Unit 4 Graphing and Analyzing Linear Functions

1. Unit 4 Graphing and Analyzing Linear Functions Michelle A. O’Malley League Academy of Communication Arts Greenville, South Carolina

2. Standards for Learning Goal 4.1 • EA1.5: Demonstrate an understanding of algebraic relationships by using a variety of representations (including verbal, graphic, numerical, and symbolic) • EA3.1: Classify a relationship as being either a function or not a function when given data as a table, set or ordered pairs, or graph.

3. Essential Question for Learning Goal 4.1 • What does a function look like?

4. Learning Goal 4.1 Notes • In order to determine whether a relationship is linear, you should focus on the rate of change in the relationship. • Linear relationships are characterized by a constant change in one variable associated with a constant change in the other variable. • That is, for each unit change in the independent quantity (variable), there is a constant change in the dependent quantity (variable).

5. Learning Goal 4.1 Notes • A constant rate of change is what makes the line “straight.” • A constant increase in one variable compared to the other is associated with straight lines having a positive slope. • A constant decrease in one variable as compared to the other is associated with straight lines having a negative slope. • Some linear functions are proportional and take the form of y=mx. • Some linear functions are non-proportional and take the form y=mx + b.

6. Learning Goal 4.1 Example 1: Linear or Non Linear? • Phil is making a 3 foot by 4 foot banner for the math club. Realizing that the banner is too small, he decides to increase each side. Phil must decide how the new dimensions will affect the cost of the materials. (cost versus area) Non-Linear because the change in area is not constant

7. Learning Goal 4.1 Example 2: Linear or Non Linear? • A scuba diver is 120 feet below sea level. She knows that to avoid suffering from the bends, she must come up at a rate of 7 feet per minute (depth versus time). Linear because the rate of change is constant

8. Learning Goal 4.1 Example 3: Linear or Non Linear? • The pattern in the first table is linear: The constant rate of change in the y is zero and the function is y=3. • The second pattern is also linear; the constant rate of change in y is -3 (for a unit change in x) and the function is y=-3x+1. Note: when finding the rate of change if the x-values do not increase in equal increments, as long as the rates of change (change in y/change in x) are equivalent (i.e. (-9/3 = -3/1 = -6/2 = -3), the function is linear. • The third pattern is not linear: for each 1 unit increase in x, the change in y is not constant. Table 1 Table 3 Table 2

9. Learning Goal 4.1 Essential Knowledge • Students should be able to identify that a constant rate of change is the criteria used to determine whether or not a relationship is linear. • Students should be able to determine if a relationship given in tabular form represents a linear function. • Students should be able to determine if a relationship given in verbal form represents a linear function.

10. Standards for Learning Goal 4.2 • EA-5.6 Carry out a procedure to determine the slope of a line from data given tabularly, graphically, symbolically, and verbally. • EA-5.7 Apply the concept of slope as a rate of change to solve problems.

11. Essential Question for Learning Goal 4.2 • What is slope?

12. Learning Goal 4.2 Notes • Slope is usually represented by the letter m (from the French, monter, which means “to go up”). • A constant rate of change is called the slope of a line. • Slope is the ratio of vertical change to horizontal change between points on the graph. • Slope is defined as m = (y2 – y1)/(x2 – x1), where (x1, y1) and (x2, y2) are points on the line.

13. Learning Goal 4.2 Notes • Slope should also be considered in terms of a ratio. • Slope is the ratio for the differences in the x and y coordinates of two points on the line. • The ratio of vertical rise to horizontal run (m=rise/run) • Slope is a rate of change.

14. Learning Goal 4.2 Notes • Slopes can be positive, which means rising to the right. • Slopes can be negative, which means falling to the right. • Slopes can be zero, which means the line will be horizontal. • Slopes can be Undefined, which are sometimes referred to as “no slope” which should not be confused with zero slope.

15. Learning Goal 4.2 Notes • For a line to have a positive slope, as the independent quantity increases (or decreases), the dependent quantity increases (or decreases). • For a line to have a negative slope, as the independent quantity increases (or decreases), the dependent quantity decreases (or increases). • For a line to have a zero slope (equation y=constant), there is no change in the dependent quantity associated with any change in the independent quantity. (horizontal line) • For a line to have an undefined slope (equation x=constant), there is no change in the independent quantity associated with changes in the dependent quantity (no division by zero) (vertical line)

16. Learning Goal 4.2 Notes • Slope has a proportional nature. • For example, for Y=3x+1 • For each unit increase in x, y increases by 3 • If x decreases by 10, y will decrease by 30

17. Learning Goal 4.2 Essential Knowledge • Students should know how to find slope when given a table, graph, set of ordered pairs or algebraic representation. • Students should be able to determine whether the slope is positive, negative, zero, or undefined (not a function) when given the graph of a line. • Students should be able to sketch an appropriate graph when given slope values that are positive, negative, zero, or undefined. • Students should be able to interpret the slope as the rate of change in the context of a problem.

