 =1 center co-vertices: 𝑊1 (−7.9,5), 𝑊2 (1.9, 5) 𝑏 units from the center 3. Give the coordinates of the foci, vertices, and covertices of the ellipse with equation 9𝑥 2 + 16𝑦 2 − 126𝑥 + 64𝑦 = 71. We first change the given equation to standard form. To write the equation of an Ellipse in standard form, follow the following steps: 1. Group the same variables together on one side of the equation and position the constant on the other side. 2. Complete the square in both variables as needed. 3. Divide both sides by the constant term so that the constant on the other side of the equation becomes 1. 9(𝑥 2 − 14𝑥) + 16(𝑦 2 + 4𝑦) = 71 9(𝑥 − 14𝑥 + 49) + 16(𝑦 2 + 4𝑦 + 4 = 71 + 9(49) + 16(4) 2 9(𝑥 − 7)2 + 16(𝑦 + 2)2 = 576 (𝑥−7)2 (𝑦+2)2 64 + 36 =1 Solution: 𝑎2 = 64 𝑏 2 = 36and 𝑐 = √𝑎2 − 𝑏 2 𝑎=8 𝑏=6 𝑐 = √64 − 36 = √28 = 5.3 x-part has the longest side thus, the major axis of the ellipse is horizontal. Center: (7, -2) foci: 𝐹1 (1.7, −2), 𝐹2 (12.3, −2) 𝑐 units from the Figure 9: Graph of 9(𝑥 2 − 14𝑥) + 16(𝑦 2 + 4𝑦) = 71 center 11 vertices: 𝑉1 (−1, −2), 𝑉2 (15, −2) 𝑎 units from the center co-vertices: 𝑊1 (7, −8), 𝑊2 (7, 4) 𝑏 units from the center 4. The foci of an ellipse are (3,6) and (3,2). For any point on the ellipse, the sum of its distances from the foci is 14. Find the standard equation of the ellipse. Solution: The midpoint (3,2) of the foci is the center of the ellipse. The ellipse is vertical (because the foci are vertically aligned) and 𝑐 = 4. From the given sum, 2𝑎 = 14 so 𝑎 = 7. Also, 𝑏 = √𝑎2 − 𝑐 2 = √33. The equation is, (𝑥 + 3)2 (𝑦 + 2)2 + =1 33 49 What’s More Activity 1.2: NOW IT’S YOUR TURN! Instruction: Complete the table by filling out the center, foci, vertices, co-vertices, and standard equation of ellipse. Number 1 is already done for your reference. No equation center foci vertices co- vertices 1 𝑥2 𝑦2 (0,0) 𝐹1 (12, 0), 𝑉1 (13, 0), 𝑊1 (0, −5), + =1 𝐹2 (−12, 0) 𝑉2 (−13, 0) 𝑊2 (0, 5) 169 25 2 (𝑥 − 2)2 (𝑦 + 3)2 (1) (2) 𝑉1 (2, 0), (4) + =1 (3) 𝑉2 (2, −6) (5) 4 9 3 9𝑥 2 + 25𝑦 2 − 54𝑥 − 50𝑦 − 119 = 0 (3, 1) (6) (7) 𝑉1 ________, 𝑊1 (3, 4), 𝑉2 (−2, 1) (𝟖)𝑊2 _________ 4 (𝑥 + 2)2 (𝑦 − 5)2 (9) 𝐹1 (−2, 7), (11) 𝑊1 (−6, 5), + =1 (𝟏𝟎)𝐹2 ________ (12) 𝑊2 (2, 5) 16 20 What I Need to Remember When x-part has the largest denominator, the major axis of ellipse is horizontal. When the y-part has the largest denominator, the major axis of ellipse is vertical. To solve the co-vertices of an ellipse, use the equation 𝑏 = √𝑎2 − 𝑐 2. horizontal parabolas. 12 What I Can Do Activity 1.3: LET’S GET REAL Instruction: Answer the following. 1. Give the coordinates of the center, foci, vertices, and covertices of the ellipse with equation 41𝑥 2 + 16𝑦 2 + 246𝑥 + 192𝑦 = −289. Locate these coordinates in a cartesian plane and sketch its graph. 2. Find the standard equation of the ellipse which satisﬁes the given conditions. foci (7,6) and (1,6), the sum of the distances of any point from the foci is 14. 3. 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. Figure 10: Illustration of a tunnel in a shape of a semi-ellipse 13 Assessment (Posttest) Instruction: Choose the letter of the correct answer. Write your chosen answer on a separate sheet of paper. 