UNIT 4 Math 621. Forms of Lines and Modeling Using Linear Equations

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1 UNIT 4 Math 621 Forms of Lines and Modeling Using Linear Equations Description: This unit focuses on different forms of linear equations. Slope- intercept, point-slope and standard forms are introduced. Students will write all three forms using short descriptions, tables of values and/or graphs. In addition, students will be able to transform one form into another and construct linear graphs for each form. They will be able to graph lines using the x and y- intercepts, a slope and y-intercept, a point and a slope or two points, accordingly to the given form of the equation. The unit will be concluded with an introduction to linear modeling, where students will recognize linear relationship between independent and dependent variables. Students will derive equations, graph functions and use their equations and graphs to make predictions. In the process of modeling, students will determine the constant rate of change and initial value of a function from a description of a relationship. They will interpret slope as rate of change, relate slope to the steepness of a line, and learn that the sign of the slope indicates that a linear function is increasing if the slope is positive and decreasing if the slope is negative. 4.1 Slope and slope-intercept form of lines 4.2 Ivy Smith Performance Task 4.3 Point-Slope form of lines 4.4 Standard Form for lines Quiz on sections Modeling using linear equations 4.6 More on Modeling 4.7 Unit Review Unit 4 TEST

2 RESOURCES 4.1 Slope-intercept form of lines Standard Form for lines Point-Slope form of lines Review of linear forms Modeling using linear equations More on Modeling 2

3 4.1 Slope Formula and Slope-Intercept Form Review The slope formula is: m = y 2 y 1 x 2 x 1 for the points (x 1, y 1 ) and (x 2, y 2 ) Example: Find the slope of the line that goes through the points (5, -3) and (2, 3) 1: Use the slope formula to find the slope of the line through the given points. Leave answers in simplified fractional form. 1. A(3, 1) and B(5, -1) 2. C(7, 0) and D(5, -3) 3. E(7, 8) and F(3, 8) 4. G(-3, 5) and H(-3, 2) 5. (2, 6) and (7, 1) 6. (4, -1) and (0, 9) 7. (3, 7) and (4, 7) 8. (7, 2) and (7, -4) 3

4 9. (3, -3) and (2, -2) 10. (7, 0) and (2, 10) Rewrite each of these equations in slope-intercept form x 4y = x + 4y = 12 Use the slope formula to find the slope of the line going through each of these pairs of points. 13. A(7, 4) and B(-2, 1) 14. C(0, 3) and D(4, 0) Identify the type of line (vertical, horizontal or oblique) for the given equations. 15. y = 3x + 4 vertical horizontal oblique 16. 3x = 7y vertical horizontal oblique 17. 3y = 9 vertical horizontal oblique 18. x = 3 vertical horizontal oblique 4

5 Identify the slope and y-intercept of the lines represented by each of these equations. 19. y = 4x x + 2y = 12 slope = y-intercept = slope = y-intercept = 21. x = y = 3 slope = y-intercept = slope = y-intercept = Find the equation of each line in slope-intercept form 23. The line has a slope of 7 and a y-intercept at The slope is -2 and it goes through the point (0, 3). 25. The slope is 5 and it goes through the point (2, -1) Write the equation of each line in slope-intercept form

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7 Section 4.3 Point-Slope Form Notes and Practice Lines of the form are written in form. Example 1: Write an equation for the line with slope = 2, that goes through the point (7, 3) Example 2: Write an equation for the line with slope = 2, that goes through the point (5, -6) 3 Example 3: Write an equation for the line with slope = -4, that goes through the point (-2, 1) In order to write the equation of any line you need and the. If you are given both of those things, as you were above, then it s easy to use the point-slope form of the line. Sometimes you aren t actually given the information out-right, but you can figure it out from other information. Example 4: Write an equation for a line that goes through the points (1, 3) and (-2, 2). You have a point in fact, you have two points from which to choose, but you need. Fortunately, you can compute it: Let s use the first point: with the slope. The equation is: 7

8 Example 5: Write an equation for a line that goes through the points (4, -5) and (2, 3): Example 6: Write an equation for a line that goes through the points (1, 3) and (7,3): Example 7: Write an equation for a line that goes through the points (1, 3) and (1,4): It is easy to graph lines that are written in point-slope form. Just plot the point and use the slope. Example 8: a. Graph the line y 2 = 1 (x + 5) 2 b. Graph the line y 3 = 2(x 1) The slope is: a point is The slope is: a point is 8

