1. Chapter 3 Modeling Functions Math 10 – Ms. Lilian Albarico

2. Students are expected to: demonstrate an understanding of and apply proper use of discrete and continuous number systems demonstrate an understanding of and apply proper use of discrete and continuous number systems express problems in terms of equations and vice versa express problems in terms of equations and vice versa model real-world phenomena with linear, quadratic, exponential, and power equations, and linear inequalities model real-world phenomena with linear, quadratic, exponential, and power equations, and linear inequalities gather data, plot the data using appropriate scales, and demonstrate an understanding of independent and dependent variables and of domain and range gather data, plot the data using appropriate scales, and demonstrate an understanding of independent and dependent variables and of domain and range identify, generalize, and apply patterns identify, generalize, and apply patterns construct and analyze graphs and tables relating two variables construct and analyze graphs and tables relating two variables develop and apply strategies for solving problems develop and apply strategies for solving problems solve problems using graphing technology solve problems using graphing technology determine if a graph is linear by plotting points in a given situation determine if a graph is linear by plotting points in a given situation describe real-world relationships depicted by graphs and tables of values and written descriptions describe real-world relationships depicted by graphs and tables of values and written descriptions

3. Vocabulary Algebra Irrational number Real number Rational number Imaginary number Radical number Refer, Consist

4. Review

6. The amount of fuel remaining in your gas tank is plotted for each hour you drive. However, the amount of fuel in your gas tank decreases every minute/second you drive. All values on the graph are meaningful and therefore can be joined. This is continuous data. The cost of CDs is plotted for each CD you purchase. The cost to you changes only when another CD is purchased. The values between the plotted points are not meaningful and therefore are not joined. This is discrete data.

7. Real Numbers System (Q, I (or Z), N, W, ) The Real number system is the set of numbers consisting of the positive numbers, negative numbers, and zero. It can be divided into the following groups. Whole Numbers (W) → {0, 1, 2, 3, ….} Whole Numbers (W) → {0, 1, 2, 3, ….} Natural Numbers (N) → {1, 2, 3, 4, …} Natural Numbers (N) → {1, 2, 3, 4, …} Integer Numbers (I or Z) → {… -2, -1, 0, 1, 2, …} Integer Numbers (I or Z) → {… -2, -1, 0, 1, 2, …} Rational Numbers (Q) → {---------------------- } Rational Numbers (Q) → {---------------------- } Irrational Numbers (---------------- ) → a number that cannot be written as a ratio of two numbers, Irrational Numbers (---------------- ) → a number that cannot be written as a ratio of two numbers,

8. . The symbols used in writing set notation are: { } means “set” { } means “set” | means “such that” | means “such that” ≤ means “less than or equal to” ≤ means “less than or equal to” ≥ means “greater than or equal to” ≥ means “greater than or equal to” R means “Real numbers” R means “Real numbers” Є means “is a member of” Є means “is a member of” I means “Integers” I means “Integers” 3 ≤ x ≤ 7 means “x is between 3 and 7 inclusively”

9. Domain and Range When we talked about input (x) values in chapter 1, we were referring to the domain. All of the x values are the domain. Therefore the output values, or the y values, are the range. When we talked about input (x) values in chapter 1, we were referring to the domain. All of the x values are the domain. Therefore the output values, or the y values, are the range. A relation is a mathematical statement, that if evaluated, each value chosen will produce at least one answer. A relation has a domain and a range. A domain is the set { }of all values that were chosen when the relation was evaluated. These values are usually the x-values. A range is the set { } of all the results obtained when the relation was evaluated. These values are usually the y-values. The domain is the set of independent values and the range is the set of dependent values. A relation is a mathematical statement, that if evaluated, each value chosen will produce at least one answer. A relation has a domain and a range. A domain is the set { }of all values that were chosen when the relation was evaluated. These values are usually the x-values. A range is the set { } of all the results obtained when the relation was evaluated. These values are usually the y-values. The domain is the set of independent values and the range is the set of dependent values.

10. Domain and Range

11. Examples State the domain and the range for each of the following. a) –(y – 4) = (x + 2) 2

12. Examples State the domain and the range for each of the following. a) –(y – 4) = (x + 2) 2 Domain: {x  x R} Range: {y  y ≤ 4, y R}

13. b)

14. c) x0123 y-7-5-3

15. d)

16. Lets Investigate!!!

19. Equation – a mathematical sentence showing that expressions are equal in value. ( x+2 = 10). When equations are written, letters are used to represent variables. Expression - is a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators (like add, subtract, multiply, and divide). (2x, 4y-1) y- intercept – the value of y in x- y coordinate when x = 0 in a linear equation. Also represented as b in the general form y=mx + b.

20. Linear Equation An equation that makes a straight line when it is graphed. Often written in the form: y = m x +b Where m = slope b = y-intercept

21. Graph of Linear Equation If we look at the x-y coordinate system above, we can see that the y-axis is the vertical axis or is rising and the x-axis is horizontal or running across the page.

22. Rise/Run (Slope) The slope (m) of the line gives, using numbers, a way to tell us the steepness of the line. The slope (m) of the line gives, using numbers, a way to tell us the steepness of the line.

23. Rise/Run (Slope) The slope (given the symbol m) is then the rise over the run or The rise is the difference in two of the y values that lie on the line (y 2 – y 1 ) and the run is the difference in two corresponding x values that lie on the line (x 2 - x 1 ). So our equation now becomes: {Δ means the change in}.

24. a) Find the slope, y-intercept and equation of the following:

25. b)

26. c)

27. d)

28. Question? What did you notice with the different slopes of the four graphs?

29. Let’s answer this! 1) The “Town Clean-Up” committee has initiated a project to clean up the town and to raise extra funds at the same time. The committee is advertising for people to get involved in the project by offering a $15.00 member sign in bonus plus 3¢ for each recyclable they bring to the local fire hall. a) Is this a linear relationship? How can you tell? b) State the independent and dependent variables. c) Create a table of values and graph to represent this scenario. Is the data discrete or continuous? d) Write an equation to represent the graph. e) State the domain and range.

31. Let’s answer this! 2) Chit Chat Cell Phones is offering college students a cell phone package that includes a monthly fee of $20 and $0.03 per minute for air time. a) State the independent and dependent variables. b) Create a table of values and graph to represent this scenario. Is the data discrete or continuous? c) What is the slope and y-intercept of the graph? d) Write an equation for the relationship. e) State the domain and range.

33. Practice Exercises Please make sure you have the copy of two exercises hand outs.

34. Homework: CYU # 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, and 25 on page 100- 103 of your textbook. Check this website for more Algebra lessons – (http://www.mathsisfun.com/algebra/index.html)