1. Which of the following does NOT describe an ellipse? A. It is formed when the tilted plane intersects only one cone to form a bounded curve. B. It is formed when the tilted plane intersects only one cone to form a bounded curve. C. It has set of all points P, whose distances from 𝐹1 and 𝐹2 add up to a certain constant. D. It is a shape that looks like an oval or a flattened circle. 𝑥2 𝑦2 2. Given the equation of an ellipse 36 + 4 =1. Determine the foci of the ellipse. A. 𝐹1 (-5.66,0) and 𝐹2 (5.66,0) C. 𝐹1 (6,0) and 𝐹2 (-6,0) B. 𝐹1 (0, -5.66) and 𝐹2 (0,5.66) D. 𝐹1 (4, 0) and 𝐹2 (0,4) 3. What is the equation in standard form of the ellipse whose center is at (5,3), horizontal major axis of length 20 and minor axis of length 16? (𝑥−5)2 (𝑦−3)2 (𝑥+5)2 (𝑦+3)2 A. 100 + 64 =1 C. 100 + 64 =1 (𝑥+5)2 (𝑦−3)2 (𝑥−3)2 (𝑦+5)2 B. + =1 D. + =1 36 100 100 64 4. Which of the following statement is TRUE? A. The longer axis is called the minor axis. B. The shorter axis is called the major axis. C. The end point of the major axis is called the co-vertices. D. The center of an ellipse is the midpoint of both the major and minor axes. 5. A semielliptical tunnel has height 9 ft and a width of 30 ft. Which of the following truck will be able to pass through the tunnel? A. A truck that is 12 ft wide and 8.3 ft high. B. A truck that is12 ft wide and 7.9 ft high C. A truck that is10 ft wide and 11.9 ft high D. A truck that is 10 ft wide and 8.3 ft high 14 6. Given the graph of an ellipse below, what are the coordinates of the co- vertices? A. (±2, 0) B. (±5, 0) C. (0, ±2) D. (0, ±5) 7. Which of the following has the shape of an ellipse? A. a basketball C. a satellite dish B. a whispering gallery D. a badminton court 8. What is the standard equation of a vertical ellipse with center (ℎ, 𝑘)? (𝑥−ℎ)2 (𝑦−𝑘)2 (𝑥−𝑘)2 (𝑦−ℎ)2 A. 𝑎2 + 𝑏2 =1 C. 𝑏 2 + 𝑎2 =1 (𝑥−ℎ)2 (𝑦−𝑘)2 (𝑥−𝑘)2 (𝑦−ℎ)2 B. + 2 =1 D. + 2 =1 𝑏2 𝑎 𝑎2 𝑏 9. Find the standard equation of an ellipse with major axis of length 22, foci 9 units above and below the center (2,4). (𝑥−2)2 (𝑦−4)2 (𝑥+2)2 (𝑦+4)2 A. 40 + 121 =1 C. 40 + 121 =1 (𝑥−2)2 (𝑦−4)2 (𝑥+2)2 (𝑦+4)2 B. 121 + 40 =1 D. 121 + 40 10. Given the graph of an ellipse below, which of the following is the corresponding standard equation? (𝑥+7)2 (𝑦−4)2 A. + =1 25 16 (𝑥−7)2 (𝑦+4)2 B. 16 + 25 =1 (𝑥+4)2 (𝑦−7)2 C. 16 + 25 =1 (𝑥+7)2 (𝑦−4)2 D. 16 + 25 =1 11. A big room is constructed so that the ceiling is a dome that is semi-elliptical in shape. If a person stands at one focus and speaks, the sound that is made bounces off the ceiling and gets reﬂected to the other focus. Thus, if two people stand at the foci (ignoring their heights), they will be able to hear each other. If the room is 34 m long and 8 m high, how far from the center should each of two people stand if they would like to whisper back and forth and hear each other? A. 20 m B. 15 m C. 16 m D. 18 m 15 12. The vertices of an ellipse are (±13,0) and the foci are (±12,0). What is the