HW Unit 7: Connections (Graphs, Equations and Inequalities)

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1 Math Fundamentals for Statistics I (Math 5) HW Unit 7: Connections (Graphs, Equations and Inequalities) By Scott Fallstrom and Brent Pickett The How and Whys Guys This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike 4.0 International License 3 rd Edition (Summer 016) Math 5 Homework Unit 7 Page 1

2 Table of Contents This will show you where the homework problems for a particular section start. 7.1 Graphing and Functions Revisited Basics of Equations and Solutions Multi-Variable Equations and Solutions Multi-Variable Equations; Slope and Intercepts Making Linear Equations Part Solving equations Part 1 (The Magic Boxes) Solving Equations Part (The Magic of Algebra) Making Linear Equations Part Modeling Two-Variable Functions Inequalities : Connections Wrap-up (Practice) Graphing and Functions Revisited x-coordinate y-coordinate x-axis y-axis axis 1. Marcus claims that the point (, 9) has an x-intercept of and a y-intercept of 9. What is mixed up?. Consider an arithmetic sequence starting with 10 and with a common difference of 3. a. As an arithmetic sequence (write the equation) b. As an equation representing x and y-coordinates. c. As a function using function notation. 3. Consider f x 4x 3 and x g x 3. Use your calculator to determine the following: a. f 0 b. f 73 c. f,137 d. g 8 e. g 1 f. g 15 Math 5 Homework Unit 7 Page

3 4. On the grid below, plot the points (, 1), ( 1, 4), ( 3, ), (, 4), and ( 5, 0). y A D x C B 5. On the grid above, identify the location of the labeled points. A: B: C: D: 6. Shawn is looking at the weight of his miniature horse. The function t 4.1t 15 w shows the weight in pounds (w) where t measures years of age of the horse. Find and interpret the following: a. w 3 b. w 0 c. w 11 d. w 7 e. Use any two points above to determine the slope of this line. f. Interpret the slope using the appropriate units. g. Ivana used her calculator and found w Explain (without calculating anything else) why this could not be true in the context of the problem. 7. A new function is given by: x j x. Use this function to fill out the rest of the table below. Then graph those points and connect them smoothly. Remember to use parenthesis when plugging in numbers ( ). y x y 1 0 x 1 Math 5 Homework Unit 7 Page 3

4 a. Is the function one of our traditional graphs? Does it look like an arithmetic or geometric graph? Explain. b. This type of function is often called exponential. Based on the location of the variable, explain why this is a good name. x c. Compare and contrast the graphs from hx x and j x. What is similar about the functions and graphs, and what is different? d. The output from a function is the y-coordinate on a graph. Explain how this relates to someone who chooses to write Wrap-up and look back: h x x as y x. 8. Arithmetic functions are graphed as what type of shape? 9. Geometric functions are graphed as what type of shape? 10. Did you have any questions remaining that weren t covered in class? Write them out and bring them 7. Basics of Equations and Solutions solution 1. If Maria plugs x = 5 into an equation and ends up with 13 = 13, what can you say about the number 5?. Jane finds that 7 is a solution to an equation and Jason found that 9 is a solution to the same equation. Can this ever happen? 3. Is it possible that an equation has no solution? Explain why it is not possible or create an equation that has no solution. 4. Determine if the following numbers are solutions to the equation: 3x 111. a. x = 3 c. x = 4 b. x = 1 d. x = 0 5. Determine if the following numbers are solutions to the equation: 8 5x. a. x = c. x = 11 b. x = 6 d. x = Math 5 Homework Unit 7 Page 4

5 6. Determine if the following numbers are solutions to the equation: 7 3x 5. 3 c. x = 11 a. x 8 6 d. x 7 b. x 3 7. Determine if the following numbers are solutions to the equation: 7x 3 0 x. a. x = c. x = 0 b. x = 8 d. x = 1 Wrap-up and look back: 8. Did you have any questions remaining that weren t covered in class? Write them out and bring them 7.3 Multi-Variable Equations and Solutions solution 1. When we have a linear equation, like y x 8, how many solutions are there?. If you plug in the point (4, ) into a linear equation and end up with 3 = 15, what does this mean? 3. Find 3 solutions to y x Find 3 solutions to x 3y Determine if the following are solutions to the equation: x 3y 1. a. (0, 4) d. (30, 16) b. (5, 3) e. (10, 5) c. (15, 6) f. (6, 1) 6. Determine if the following are solutions to the equation: 5x 3y 3 5 a. 5, d., b. (1, 1) c. (1, 6) e. (8., 6) f. ( 3, 13) Math 5 Homework Unit 7 Page 5

