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- 1. SARAH B. BALIGOD,LPT Subject Teacher, SHS Teacher III General Mathematics Senior High School Grade 11
- 2. PRELIMINARIES Class Schedule Engagement in Activities Triad – Learners will form a three member group. Incentives Discipline and Respect Subject Requirements Key Result Area’s Student’s Assignment Monitoring Sheet
- 3. Representing Real- Life Situations Using Functions General Mathematics Lesson 1 Senior High School Grade 11
- 4. LEARNING OUTCOMES Representing Real-life Situations Using Functions • Recall the concepts of relations and functions • Define and explain functional relationship as a mathematical model of situation • Represent real-life situations using functions, including piece-wise functions
- 6. Is any set of ordered pairs. The set of all first elements of the ordered pairs of the relation. The set of all second elements. Relation Domain Range
- 7. . Is a relation or rule of correspondence between two elements (domain and range) such that each element in the domain corresponds to exactly one element in the range. Function
- 9. Which is Which? Function 𝑨 = 𝟏, 𝟐 , 𝟐, 𝟑 𝟑, 𝟒 𝟒, 𝟓
- 10. Which is Which? Function 𝑩 = 𝟑, 𝟑 , 𝟒, 𝟒 𝟓, 𝟓 𝟔, 𝟔
- 11. Relation Which is Which? 𝑪 = 𝟏, 𝟎 , 𝟎, 𝟏 −𝟏, 𝟎 𝟎, −𝟏
- 12. Relation Which is Which? 𝑫 = 𝒂, 𝒃 , 𝒃, 𝒄 𝒄, 𝒅 𝒂, 𝒅
- 13. Which is Which? Function X 1 2 3 4 5 6 Y 2 4 6 8 10 12
- 14. Which is Which? Function X 4 -3 1 2 5 Y -5 -2 -2 -2 0
- 15. Which is Which? X 0 -1 4 2 -1 Y 3 4 0 -1 1 Relation
- 16. Which is Which? DOMAIN RANGE a b c x y Function
- 17. Which is Which? DOMAIN RANGE x y a b c Relation
- 18. Which is Which? DOMAIN RANGE Jan Feb Mar red blue pink Function
- 19. . A relation between two sets of numbers can be illustrated by graph in the Cartesian plane, and that a function passes the vertical line test. A graph of a relation is a function if any vertical line drawn passing through the graph intersects it at exactly one point.
- 20. Which is Which? Function Function Relation Relation
- 23. Which is Which? Scenario 3: As part of their requirements in Statistics class, Andrei made a survey on the religion of his classmates and here’s what he found out. Andrei: Good morning classmates, as our requirement in Statistics may I know your religion. This data will be part of my input in the survey that I am doing. Ana : I am a Catholic. Kevin: I am also a Catholic. Sam: I am a member of the Iglesia ni Cristo. Joey: I am a Born Again Christian. Lanie: My family is a Muslim. Jen: We are sacred a Catholic Family. Andrei: Thank you classmates for your responses. Function
- 25. Real-Life Situations If height (H) is a function of age (a), give a function H that can represent the height of a person in a age, if every year the height is added by 2 inches. Solution: Since every year the height is added by 2 inches, then the height function is 𝑯(𝒂)=𝒂+2
- 26. Real-Life Situations If distance (D) is a function of time (t), give a function D that can represent the distance a car travels in t time, if every hour the car travels 60 kilometers. Solution: Since every hour, the car travels 60 kilometers, therefore the distance function is given by 𝑫(𝒕)=𝟔𝟎𝒕
- 27. Real-Life Situations Give a function B that can represent the amount of battery charge of a cellular phone in h hour, if 12% of battery was loss every hour. Solution: Since every hour losses 12% of the battery, then the amount of battery function is 𝑩(𝒉)=𝟏𝟎𝟎−𝟎.𝟏𝟐𝒉
- 28. Real-Life Situations Squares of side x are cut from each corner of a 10 in x 8 in rectangle, so that its sides can be folded to make a box with no top. Define a function in terms of x that can represent the volume of the box. Solution: The length and width of the box are 10 - 2x and 8 - 2x, respectively. Its height is x. Thus, the volume of the box can be represented by the function. 𝑽(𝒙) = (𝟏𝟎 − 𝟐𝒙)(𝟖 − 𝟐𝒙)(𝒙) = 𝟖𝟎𝒙 − 𝟑𝟔𝒙𝟐 + 𝟒𝒙𝟑
- 29. Piecewise Functions 𝑓 𝑥 = formula 1 if x is in domain 1 formula 2 if x is in domain 2 formula 3 if x is in domain 3
- 30. Piecewise Functions A user is charged ₱250.00 monthly for a particular mobile plan, which includes 200 free text messages. Messages in excess of 200 are charged ₱1.00 each. Represent the monthly cost for text messaging using the function t(m), where m is the number of messages sent in a month. 𝒕 𝒎 = 𝟐𝟓𝟎 𝒊𝒇 𝟎 < 𝒎 ≤ 𝟐𝟎𝟎 𝟐𝟓𝟎 + 𝒎 − 𝟐𝟎𝟎 𝒊𝒇 𝒎 > 𝟐𝟎𝟎
- 31. Piecewise Functions A certain chocolate bar costs ₱50.00 per piece. However, if you buy more than 5 pieces they will mark down the price to ₱48.00 per piece. Use a piecewise function to represent the cost in terms of the number of chocolate bars bought. 𝑷 𝒄 = 𝟓𝟎 𝒊𝒇 𝟏 < 𝒄 ≤ 𝟓 𝟒𝟖 𝒄 𝒊𝒇 𝒄 > 𝟓