standard equation? (𝑥−2)2 (𝑦+4)2 𝑥2 𝑦2 A. 16 + 12 =1 C. 12 + 16 =1 𝑥2 𝑦2 (𝑥+4) 2 (𝑦−2)2 B. 169 + 25 = 1 D. 16 + 12 =1 13. What is the center of an ellipse with general equation: 36𝑥 2 +20𝑦 2 + 144𝑥 − 120𝑦 − 396 = 0? A. (3,2) B. (2, -3) C. (-2,3) D. (-2, -3) 14. Which of the following general equations is an ellipse? A. 9𝑥 2 + 9𝑦 2 + 42𝑥 + 84𝑦 + 65 = 0 B. 16𝑥 2 + +72𝑥 − 112𝑦 + 211 = 0 C. 6𝑥 2 + 5𝑦 2 − 24𝑥 + 20𝑦 + 14 = 0 D. −6𝑥 2 + 5𝑦 2 − 24𝑥 + 4𝑦 + 26 = 0 𝑥2 𝑦2 15. What are the foci of an ellipse with the equation 169 + 144 = 1? B. (±13,0) B. (0, ±13) C. (±5,0) D. (0, ±5) 16 References Book Garces, I. J. (2016). Conic Sections. In I. J. Garces, Precalculus: Teaching Guide for Senior High School (pp. 31-40). Quezon City. PDF File James Stewart et.al, Pre-Calculus: Mathematics for Calculus 6th edition (Belmont CA, USA: Brooks/Cole, Cengage Learning © 2012), 732-740 Zeager, J. (2010). Hooked in Conics. In J. Zeager, Precalculus: Version ⌊𝜋⌋ = 3Corrected editon (pp. 516-527). Lorain County Community College Activities Drawing an Ellipse: The String Method https://mk0thisiscarpen83ynj.kinstacdn.com/wpcontent/uploads/2013/06 /Full-Ellipse-string-method.pdf Figures Figure 1: Garces, I. J. (2016). Conic Sections. In I. J. Garces, Precalculus: Teaching Guide for Senior High School (pp. ). Quezon City. Figure 2: Zeager, J. (2010). Hooked in Conics. In J. Zeager, Precalculus: Version ⌊𝜋⌋ = 3Corrected editon (pp. 517). Lorain County Community College Figure 3: Licensed under Creative Commons Attribution-Share Alike 3.0 (https://upload.wikimedia.org/wikipedia/commons/f/f4/Parts_of_Parabola. svgFigure Figure 4: Reis, Marcelo 2004, Imagem importada da pt: Wikipedia(https://upload.wikimedia.org/wikipedia/commons/1/12/Conica s1.PNG) Figure 5: Garces, I. J. (2016). Conic Sections. In I. J. Garces, Precalculus: Teaching Guide for Senior High School (pp. 32). Quezon City. Figure 6: Garces, I. J. (2016). Conic Sections. In I. J. Garces, Precalculus: Teaching Guide for Senior High School (pp. 32). Quezon City. Figure 7: Garces, I. J. (2016). Conic Sections. In I. J. Garces, Precalculus: Teaching Guide for Senior High School (pp. 34). Quezon City. Figure 8: Garces, I. J. (2016). Conic Sections. In I. J. Garces, Precalculus: Teaching Guide for Senior High School (pp. 37). Quezon City. Figure 9: Garces, I. J. (2016). Conic Sections. In I. J. Garces, Precalculus: Teaching Guide for Senior High School (pp. 38). Quezon City Figure 10: Garces, I. J. (2016). Conic Sections. In I. J. Garces, Precalculus: Teaching Guide for Senior High School (pp. 38). Quezon City. 18 Congratulations! You are now ready for the next module. Always remember the following: 1. Make sure every answer sheet has your ▪ Name ▪ Grade and Section ▪ Title of the Activity or Activity No. 2. Follow the date of submission of answer sheets as agreed with your teacher. 3. Keep the modules with you AND return them at the end of the school year or whenever face-to-face interaction is permitted. 19

Pre-Calculus Quarter 1 - Module 3 PDF

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