18. Standards for Learning Goal 4.3 • EA 1.3 Apply algebraic methods to solve problems in real-world contexts. • EA 1.5 Demonstrate an understanding of algebraic relationships by using a variety of representations (including verbal, graphic, numerical, and symbolic). • EA 3.6 Classify a variation as either direct or inverse.

19. Essential Questions for Learning Goal 4.3 • How do I distinguish between direct and inverse variation?

20. Learning Goal 4.3 Notes • Direct Variation • Two variables, x and y, vary directly if there is a nonzero number k such that y = kx. • Just because one quantity increases when the other increases does not mean that x and y vary directly. • When one quantity always changes by the same factor as another, the two quantities are in direct proportion; “k” is the constant of proportionality as well as the slope of the linear function.

21. Learning Goal 4.3 Notes • Inverse Variation • Two variables, x and y, vary inversely if there is a non-zero number k such that y = k/x, or xy = k (x ≠ 0) • Just because one quantity decreases as the other increases does not mean that the two quantities are inversely proportional. • When one quantity always decreases by the same factor as the other increases, the two quantities are inversely proportional. • If xy=9 and x is multiplied by 2, then y is divided by 2 in order to preserve the constant 9.

22. Learning Goal 4.3 Notes • Although Direct Variation is a linear function, it will be of the form y = mx, not y = mx + b because the y-intercept must be zero. • Given that the y-intercept must be zero, the graph will always go through the origin (0,0) • The Graphical representation of inverse variation is a hyperbola (curved line). • Inverse variation – the product of the two values remains constant as x increase, y decreases, or as x decreases, y increases.

23. Learning Goal 4.3 Essential Knowledge • Students should be able to determine whether the data demonstrates direct variation, inverse variation, or neither when given a graph, table, or real world application. • Students should be able to find the constant of variation, k, and write an equation that relates x to y when given function values. • Students should be able to find the constant of variation when given a direct or inverse variation graph. • Students should be able to sketch a graph of the function, when given a direct or inverse variation equation. • Students should be able to write an inverse or direct variation equation when given a verbal description.

24. Standards for Learning Goal 4.4 • EA 1.5 Demonstrate an understanding of algebraic relationships by using a variety of representations (including verbal, graphic, numerical, and symbolic). • EA 1.6 Understand how algebraic relationships can be represented in concrete models, pictorial models, and diagrams. • EA 4.6 Represent linear equations in multiple forms (including point-slope, slope-intercept, and standard).

25. Essential Questions for Learning Goal 4.4 • How do I translate between the various algebraic forms of a linear function?

26. Learning Goal 4.4 Notes • Equations of the form ax + by = c (standard form) and/or y = mx + b (Slope-intercept form), or any equations that can be transformed into either of these two forms are linear.

27. Learning Goal 4.4 Notes • In the equation ax + by = c, if a = 0 then the equation becomes y = c/b; this is the equation of a horizontal line. • In equation ax + by = c, if b = 0 then the equation becomes x = c/a; this is the equation of a vertical line.

28. Learning Goal 4.4 Notes • A solution of an equation in two variables, x and y, is an ordered pair (x, y) that makes the equation true. • The graph of an equation in x and y is the set of all points (x, y) that are solutions of the equation.

29. Learning Goal 4.4 Quiz

30. Learning Goal 4.4 Essential Knowledge • Students should be able to determine if an equations represents a linear function. • Students should be able to translate between slope-intercept {y=mx + b}, point-slope {y – y1 = m(x – x1)}, and standard forms {Ax + By = C} of linear functions.

31. Standards for Learning Goal 4.5 • EA 1.1 Communicate a knowledge of algebraic relationships by using mathematical terminology appropriately. • EA 1.3 Apply algebraic methods to solve problems in real-world contexts. • EA 5.5 Carry out a procedure to determine the x-intercept and y-intercept of lines from data given tabularly, graphically, symbolically, and verbally.

32. Essential Questions for Learning Goal 4.5 • What do intercepts mean?

33. Learning Goal 4.5 Notes • Essential knowledge and skills for this unit: • Be able to find the x and y intercepts of al ine given the equation, table, or graph • Be able to graph a line using the intercepts • Be able to interpret the real world meaning of the x and y intercepts

34. Learning Goal 4.5 Notes • How do you find the x and y intercepts? • First you must have an equation • For example, y = -3x + 48 • The slope is -3 • The y intercept is 48 – you know this because you know in y=mx + b, b is the y intercept. Also, if you replace x with a 0, your equation will change to y=-3(0) + 48, which gives you y=48. • The x intercept can be found by substituting 0 for y; for example, 0 = -3x + 48, -48=-3x, divide by three to isolate x and your answer would be x = 16, which is the x intercept.

35. Learning Goal 4.5 Notes • If you are looking at coordinate pairs, (x,0) represents the x intercept and (0,y) represents the y intercept. • Which one of the numbers in the table on the right represent the x and y intercepts.

