pre-calculus-lesson-for-high-school-vectors.pptx

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1. Pre-Calculus : Analytic Geometry (Introduction of Conic Section) Here is where your presentation begins

2. Learning Competency This lesson has been designed to help you gain the skill to: Illustrate the different types of conic sections: circle, parabola, ellipse , hyperbola and degenerate cases. Define a circle, parabola, ellipse and hyperbola. D etermine the standard form of conic sections and its equation. Graph the different types of conic section in a rectangular coordinate system. .

3. Table of contents 01 02 04 05 Introduction to Conics 03 Circle Parabola Ellipse Hyperbola

4. INTRODUCTION OF CONIC SECTION 01 Illustrate the differe n t types of conic sections: circle, parabola, ellipse, hyperbola and degenerate cases.

5. Conic Section A curve, generated by intersecting a right circular cone with a plane is termed as ‘conic’. It has distinguished properties in Euclidean Geometry . The vertex of the cone divides it into two nappes referred to as the upper nappe and the lower nappe.

6. Euclidean geometry , the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician  Euclid  (c. 300  bce ). Take Note:

7. Double Right Circular Cone Composed of two identical parts with a common points through which an axis whose endpoints are the centers of the circular bases lies perpendicular to both bases. The identical parts are called the nappes and the common point is called the apex.

8. Parts of Conic Section

9. Different Types of Conic Section Conic sections are generated by the intersection of a plane with a cone.

10. Different Types of Conic Section The conic sections are the nondegenerate curves generated by the intersections of a plane with one or two nappes of a cone. For a plane perpendicular to the axis of the cone, a circle is produced. For a plane that is not perpendicular to the axis and that intersects only a single nappe, the curve produced is either an ellipse or a parabola . The curve produced by a plane intersecting both nappes is a hyperbola .

11. Different Types of Conic Section If the plane is parallel to the axis of revolution (the y-axis), then the conic section is a hyperbola . If the plane is parallel to the generating line, the conic section is a parabola . If the plane is perpendicular to the axis of revolution, the conic section is a circle . If the plane intersects one nappe at an angle to the axis (other than 90°), then the conic section is an ellipse .

12. Different Types of Conic Section

13. CIRCLE 02 Define a circle. Determine the standard form its equation. Graph a circle in a rectangular coordinate system.

14. A circle is a set of all points that are equidistant to a fixed point . The fixed distance is called the radius of the circle and the fixed point with the coordinates is called the center .   What is Circle?

15. A circle is a set of all points that are equidistant to a fixed point . The fixed distance is called the radius of the circle and the fixed point with the coordinates is called the center .   What is Circle?

16. The Equation of the Circle in Standard Form The standard form of the equation of a circle with center at , and radius :  

17. Illustrative Examples Determine the equation of the circle described in each of the items below. Center at the origin with the radius of 13 units. Center at , Radius is Center at and containing the point Has a diameter whose endpoints are and .  

18. In geometry, the midpoint of a line segment is the point that is equidistant to both the endpoints. Its coordinates are the mean of the coordinates of the endpoints, that is, if the endpoints are and Q , then its midpoint is at we call this the MIDPOINT FORMULA .   Take Note:

19. The Equation of the Circle in General Form The general form of the equation of a circle is written in Where , , are all real numbers.  

20. Illustrative Examples Determine the equation of the circle described in each of the items below and transform it into general form. Center at the origin with the radius of 13 units. Center at , Radius is Center at and containing the point Has a diameter whose endpoints are and .  

21. PARABOLA 02 Define a parabola. Determine the standard form its equation. Graph a parabola in a rectangular coordinate system.

22. A parabola is a set of all points such that the distance between every point and fixed point is equal to its distance to line not containing the point . The distance from a fixed point , called the focus , is equal to the distance from a fixed line, called the directrix . The point halfway between the focus and the directrix is called the vertex of the parabola.   What is Parabola?

23. Orientation of the Graphs of a Parabola Whenever the parabola opens upward or downward, the directrix is a horizontal line and the focus has for its -coordinate. Consequently, the directrix is vertical and the -coordinate of the focus is whenever it opens sideways.  

24. Graph of the Parabola opens Upward and Downward Mercury is the closest planet to the Sun and the smallest of them all Venus has a beautiful name and is the second planet from the Sun Despite being red, Mars is actually a cold place. It’s full of iron oxide dust Algebraic Functions Graphing

25. Graph of the Parabola opens Sideways Mercury is the closest planet to the Sun and the smallest of them all Venus has a beautiful name and is the second planet from the Sun Despite being red, Mars is actually a cold place. It’s full of iron oxide dust Algebraic Functions Graphing

26. Illustrative Examples The graphs of three parabola are shown at the right. Determine the coordinates of the vertex, the coordinates of the focus, its axis of symmetry, and the equation of the directrix for each parabola.