9 PRACTICE Find an equation for the line with the given information. Write your answer in whatever form is easiest. Note: Parallel lines have the same slope! 1. slope -1, through (5, -2) 2. slope: 5, through (7,2) 4 3. through (1, 3) and (-2, -5) 4. through (1, 5) and (-2, 5) 5. slope: -2; y-intercept 5 6. slope = 4, through (0, 0) 7. through (3, 4) with undefined slope 8. parallel to y = 2x 7 with y-intercept 5 9

10 9. Graph the following lines. a. y +1 = 2(x 4) b. y + 4 = 2 (x 2) Write the equations of these graphs in point-slope form. a. b. 10

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13 4.4 - Standard Form for Lines Notes There are three common forms for the equations of lines that we will be using this year. For the purposes of definition, coordinates of points are indicated by the letters &. Constants are indicated by the letters, and. We use the letter for slope. Note that capital B stands for and lower case b stand for the. It is important that you be consistent with how you write the letters. Name General Form Specific Example Slope-intercept Today we will focus on STANDARD form. If you are asked to write a linear equation in STANDARD form, you must do three things: 1. Eliminate fractions by multiplying across by the least common multiple of the denominators. You may have to distribute first, if the equation is in point-slope form to start. 2. Move terms with x and y to the left and all constants to the right using addition/subtraction. 3. Make the first term on the left the term with the x and make it positive if it is negative by multiplying everything by negative 1. Example 1. Write the equation of the line y = 2 x 4 in Standard form. 3 13

14 Example 2. Write the equation of the line y 3 = 1 (x 4) in Standard form. 2 Example 3: Write the equation of the line 2y 2 3 x = 4 in Standard form. Given the graph find equation in Ax + By = C form

15 6. Standard form is particularly good for finding x-intercepts and y-intercepts of lines. Remember that an intercept is the point where a crosses an. An x-intercept is the point where a crosses the - axis. A y-intercept is the point where a crosses the - axis. Example 7. Find the x-intercept and the y-intercept of the line 4x + 6y = 12 To find the x-intercept, substitute 0 in for the value. The x-intercept is. To find the y-intercept, substitute 0 in for the value. The y-intercept is. Example 8. Find the x-intercept and the y-intercept of the line 2x 3y = 24 To find the x-intercept, substitute 0 in for the value. The x-intercept is. To find the y-intercept, substitute 0 in for the value. The y-intercept is. 15

16 Example 9. Find the x-intercept and the y-intercept of the line -3x + 6y = 36 The x-intercept is. The y-intercept is. Example 10. Graph the line x - 5y = 10. Hint: find the x-intercept (make y= 0 in equation and solve for x) and y-intercept (make x= 0 in equation and solve for y) y-intercept! x-intercept! X 0 Y Graph the line 3x 2y = 6 y-intercept! x-intercept! X 0 Y 0 16

17 Section 4.4 Classwork State the name for the form of the line shown below. 1. y = mx + b 2. Ax + By = C 3. y y 1 = m( x x 1 ) Rewrite these equations in Standard Form 4. y = 2x 5 5. y = 1 5 x + 3 Find the x-intercept and the y-intercept of these lines. 6. 6x 2y = 12 x-intercept y-intercept 7. 3x + 4y = 9 8. y = 6 9. x = 2 17

18 Use the x and y intercepts to graph these two lines x 3y = x + 2y = 15 18

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21 Section Quiz Review & Study Guide 1. You should know the names (including spelling) and general forms for each of the three forms of linear equations we have studied in this unit. You should also be able to provide a specific example, using numerical values as constants. You should know the difference between a general form and specific example of an equation. Fill in the missing information: Name General Form Specific Example a. Slope-intercept b. (y y 1 ) = m(x x 1 ) c. 2x + 3y = Given the Standard Form of a line, you should be able to find the slope, x-intercept and y- intercept of the line it represents. Find the slope, x-intercept and y-intercept for each of these lines. Linear Equation Slope x-intercept y-intercept a. 3x + 4y = 24 b. 5x 3y = 30 c. x + 2y = 1 d. 6x 10y = 30 e. 8x + 4y = 64 f. 7x 7y = 21 21

22 3. You should be able to convert from one form of a linear equation to another form. Convert these linear equations to Standard Form. a. y = 7x - 6 b. y + 12 = -2(x+3) c. y x = 1 3 d. 2y x = 5 4. Given a line in standard form, you should be able to draw its graph using the x-intercept & y-intercept. a. 6x -10y = 30 b. 7x + 3y =