6 Wrap-up and look back: 7. Can there be more than one solution to an equation? Explain. 8. Did you have any questions remaining that weren t covered in class? Write them out and bring them 7.4 Multi-Variable Equations; Slope and Intercepts Solution Linear Equation Slope x-intercept y -intercept 1. How many solutions does y x 8 have?. How many solutions does y x 8 have where x = 0? 3. Where is an x-intercept located? 4. For the x-intercept, what is the value of the y-coordinate? 5. Where is a y-intercept located? 6. For the y-intercept, what is the value of the x-coordinate? 7. In the equation y mx b, what do all the variables represent? 8. Find 3 solutions to y x 8. Graph those on a graph like the one below and connect them using a ruler. a. Look at the points you found as solutions: describe how to get from one solution to the next. b. What is the slope of the line? c. What is the y-intercept of the line? 9. Find 3 solutions to x 3y 1. Graph those on a graph like the one below and connect them with a ruler. a. Look at the points you found as solutions: describe how to get from one solution to the next. b. What is the slope of the line? c. What is the y-intercept of the line? 10. Find 3 solutions to y x 1. Graph those on a graph like the one below and connect them with a 3 ruler. a. Look at the points you found as solutions: describe how to get from one solution to the next. b. What is the slope of the line? c. What is the y-intercept of the line? Math 5 Homework Unit 7 Page 6

7 11. Find the slope and y-intercept of the following equations. Then graph the line using this information on a grid. a. 4 y x 3 b. y x c. y 6x 4 y d. 3 y x 1 e. 1 y x f. y 6 x Wrap-up and look back: 1. How can you use the slope and y-intercept to graph a line? 13. Molly says the slope of y 6x 4 is 6x. Is she correct? Explain. 14. Did you have any questions remaining that weren t covered in class? Write them out and bring them 7.5 Making Linear Equations Part 1 None 1. Explain how to find the slope of a line from the graph.. If you find two points of (1, 6) and (7, 3), does it matter which point you start with to find the slope? Show both ways in your explanation. 3. John finds the slope between (1, 6) and (7, 3) by noticing that 1 to 6 goes up 5, and 7 to 3 goes 10 down 10, so he writes the slope as m. Is his technique correct? Explain Jalissa finds the slope between (1, 6) and (7, 3) by noticing that 1 to 7 goes up 6, and 3 to 6 goes 9 3 up 9, so she writes the slope as m. Is her technique correct? Explain. 6 Math 5 Homework Unit 7 Page 7

8 Inside Temperature in F For the following exercises, practice writing in terms of x and y, but also as a function like f x. 5. Find an equation of the line with slope of 5 and y-intercept of (0, ) Find an equation of the line with slope of and y-intercept of (0, 4) Find an equation of the line with slope of and y-intercept of (0, 13) Find an equation of the line passing through (0, 5) and (, 14). 9. Find an equation of the line passing through (0, ) and (, 8). 10. Find an equation of the line passing through (0, 5) and (1, 3). 11. Find an equation of the line passing through (0, 7) and (, 15). 1. Find an equation of the line passing through (0, 8) and ( 4, ). 13. Find an equation of the line passing through (0, 1) and ( 3, 7). 14. The scatterplot below measures the cost to keep a home at a fixed temperature on a 100 degree day. a. For the scatterplot, draw a line that best fits the data points. b. Write the equation of the line as a function (linear equation). c. Interpret the slope (with the appropriate units). AC cost and temperature Cost of Daily AC in Dollars Math 5 Homework Unit 7 Page 8

9 Wrap-up and look back: 15. If you have a scatterplot, how can you find the y-intercept of the line that best fits the data? 16. If you have a scatterplot, how can you find the slope of the line that best fits the data? 17. Did you have any questions remaining that weren t covered in class? Write them out and bring them 7.6 Solving equations Part 1 (The Magic Boxes) None 1. How does the order of operations come up when writing the operations above the arrows?. When solving an equation with these boxes, where do you put the number that it is equal to? 3. Solve these box problems: a b c Practice using these boxes to solve some equations. Write down the operations you re doing in the order you re doing them. Then solve with the box technique. x 3 a. 7 4 b. 4x Math 5 Homework Unit 7 Page 9

10 c. x 1 d. 6x 4 10 x 3 e. 4 5 Wrap-up and look back: 5. What s the coolest thing about solving equations with the boxes? 6. Did you have any questions remaining that weren t covered in class? Write them out and bring them 7.7 Solving Equations Part (The Magic of Algebra) Properties of Equality Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality 1. How do the properties of equality relate to a balance?. Solve these equations using the properties of equality. Remember, each step needs a reason! Check your work too! a. 5x 7 = 8 f. x 13 = 4x b. 3x + 11 = 50 c. 3(x 7) = 7 x 7 d. 9 3 e. 3x + 4 = 5x 14 g. (x ) = 5x 6 h. 3x = 14 11x i. x 8 x 4 7 j. 6.x x 3 Math 5 Homework Unit 7 Page 10

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12 Wrap-up and look back: 3. When you have fractions in an equation, what could you do to both sides so that the fractions go away? 4. When you have decimals in an equation, what could you do to both sides so that the decimals go away? 5. Did you have any questions remaining that weren t covered in class? Write them out and bring them 7.8 Making Linear Equations Part None 1. If we are given any two points, can we create a line between them?. Solve for b: 3 b Solve for b: 5 b In y mx b, solve for b when x = 3, y = 11, and m =. 5. In y mx b, solve for b when x = 3, y = 11, and m =. 6. In y mx b, solve for b when x = 3, y = 11, and m = In y mx b, solve for b when x = 3, y = 4, and m = In y mx b, solve for b when x = 3, y = 4, and m = In y mx b, solve for b when x = 3, y = 4, and m = 10. Find an equation of the line through the two points a. (0, 7) and ( 4, 8) b. ( 3, 0) and ( 5, 8) Wrap-up and look back: c. (6, 0) and (7, 3) d. (0, 8) and (, 0) 11. Did you have any questions remaining that weren t covered in class? Write them out and bring them Math 5 Homework Unit 7 Page 11

13 7.9 Modeling Two-Variable Functions None 1. If you model an equation, what part is the constant? (slope or y-intercept). If you model an equation, what part is the change? (slope or y-intercept) 3. If an equation takes pennies as the input an gives ounces as the output, what are the units on the slope? 4. If an equation takes seconds as the input an gives speed (in miles per hour) as the output, what are the units on the slope? 5. When we have a function like x f, what does 0 f represent? 6. Create a function that measures the speed of a car in miles per hour, S, based on time in seconds, t. At time 0 (when we start), the speed is 65 mph. 3 seconds later the speed is 0 mph. Find and interpret S. S t, then find 7. Create a function that measures the cost of a plumber in dollars, C, based on time working in hours, t. A 4 hour job costs $380 and an 11 hour job costs $905. Find C 8 and C 0. C t, then find and interpret 8. Create a function that measures the wage of a tutor in dollars per hour (per student), W, based on the number of students at a tutoring session, n. For 1 student, the tutor charges $30/hr, but for 5 students, the tutor charges $0/hr (per student). Find W n, then find and interpret 7 W. 9. Create a function that measures the salary of a sociologist in dollars per year, S, based on the number of years working, x. The salary after working 4 years is $45,000 and after 13 years is $63,000. Find S x, then find and interpret 5 S and 0 S. 10. The starting salary for a person with a Bachelor s degree in engineering is $64,000. It is expected that they will receive a $,000 raise each additional year they work. Write a function that models this salary structure. Let x represent the number of additional years and S(x) represent the salary in dollars. a. Find a formula (function form) for S x. b. Based on the formula found above, find and interpret the following: i. S 1 ii. S 0 iii. S 5 iv. What value of x makes x 80, 000 S? Math 5 Homework Unit 7 Page 1