27. The Equation of the Parabola in Standard Form Given a parabola opening upward with vertex located at and  focus located at , where is a constant, the equation for the parabola is given by  

28. The Equation of the Parabola in General Form The general form of a parabola is written as o r  

29. Four concepts Venus has a beautiful name and is the second planet from the Sun. It’s terribly hot, even hotter than Mercury Earth is the third planet from the Sun and the only one that harbors life in the Solar System Despite being red, Mars is actually a cold place. It’s full of iron oxide dust, which gives the planet its reddish cast Jupiter is the biggest planet in the Solar System. It’s the fourth-brightest object in the night sky Geometry Trigonometry Exponents Logarithms

30. Venus has a beautiful name, but also high temperatures Neptune is the fourth-largest planet in the Solar System Reviewing concepts Despite being red, Mars is actually a very cold place Earth is the third planet from the Sun and has life Saturn is the second-largest planet in the Solar System Jupiter is a gas giant and has around eighty moons Graphing Exponential Polynomial Rational Sequences Series

31. Awesome words

32. “This is a quote, words full of wisdom that someone important said and can make the reader get inspired” —Someone Famous

33. A picture is worth a thousand words

34. 98,300,000 Big numbers catch your audience’s attention

35. Jupiter’s rotation period 9h 55m 23s 333,000 The Sun’s mass compared to Earth’s 386,000 km Distance between Earth and the Moon

36. Mercury is the closest planet to the Sun and the smallest of them all Category A Venus has a beautiful name and is the second planet from the Sun Category B Despite being red, Mars is actually a cold place. It’s full of iron oxide dust Category C Percentage breakdown 50% 75% 25%

37. Computer mockup You can replace the image on the screen with your own work. Just right-click on it and select “Replace image”

38. Tablet mockup You can replace the image on the screen with your own work. Just right-click on it and select “Replace image”

39. Phone mockup You can replace the image on the screen with your own work. Just right-click on it and select “Replace image”

40. Venus Geographical vector analysis Venus is the second planet from the Sun. It’s terribly hot, and its atmosphere is extremely poisonous. It’s the second-brightest natural object in the night sky after the Moon

41. Domain and range 02 You can enter a subtitle here if you need it

42. Lesson timeline Venus Venus is the second planet from the Sun Mercury Mercury is the closest planet to the Sun Mars Despite being red, Mars is a cold place Jupiter Jupiter is the biggest planet of them all 1st semester Section 1 Section 2 Section 3 Section 4 2nd semester

43. Key concepts Pre-calculus By studying pre-calculus, students develop critical thinking skills, logical reasoning, and the ability to analyze and interpret mathematical models Functions & graphs Define functions and emphasize their importance Trigonometry Introduce trigonometric ratios and their applications Logarithmic functions Explain the properties and applications Rational functions Rational functions, emphasizing simplifying and solving Equations and inequalities The systems of linear and nonlinear equations

44. Pre-calculus lesson plan Lesson no. Topic Key concepts 01 Introduction to pre-calculus Definition and importance 02 Functions and graph Domarin, range, graphing 03 Trigonometry Ratios, functions, identities 04 Polynomial and rational functions Factoring, simplifying, rational 05 Exponential and logarithmic Properties, equations, graphing 06 Systems of equations Applications, solving systems

45. Vector options distribution Follow the link in the graph to modify its data and then paste the new one here. For more info, click here Magnitude Mercury is quite a small planet Direction Jupiter is an enormous planet Scalar Venus has very high temperatures Cross Saturn is a gas giant with rings

46. Timmy Jimmy Susan Smith Our team You can speak a bit about this person here You can speak a bit about this person here You can speak a bit about this person here Jenna Doe

47. Graphing functions 03 You can enter a subtitle here if you need it

48. Some tips Here are some tips for solving equations in high school: Properties Mercury is the closest planet to the Sun Check Jupiter is the biggest planet in the Solar System Order Despite being red, Mars is actually a cold place Combine Neptune is the farthest planet from the Sun Isolate Venus is the second planet from the Sun Attention Saturn has a high number of moons, like Jupiter

49. Introduction to the exercises You can give a brief description of the topic you want to talk about here. For example, if you want to talk about Mercury, you can say that it’s the smallest planet in the entire Solar System

50. Graph of y = sin x The following table presents the values of the sine function (sin x) for angles ranging from 0 to 360 degrees You can represent these values visually by creating a graph . Here is an illustration of how the graph would look like 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,4 1,2 1,0 0,8 0,6 0,4 0,2 30 60 90 120 150 180 210 240 270 310 330 360 X Y 30 0,5 60 0,8 90 1,0 120 0,8 150 0,5 180 210 -0,5 240 -0,8 270 -1,0 300 -0,8 330 -0,5 360

51. Identifying functions To determine whether each table of values represents a function , we need to check if there is a unique output (y-value) for every input (x-value) in the table. If there is no repetition of x-values and each x-value corresponds to a single y-value, then the table represents a function. State whether each table of values represents a function X Y -12 2 -10 10 -2 5 -6 8 -11 15 -15 X Y 9 -18 -20 -6 1 -17 16 9 17 11 19 X Y 4 -20 1 -17 4 -14 16 5 10 -19 -16 X Y -15 18 -11 18 -14 18 -9 18 -1 18 -5 18

52. Some equations Determine the relationship between the equations . Place greater than (>), less than (<) or equal to (=) in the space provided Where x=3 a) 5x + 4 ____ 3x + 15 b) x + 23 ____ 5x - 4 c) 7x - 2 ____ 4x + 4 d) 2x + x ____ 6x - 5 e) 6x + 2 ____ 4x + 4 f) 3x + 5 ____ 6x - 4 Where x=7 a) 3x - x ____ 4x + 14 b) 2x + 12 ____ 3x - 4 c) x + x + 7 ____ 5x d) 2x + 10 ____ 5x - 5 e) 6x - 18 ____ 4x - 4 f) 8x ____ 3x + 2x + 15

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