23 5. Given the graph of a line, you should be able to identify its x-intercept, y-intercept and slope. You should also be able to write its equation in all 3 linear forms. a) x-intercept = b) y-intercept = c) slope = d) Write the equation of the line in point-slope form. e) Write the equation for the line in slope-intercept form. f) Write the equation for the line in standard form. 6. Given information about a line, you should be able to write its equation in point-slope, slopeintercept and Standard forms. a. Write an equation for the line that passes through (1, -3) and has slope = 6 in point-slope form. b. Write an equation for the line that passes through (3, 2) and has slope = 1 3 form. in point-slope c. Write an equation for the line passing through the points (2, 5) and (-3, 1) in point-slope form. d. Write an equation for the line that passes through (-1, 3) and has slope = 2 in slope-intercept form. 23

24 7. You should be able to graph lines presented in all three forms of linear equations. Graph the following lines. a. 4x + 3y = 12 b. y = 3 4 x 2 c. (y 3) = 1 2 (x + 4) d. (y +1) = 2(x + 5) e. y = 4 f. 2x + 3y = 6 24

25 ANSWERS 1a. Slope-intercept y = mx + b y = 4x + 3 1b. Point-slope (y y 1 ) = m(x x 1 ) (y 1) = 2(x 3) 1c. Standard Ax + By = C 2x + 3y = 4 Linear Equation Slope x-intercept y-intercept 2a. 3x + 4y = 24-3/ b. 5x 3y = 30 5/ c. x + 2y = 1-1/2 1 ½ 2d. 6x 10y = 30 3/ e. 8x + 4y = f. 7x 7y = a. 7x y = 6 3b. 2x + y = -18 3c. 9x +12y = 4 3d. x 6y = 15 4a. 4b a. 2.5 b. 5 c. -2 d. y 3 = 2(x 1) e. y = 2x + 5 f. 2x + y = 5 a. y + 3 = 6(x 1) b. y 2 = 1 (x 3) 3 c. y 1 = 4 (x + 3) 5 y 5 = 4 (x 2) 5 d. y = 2x + 5 7a. 7b. 7c. 7d. or 7e. 7f. 25

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27 4.6 Linear Functions as a Model We will continue to use these formulas in this section: Slope Formula: Slope-intercept form for a line: Standard form for a line: Point-Slope form for a line: m = y 2 y 1 x 2 x 1 y = mx + b, where m is slope and b is the y intercept Ax + By = C, where A is positive y y 1 = m(x x 1 ), where (x 1, y 1 ) is a point on the line Definitions: Independent Variable: Dependent Variable: Example A Identify the independent (x) and dependent (y) variables in each situation. 1) Ben collects 3 stamps every month. The number of stamps collected depends on the number of years he keeps his hobby. x = y = 2) Masha plans to decorate her living room area with a carpet, but she is undecided on the length of carpet she will buy. The carpet she likes costs $10 per square foot. x = y = 3) Stephan is placing masking tape around circular mirrors. He wants to find the amount of tape to purchase by measuring the radii for each mirror. x = y = 27

28 4) A woman plans to travel to the Dominican Republic in 6 months for a family reunion. She wants to estimate the travel cost by surveying her family members who have already bought airline tickets about the total cost. She also asked them how many people are traveling in their groups. x = y = 5. Axel s Warehouse has banquet facilities to accommodate a maximum of 250 people. When the manager quotes a price for a banquet she is including the cost of renting the room plus the cost of the meal. A banquet for 70 people costs $1300. For 120 people, the price is $2200. Let p to be the number of people and c the total cost. (a) Determine the dependent variable and independent variable. Independent = Dependent = (b) Identify two points discussed in the problem above. and (c) Determine the slope of the line. (d) What does the slope of the line represent? (e) Construct an equation and rewrite it in slope-intercept form. (f) Plot a graph of cost versus the number of people. (g) What is the y-intercept? What does it mean? (h) Use your equation from part (e) to find the cost of a banquet for the maximum capacity of people for this facility. 28

29 6. When a 40 gram mass was suspended from a coil spring, the length of the spring was 24 inches. When an 80 gram mass was suspended from the same coil spring, the length of the spring was 36 inches. (a) Identify the independent variable and dependent variable in this problem x = y = (b) Graph the line, label the x and y axes. (c) Estimate the length of the spring for a mass of 70 grams and 90 grams based on the graph. for 70 grams: for 90 grams: (d) Determine an equation that models this situation. Write the equation in slope-intercept form. (e) Use the equation to find the length of the spring for a mass of 70 grams and 90 grams. Are these exact answers the same as your estimated ones from part (b)? for 70 grams: for 90 grams: (f) What is the x-intercept? What does it represent? 29

30 7. Jason leaves his summer cottage and drives home. After driving for 5 hours, he is 112 km from home, and after 7 hours, he is 15 km from home. Assume that the distance from home and the number of hours driving form a linear relationship. a. State the dependent and the independent variables. dependent variable: independent variable: b. What are the two points given? point 1: point 2: c. What is the slope and what does it mean in this problem? d. Determine an equation to model this situation. e. Find the distance-intercept (when distance = 0) and explain what it represents. f. How long did it take Jason to drive from his summer cottage to home? 30

31 4.7 More Modeling We will continue to use these formulas in this section: (Fill the blanks) Slope Formula: Slope-intercept form for a line: Standard form for a line: Point-Slope form for a line: 1. A rental car company offers a rental package for a midsize car. The cost is comprised of a fixed $30 administrative fee for the cleaning and maintenance of the car plus a rental cost of $35 per day. a. What is the dependent variable? b. What is the independent variable? c. What is the slope? d. What does the slope mean in the context of this problem? e. Determine an equation to model the relationship between the number of days and the total cost of renting a midsize car. 31

32 2. The same company is advertising a deal on compact car rentals. The linear function y= 30x + 15 can be used to model the relationship between the number of days (x) and the total cost (y) of renting a compact car. a. What is the fixed administrative fee? b. What is the rental cost per day? c. What is the y-intercept? d. What is the slope and what does it mean in this problem? slope = It means In 2008, a collector of sports memorabilia purchased 5 specific baseball cards as an investment. Let y represent each card s resale value (in dollars) and x represent the number of years since purchase. Each of the cards resale values after 0, 1, 2, 3, and 4 years could be modeled by linear equations as follows: Card A: y = 5 0.7x Card C: y = x Card E: y = x Card B: y = x Card D: y = x a. Which card(s) are decreasing in value each year? How can you tell? b. Which card(s) had the greatest initial values at purchase (at 0 years)? c. Which card(s) is increasing in value the fastest from year to year? How can you tell? 32

33 d. If you were to graph the equations of the resale values of Card B and Card C, which card s graph line would be steeper? Explain. e. Write a sentence that explains the 0.9 value in Card C s equation 4. A car starts a journey with 18 gallons of fuel. Assuming a constant rate, the car will consume 0.04 gallons for every mile driven. Let A represent the amount of gas in the tank (in gallons) and m represent the number of miles driven. a. State the dependent and the independent variables. independent: dependent: b. How much gas is in the tank if 0 miles have been driven? How would this be represented on the axes above? c. What is the rate of change that relates the amount of gas in the tank to the number of miles driven? Explain what it means within the context of the problem. 33

34 d. On the axes below, draw the line, or the graph, of the linear function that relates A to m. e. Write the linear function that models the relationship between the number of miles driven and the amount of gas in the tank. 34

35 5. Andrew works in a restaurant. The graph below shows the relationship between the amount Andrew earns and the number of hours he works. a. If Andrew works for 7 hours, approximately how much does he earn? b. Estimate how long Andrew has to work in order to earn $64. c. What is the rate of change of the function given by the graph? d. Interpret the value of the rate of change within the context of the problem. e. Record the coordinates of any two points from the graph. Point 1: Point 2: f. Compute the slope using the two points. (check your answer with your answer in (c)). g. Write linear function that models the relationship in the story using two points written in (e). h. What is the name of the form of the linear function you used? 35

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37 4.8 Unit 4 Review You should know the names and forms of each of the three types of equations we studied in this unit. State the name for the form of the line shown below. 1. y = mx + b 2. Ax + By = C 3. y y 1 = m( x x 1 ) You should be able to identify the slope, x-intercept and y-intercept for an equation written in any of the formats we have studied Find the slope, x-intercept and the y-intercept of these lines. slope x-intercept y-intercept 4. 5x 2y = x + 6y = x = y = 7 8. y + 7 = 5(x + 1) 9. 3x + 6y = x = y x = 0 37

38 You should be able to compute the slope of a line that goes through two given points. Find the slopes of the lines that go through these points. 12. (6, 0) and (2, -12) 13. (-5, 0) and (0, 15) 14. (8, 1) and (12, 1) 15. (3, -5) and (3, 6) You should be able to recognize the most efficient form to use to write an equation for a line given information about it and then write the equation correctly. Give an equation for each of these lines, given the following characteristics: 16. Horizontal line through the point (7, 4) 17. Vertical line through the point (9, 10) 18. A line through the points (7, 4) and (9, 10) 19. A line parallel to the line y = 4x + 5 and through the point (1, 8) 20. A line with slope = 2, through the point (8, 3) A line with slope = 4, with y-intercept (0, 5) 22. A line parallel to the line 2y 5x = 14 and through the point (-5, -2) 38