14 11. The starting salary for an person with a Bachelor s degree in Liberal Arts is $36,000. It is expected that they will receive a $1,500 raise for each additional year they work. Write a function that models this structure. Let x represent the number of additional years and S(x) represent the salary in dollars. a. Find a formula (function form) for S x. b. Based on the formula found above, find and interpret the following: i. S 1 ii. S 0 iii. S 5 iv. What value of x makes x 80, 000 S? 1. Create a function that measures the cost of cheese in dollars, C, based on the number of pounds (p). 6 pounds costs $45 and 8 pounds costs $60. Find 3 C 0. Wrap-up and look back: C p, then find and interpret C and 13. If we have any two data points, does that mean we can find a linear function that goes through both? 14. Does that mean that everything is linear? Are there times when a line wouldn t really capture the data accurately? 15. Did you have any questions remaining that weren t covered in class? Write them out and bring them 7.10 Inequalities Inequalities 1. Will an inequality that involves always have the arrow going to the right? Why or why not.. How do we represent = on the number line graph? 3. How do we represent on the number line graph? 4. Will an inequality that involves always have the arrow going to the left? Why or why not. 5. Find the solution on the number line using a graph. a. x 6 b. x. 5 c. x 4 d. x 10 e. x. 1 f. x Math 5 Homework Unit 7 Page 13

15 6. Describe the solutions to the graph in words. a. b. ( ] ) ( c. ) [ 6 8 d. e. Wrap-up and look back: Did you have any questions remaining that weren t covered in class? Write them out and bring them 7.11: Connections Wrap-up (Practice) [ [ ) ( ) A group of people is creating boxes. The person training them completes a few boxes prior to everyone starting work. The equation showing the number of boxes made in h hours is B h 7h 3. a. Find and interpret the value of B. b. Find and interpret the value of B 6. c. Find the slope and interpret what it means using the appropriate units. d. Find the y-intercept and interpret what it means using the appropriate units. e. Find and interpret the value of h that makes B h 66. f. Peter says that to find how long it takes to make 50 boxes you should find B 50. Is Peter correct?. For the following equations, find the slope and y-intercept. a. y x 7 e. y x 34 b. y x 4 7 c. y 0.15x d. y 17 f. y 0.15x 4. g. 3 y x 4 8 h. y 5 Math 5 Homework Unit 7 Page 14

16 3. Based on the graph above, determine the following: a. An equation as a function for meals in terms of hours: M h b. Find and interpret the value of M 8. c. Find and interpret the value of M 3. d. Interpret what the slope means. e. Interpret what the y-intercept means. f. Find and interpret the value of h that makes M h 750. g. Use your function to determine and interpret the value of M Find at least 3 solutions to these equations: a. y x 1 b. 4x 6y 4 c. x y 3 d. 0 5y 15 x 5. Determine if the following numbers are solutions to the equation: 3x 0 11 a. x = 3 b. x = 8 6. Determine if the following are solutions to the equation: x 3y 1 a. (10, 0) c. (0, 4) b. (3, 3) d. ( 0, 15) Math 5 Homework Unit 7 Page 15

17 B A Years after Based on the graph above, a function y P is created. This equation is not a line, but we can still find information from the graph. a. Find and interpret P 4. b. Interpret the point (0,1.0). c. What is the price per gallon in 013? Write this in function notation. d. Interpret the y-intercept of the graph. e. Does this graph have an x-intercept? Explain. f. What is the slope of the line between point A and point B? Interpret what this slope means. 8. Solve the equation for y. When finished, find the slope and y-intercept. a. 4x 3y = 16 b. 30x y = Use the box method to solve these equations: a. 5x 9 16 b. 6x 5 7 c. 3x 8 16 d. 6x 5 48 Math 5 Homework Unit 7 Page 16

18 y 10. Graph the equation y 3x Graph the equation y x. 4 x 1. Find an equation of the line through the two points. a. ( 5, 5) and ( 8, 1) b. ( 5, 5) and ( 6, ) c. (0, 4) and ( 3, 11) d. (0, ) and ( 3, 11) 13. Graph the solutions on the number line.. a. x 4. 7 b. x Describe the solutions to the graph in words. a. b. [ ) ) ( c. ) [ 3 0 Math 5 Homework Unit 7 Page 17