- 32. Piecewise Functions The cost of hiring a catering service to serve food for a party is ₱250.00 per head for 50 persons or less, ₱200.00 per head for 51 to 100 persons, and ₱150.00 per head for more than 100. Represent the total cost as a piecewise function of the number of attendees to the party. 𝑪 𝒉 = 𝟐𝟓𝟎𝒉 𝒊𝒇 𝒉 ≤ 𝟓𝟎 𝟐𝟎𝟎𝒉 𝒊𝒇 𝟓𝟏 < 𝒉 ≤ 𝟏𝟎𝟎 𝟏𝟎𝟎𝒉 𝒊𝒇 𝒉 > 𝟏𝟎𝟎
- 33. Seatwork Time
- 34. Quiz Time
- 35. Homework 1. In three to five sentences, write the significance of function in showing real-life situations. 2. Cite 2 real-life situations that show functions.
- 36. Performance Task At home or in your community, look for the at least three (3) situations that could represent functions. From the identified situations, write a sample problem and its corresponding function equation. Example: Situation: The budget for food is a function of the number of family members. Problem: Reyes family has Php ₱1,500.00 food budget for each member of their family in a month. Express the total food budget (B) as a function of number of family members (n) in one month. Function: 𝐵(𝑥)=1500𝑥
- 37. . o What do you think were the key mathematical concepts addressed in this lesson? o Would you rate your level of understanding of the material covered in this lesson as high, moderate, or low? o Has the lesson helped you to gain further insight into aspects of the material covered that represent strengths or represent weaknesses? o What would you describe as the main barriers, if any, to your ongoing progress and achievement in relation to the topic area addressed in this lesson? o What do you think would best assist your ongoing progress and achievement in relation to the topic area?
- 38. Thank You!!!
Editor's Notes
- Welcome to the first lesson of your General Mathematics. This lesson will give you the practical application of functions in a real-life scenario including the piece-wise function. When you are in Grade 8, you already encountered relation and function. But in this module, let’s take into a deeper sense on how this topic can be useful in our daily life. Are you all ready? Before we proceed in representing real-life scenario using function, let’s go back to where we start. What have you remembered about relations and functions?
- Functions can often be used to model real-life situations. Identifying an appropriate functional model will lead to a better understanding of various phenomena. The above scenarios are all examples of relations that show function. Monogamous marriage (e.g. Christian countries) is an example of function when there is faith and loyalty. Let say, June is the domain and Mae is the range, when there is faithfulness in their marriage, there will be one-to-one relationship - one domain to one range.
- Nationality could also illustrate a function. We expect that at least a person has one nationality. Let say Kim is the domain and her nationality is the range, therefore there is a one-to-one relationship. Since Kim was born and live in the Philippines, she can never have multiple nationalities except Filipino. (Remember: Under RA 9225 only those naturally-born Filipinos who have become naturalized citizens of another country can have dual citizenship. This is not applicable to Kim since she was born in the Philippines and never a citizen of other country.)
- Religion is also an example of function because a person can never have two religions. Inside the classroom, three classmates said that they are Catholic. This shows a many-to-one relationship. Classmates being the domain and religion being the range indicate that different values of domain can have one value of range. One-to-one relationship was also illustrated by the classmates who said that they are Born Again, Muslim and Iglesia ni Cristo - one student to one religion.
- Function can be illustrated as a machine where there is the input and the output. When you put an object into a machine, you expect a product as output after the process being done by the machine. For example, when you put an orange fruit into a juicer, you expect an orange juice as the output and not a grape juice. Or you will never expect to have two kinds of juices - orange and grapes.
- You have learned that function can be represented by equation. Since output (y) is dependent on input (x), we can say that y is a function of x. For example, if a function machine always adds three (3) to whatever you put in it. Therefore, we can derive an equation of x + 3 = y or f(x) = x+ 3 where f(x) = y.
- There are functions that requires more than one formula in order to obtain the given output. There are instances when we need to describe situations in which a rule or relationship changes as the input value crosses certain boundaries. In this case, we need to apply the piecewise function. A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains.