PROPORTIONAL VERSUS NONPROPORTIONAL RELATIONSHIPS NOTES

Transcription

1 PROPORTIONAL VERSUS NONPROPORTIONAL RELATIONSHIPS NOTES Proportional means that if x is changed, then y is changed in the same proportion. This relationship can be expressed by a proportional/linear function (equation): f(x) = mx It is most commonly written as y = mx, where m represents the rate of change or slope of the equation. The graph of a proportional function is a straight line passing through the origin (0,0). It is possible to determine if equations, tables, or graphs are proportional if a few basic steps are followed. Let s examine these steps below with examples and practice problems. EQUATIONS: To determine if an equation is proportional the y, dependent variable, should change at a constant rate relative to the x, independent variable. This means you can multiple or divide the x variable to get the y variable. You cannot add or subtract any additional values to either variable. y = ½ x y = ¾ x + 2 y = -2x + 0 y = x 2 TABLES: To determine if a table is proportional the rate of change must be consistent or equivalent and the origin (0,0) must be one of the relations x y Adults Children Hours Dollars 2 $25 4 $ $ $ GRAPHS: To determine if a graph is proportional, the function must be a straight line that passes through the origin.

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3 NAME DATE PERIOD PROPORTIONAL VERSUS NONPROPORTIONAL RELATIONSHPS For the problems below indicate if a proportional or non-proportional relationship is represented. If proportional, write an equation for the table or graph. Justify your response by showing your work! 1. Proportional: Yes or No x y Proportional: Yes or No x y Proportional: Yes or No Meals Cost $15 $25 $35 $45 4. Proportional: Yes or No # of inches # of cm Proportional: Yes or No 6. Proportional: Yes or No 7. Proportional: Yes or No 8. Proportional: Yes or No 9. Proportional: Yes or No y = ½ x Proportional: Yes or No y = -24x

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5 NAME DATE PERIOD UNDERSTANDING PROPORTIONAL RELATIONSHIPS USING TABLES, GRAPHS AND EQUATIONS

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7 USING TRIANGLES TO FIND THE SLOPE OF LINES NOTES For any straight line the ratio of rise to run is constant. In other words the rate of change or slope of the line is always the same. -Lines that move downward from left to right have a NEGATIVE slope -Lines that move upward from left to right have a POSITIVE slope -Vertical lines have a UNDEFINED slope Question: So how do you determine the slope of a line Using triangles? -Horizontal lines have a ZERO slope Answer: Simply find the coordinates of a minimum of two exact points, create a right triangle to connect the points, and determine their rise (up/down) over their run (left/right), AKA slop,e AKA rate of change. Let s try a few examples. We will determine the slope of the line using triangles AND rather the line has a Negative, positive, undefined, or zero slope. We will also CIRCLE any proportional functions.

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9 NAME DATE PERIOD USING TRIANGLES TO FIND THE SLOPE OF LINES 1) State the slope of each line to the RIGHT of each graph. 2) Determine if the slope is P-positive, N-negative, Z-zero, or U-undefined. 3) Is the slope proportional? Yes or No m= m= m= 2) 2) 2) 3) 3) 3) m= m= m= 2) 2) 2) 3) 3) 3) m= m= m= 2) 2) 2) 3) 3) 3) m= m= m= 2) 2) 2) 3) 3) 3)

10 1. y x 3 x y (x,y) Draw two different right triangles that could be used to calculate the slope of the line. Label them Triangle A and Triangle B. Show how to use the triangles to calculate the slope. Triangle A: Triangle B: Do you get the same answer for each triangle? Why or why not? 2. y 2x 1 x y (x,y) Draw two different right triangles that could be used to calculate the slope of the line. Label them Triangle A and Triangle B. Show how to use the triangles to calculate the slope. Triangle A: Triangle B: Do you get the same answer for each triangle? Why or why not? 3. y 3x 3 x y (x,y) Draw two different right triangles that could be used to calculate the slope of the line. Label them Triangle A and Triangle B. Show how to use the triangles to calculate the slope. Triangle A: Triangle B: Do you get the same answer for each triangle? Why or why not?

11 NAME DATE PERIOD EXIT TICKET: SIMILAR TRIANGLES Find the slope using triangles. Write an equation for problems 2, 4, and 10. YOU MUST SHOW ALL YOUR WORK!! 7) (8, 10), (-7, 14) 8) (-3, 1), (-17, 2) 9) (-20, -4), (-12, -10) 10) (-12, -5), (0, -8)

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13 GRAPHING AND WRITING EQUATIONS IN SLOPE INTERCEPT FORM NOTES Slope Intercept Form y = mx + b You need to know two things to write an equation in slope intercept form, slope (m) and y-intercept (b). Write an equation in slope-intercept form of the line that has a slope of 2 and a y-intercept of 6. To write an equation, you need two things: slope (m) = We have both!! Plug them into slope-intercept form y = mx + b y intercept (b) = Write an equation in slope-intercept form for the following: 1. m = -2 b = 5 2. m = 9 b = m = ½ b = m = - ¾ b = m = 1 b =.5 6. m = -1 b = 0 When given an equation, you can easily graph it in slope-intercept form. All you need is two things slope (m) and y-intercept (b). Step 1: Step 2: Step 3: Plot your y-intercept on the y-axis Apply the slope (rise/run) starting at the y-intercept and plot this second point Draw your line through the points Let s try several problems! Determine if the slope is positive, negative, zero or undefined, AND if it is proportional. y = ½ - 2 y = - ¾ x + 1 y = -2x

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15 NAME DATE PERIOD GRAPHING AND WRITING EQUATIONS IN SLOPE INTERCEPT FORM

16 Write the equation of a line with the given information: 1. Slope of -½, through (0, -4) 2. Slope of ¾ and (0, -5) 3. Through the points (0, -6) and (4, 3) 4. Through the points (0, -2) and (7, -2) 5. Through the points (0, 8) and (-1, 7) 6. Through the points (-5, 8) and (0, -7)

17 NAME DATE PERIOD Write an equation for each graph. EXIT TICKET: WRITING EQUATIONS FROM GRAPHS

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19 NAME DATE PERIOD Rewriting Equations in Slope-Intercept Form

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21 Equation of a Line Given Slope and a Point m and (x, y) Equation of a Line Given Two Points (x 1, y 1 )(x 2, y 2 )

22 First we need to remember slope intercept form y = mx + b 1. Using the slope intercept form, PLUG IN each given piece of information into the equation to solve for b. Let s Try It! through (3, 4), slope = 2 2. Once you have solved for b, rewrite the equation in slope intercept form. EXAMPLE through (0, -4), slope = ½ through (-4, -11), slope = 7 1. Find the slope using a table 2. Use the slope and one of the ordered pair to solve for b. (Follow the ABOVE instructions for slope and a point) Let s Try It! through (2, 5) and (-3, 10) EXAMPLE through (-1, 6) and (5, -4) through (-1, -8) and (4, 12)

23 NAME DATE PERIOD WRITING EQUATIONS OF LINES GIVEN SLOPE AND A POINT Write an equation in slope intercept form given the following. 1. through (-1. 4), slope = through (-5, 1), slope = 3. through (2, -3), slope = 4. through (4, 4), slope = through (3, -4), slope = through (), slope = 7. through (2, 2), slope = 3 8. through (-4, -4), slope = 0 9. through (-3, 0), slope = 10. through (-4, -4), slope = 2

24 NAME DATE PERIOD EQUATION OF LINE GIVEN TWO POINTS Write the slope-intercept form of the equation of the line through the given points. 1) through: (, ) and (, ) 2) through: (, ) and (, ) 3) through: (, ) and (, ) 4) through: (, ) and (, ) 5) through: (, ) and (, ) 6) through: (, ) and (, ) 7) through: (, ) and (, ) 8) through: (, ) and (, ) 9) through: (, ) and (, ) 10) through: (, ) and (, )

25 NAME DATE PERIOD LINEAR EQUATIONS WORD PROBLEMS Suppose you receive $100 for a graduation present, and you deposit it in a saving account. Then each week thereafter you add $5 to the account but no interest is earned. The amount in the account is a function of the number of weeks that have passed. Write an equation for the problem, then use your equation to find when you will have $310 in the account. Marty is spending money at the average rate of $3 per day. After 14 days he has $68 left. The amount left depends on the number of days that have passed. Write an equation for the problem, than determine how much Marty will have after 36 days.

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27 NAME DATE PERIOD LINEAR EQUATION WORD PROBLEMS Fill in the table with at least three ordered pairs including the initial value. Write an equation, graph, and solve for the given value. 1. Lin is tracking the progress of her plant s growth. Today the plant is 5 cm high. The plant grows 1.5 cm per day. a. Write a linear equation and model that represents the height of the plant after d days. b. What will the height of the plant be after 20 days? 2. Mr. Thompson is on a diet. He currently weighs 260 pounds. He loses 4 pounds per month. a. Write a linear equation and model that represents Mr. Thompson s weight after m months. b. After how many months will Mr. Thompson reach his goal weight of 220 pounds?

28 3. Paul opens a savings account with $350. He saves $150 per month. Assume that he does not withdraw money or make any additional deposits. a. Write a linear equation and model that represents the total amount of money Paul deposits into his account after m months. b. After how many months will Paul have more than $2,000? 4. The population of Bay Village is 35,000 today. Every year the population of Bay Village increases by 750 people. a. Write a linear equation and model that represents the population of Bay Village x years from today. b. In approximately many years will the population of Bay Village exceed 50,000 people?

29 Vocabulary: Use the word box to fill in the blanks. Study Guide Slope Y-Intercept Slope Intercept Form Proportional Relationship Linear Equation 1. A has a graph that is a straight line. 2. An equation written in the, is y=mx+b. 3. The is the coordinate of the point where the line crosses the y-axis. Also known as b. 4. The is known as the rate of change between any two points on a line. The ratio of the rise to the run. 5. When the ratio of two variable quantities is constant, a exists where y = mx. Label if each graph is proportional or not. Explain your answer Y or N Explain: Y or N Explain: Y or N Explain: Mary earns $15 dollars for each hour she works. Show if a proportional relationship exists between the number of hours she works and the amount of money she earns by answering the following questions. 9. Write an equation representing the situation: 10. Is this proportional (does it fit the equation y=mx)? Explain: Slope: Label the slope for each graph as a positive slope, negative slope, undefined slope, and zero slope

30 Slope: Find the slope of the line. 15. m= 16. m= Slope-intercept form: Graph the linear equations. Graph at least 3 points on the line. 17. y = -2/3x + 1 m= b= 18. y = 2x 5 m= b=

31 19. y = -3/4x - 8 m= b= 20. y = -4x + 10 m= b= Convert each equation into slope-intercept form (y = mx + b) and graph the line onto the coordinate plane provided y = 6 + 8x m= b= 22. 4x = 2y 10 m= b=

32 Find the equation of each graph in slope-intercept form (y = mx + b). 23. m= b= 24. m= b= EQUATION: EQUATION: Write an equation in slope intercept form for each of the following. 25. Slope of -2 and through (8, 4) 26. Through (2, -5) and (-1, 9) 27. Through (-2, -2) and (8, 8) 28. Slope of 3 and through (2, 5) 29. Slope of -1 and through (-6, -6) 30. Through (-4, 6) and (6, 4)