39 You should be able to transform linear equations written in one form to the other forms. 23. This line is written in point-slope form. Rewrite it in Slope-Intercept Form AND in Standard Form. Point-Slope form: y 4 = 2 (x 1) 3 Slope-Intercept form: Standard form: You should be able to graph lines given their equations in any format. 24. Graph both of these lines on the same 25. Graph the line y = 2 grid: y = 3 and x = -4 3 x 3 For y = 3 slope: y-intercept: For x = -4 slope: y-intercept: For the line y = 2 3 x 3 slope: y-intercept: 39

40 26. Graph the line y + 3 = ¼ (x +7) 27. Graph the line -8x + 4y = 12 You should be able to write the equation of a line given its graph. Write the equation of each line in slope-intercept form ( y = mx +b ) Slope: y-intercept: Slope: y-intercept: Equation: Equation: 40

41 Write the equation of each line in point-slope form y y 0 = m(x x 0 ) You should be able to apply your knowledge of linear functions to applications. 32. Many cell phone companies have you pay a monthly fee that entitles you to a certain number of 'free' minutes of use during that month. You pay the fee whether you use the service for 1 minute or 1 hour. You even pay the fee if you never place a phone call at all during a particular month. However, if you use more than the included time, you are charged for your extra use. Suppose you get a cell phone plan and pay a monthly fee of $ You get 400 'free' minutes for this money. However, if you use more time, you are charged at a rate of $0.35 (35 cents) a minute. a. What will be your bill if you use the service for 3 hours one month? b. What will be your bill if you use the service for 450 minutes one month? 41

42 c. Draw the relation between the Charge (C) and the number of hours (h). Be sure to put a label and units on your vertical axis hours Remember: There are 60 minutes in an hour. 33. The Super Vine small farm has 3 lbs. of veggies in the compost bin and it is recycling 2 lbs. per day. A fruit stand nearby has place in its compost bin 4 lbs. of damage fruits and recycles 3 lbs. each day. a. Make an equation for the number n pounds of recycle material each business has after d days. b. Graph the equations from part a. (Consider if the graph is discrete or continuous) c. After 5 days, how much recycled material has been placed in each compost bin? 42

43 ANSWERS 1. slope-intercept form 2. Standard Form 3. point-slope form 4. slope = 5 2, x-intercept = (4, 0), y-intercept = (0, -10) 5. slope = 2 3, x-intercept = (6, 0), y-intercept = (0, 4) 6. slope = undefined, x-intercept = (10, 0), y-intercept = none 7. slope =0, x-intercept = none, y-intercept = (0, 7) 8. slope = 5, x-intercept = ( 2, 0), y-intercept = (0, -2) 5 9. slope = 1 2, x-intercept = (4, 0), y-intercept = (0, 2) 10. slope = undefined, x-intercept = (-5, 0), y-intercept = none 11. slope = 1, x-intercept = (0, 0), y-intercept = (0, 0) 12. m = = = m = 4 0 ( 5) = 15 5 = m = = 0 6 ( 5) = m = = 11 0 = undefined 16. y = x = y 4 = 3(x 7) or y 10 = 3(x 9) 19. y 8 = 4(x 1) 20. y 3 = 2 5 (x 8) 21. y = 4x + 5 or -4x + y = 5 22, y + 2 = 5 (x + 5) Slope-intercept: y = 2 3 x Standard: 2x 3y = for y = 3: slope: 0 y-int = (0, 3); 25. slope = 2 3 y-int = (0, -3) for x = -4: slope = undefined y-int = none 43

44 y = 3 4 x 3 ; slope = 3 4 y-int = (0, =3) 29. y = 1 4 x + 2 slope = 1 4 y-int = (0, 2) 30. y = 2 5 x 3 or y + 5 = 2 5 (x 5) 31. y = 3 5 x +1 or y 4 = 3 (x 5) 5 32a. Cost = b. Cost = $ (50) = $ c. 33.a. n 1 = 3+ 2d n 2 = 4 + 3d 33b. discrete 33c. n 1 = 3+ 2(5) = 13 n 2 = 4 + 3(5) = 19 44