36. Learning Goal 4.5 Notes • Intercepts cross the graph at a given point where one value is zero and the other is either the y or x value. • Vertical lines have no y intercepts and x is constant. (Note: A vertical line is not a function and the slope would be undefined) • Horizontal lines have no x intercept and y is constant. (Note: A horizontal line is a function and the slope would be zero.

37. Learning Goal 4.5 Notes • Today you have $48 left from money you got for your birthday. On average you have been spending $3 per day and you are not planning on changing this spending pattern. • Generate an equation that represents your spending patterns; clearly indicate what x and y represent?

38. Learning Goal 4.5 Notes • Today you have $48 left from money you got for your birthday. On average you have been spending $3 per day and you are not planning on changing this spending pattern. • Generate an equation that represents your spending patterns; clearly indicate what x and y represent? • Answer: • y will be the amount of birthday money $ and x will be the number of days • Y=-3x + 48

39. Learning Goal 4.5 Notes • Today you have $48 left from money you got for your birthday. On average you have been spending $3 per day and you are not planning on changing this spending pattern. • Next, What is the slope of the equation? What does the slope represent in general and in the context of this problem?

40. Learning Goal 4.5 Notes • Today you have $48 left from money you got for your birthday. On average you have been spending $3 per day and you are not planning on changing this spending pattern. • Next, What is the slope of the equation? What does the slope represent in general and in the context of this problem? • Answer: • The slope is – 3, which is the amount of money spent per day

41. Learning Goal 4.5 Notes • Today you have $48 left from money you got for your birthday. On average you have been spending $3 per day and you are not planning on changing this spending pattern. • Next, Identify the x intercept in this problem situation and explain what it represents in this situation.

42. Learning Goal 4.5 Notes • Today you have $48 left from money you got for your birthday. On average you have been spending $3 per day and you are not planning on changing this spending pattern. • Next, Identify the x intercept in this problem situation and explain what it represents in this situation. • Answer: • X intercept will be when the birthday money has been spent – it is all gone. • Y= -3x + 48, 0= -3x + 48, -48= -3x, isolate the variable by dividing by -3, x = 16

43. Learning Goal 4.5 Notes • Today you have $48 left from money you got for your birthday. On average you have been spending $3 per day and you are not planning on changing this spending pattern. • Next, Identify the y intercept in this problem situation and explain what it represents in this situation.

44. Learning Goal 4.5 Notes • Today you have $48 left from money you got for your birthday. On average you have been spending $3 per day and you are not planning on changing this spending pattern. • Next, Identify the y intercept in this problem situation and explain what it represents in this situation. • Answer: • The y intercept is the amount of birthday money that is left at that particular day. • Y = -3(0) + 48, y = 48

45. 50 45 40 35 30 25 20 15 10 5 0 Birthday Money $ - y axis 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Number of Days – x axis Graph of Y = - 3x + 48 X intercept = 16 And Y intercept = 48 What does your graph look like?

46. Purpose Statement based on Algebra 1 Learning Goal 4.6 • Seventh and Eighth grade Algebra 1 students will be assessed on their ability to write and graph a linear equation using the slope-intercept form, with and without technology. Relate the slope-intercept form to transformations of the parent function y=x so that the teacher, parents, instructional coach, and principal can determine if the students have met the required Algebra 1 state standards that are represented in the Graphing and Analyzing Linear Equations Unit (Greenville County School District, 2010).

47. Standards for Learning Goal 4.6 • EA 1.7 Understand how to represent algebraic relationships by using tools such as handheld computing devices, spreadsheets, and computer algebra systems (CASs). • EA 3.5 Carry out a procedure to graph parent functions (including ). • EA 4.1 Carry out a procedure to write an equation of a line with a given slope and a y-intercept. • EA 4.2 Carry out a procedure to write an equation of a line with a given slope passing through a given point.

48. Standards for Learning Goal 4.6(Continued) • EA 5.1 Carry out a procedure to graph a line when given the equation of the line. • EA 5.2 Analyze the effects of changes in the slope, m, and the y-intercept, b, on the graph of y = mx + b. • EA 5.3 Carry out a procedure to graph the line with a given slope and a y-intercept. • EA 5.4 Carry out a procedure to graph the line with a given slope passing through a given point.

49. Essential Questions for Learning Goal 4.6 • How do I use transformations to graph linear equations in slope intercept form?

50. Learning Goal 4.6 Notes • Slope-intercept form of the linear equation should be approached as transformations (slides, flips and stretches or compressions) to a parent function y = x. • Characteristics of a parent function y = x • The x and y coordinates of ordered pairs are equal and include but are not limited to (-1,-1), (0,0), (1,1), (5/2,5/2), ETC. • Slope = 1/1 = 1 • Passes through the origin • Bisect first and third quadrants creating 45 degree angles