Relationships and Graphs 7.4.A. Graphing Proportional Relationships You can use a graph to explore proportional relationships.
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1 ? LESSN.3 Proportional Relationships and Graphs ESSENTIAL QUESTIN Proportionality 7..A Represent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt. Also 7..C, 7..D How can you use graphs to represent and analyze proportional relationships? EXPLRE ACTIVITY 7..A Graphing Proportional Relationships You can use a graph to explore proportional relationships. Most showerheads that were manufactured before 199 use 5 gallons of water per minute. Is the relationship Each minute, 5 gallons between the number of gallons of water and the of water are used. So number of minutes a proportional relationship? for minutes, 5 gallons are used. A Complete the table. Time (min) 1 3 Water Used (gal) 5 35 B Based on the table, is this a proportional relationship? Explain your answer. C D E F Write the data in the table as ordered pairs (time, water used). (1, 5), (, ), (3, ), (, 35), (, ) Plot the ordered pairs. If the showerhead is used for 0 minutes, how many gallons of water will be used? What ordered pair represents this situation? What is this location called? Draw Conclusions If you continued the table to include 3 minutes, would the point (3, 15) be on this graph? Why or why not? Water used (gal) Water Use 1 0 Time (min) Lesson.3 73
2 Math n the Spot Identifying Proportional Relationships In addition to using a table to determine if a relationship is proportional, you also can use a graph. A relationship is a proportional relationship if its graph is a straight line through the origin. EXAMPLE 1 7..A A house cleaning company charges $5 per hour. Is the relationship a proportional relationship? Explain. Each hour costs $5. So for hours, the cost is $5 = $90. STEP 1 Make a table. Animated Math STEP STEP 3 Time (h) Total cost ($) Write the data in the table as ordered pairs (time, cost). (1, 5), (, 90), (3, 135), (5, 5), (, 30) Graph the ordered pairs. Total Cost ($) Connect the points with a line. Time (h) The graph is a line that goes through the origin. So, the relationship is proportional. Place time on the x-axis and total cost on the y-axis. Plot each point. YUR TURN 1. Jared rents bowling shoes for $ and pays $5 per bowling game. Graph the data. Is the relationship a proportional relationship? Explain. Games 1 3 Total Cost ($) Total Cost ($) Games Personal Math Trainer nline Assessment and Intervention 7 Unit
3 Analyzing Graphs Recall that you can describe a proportional relationship with the equation y = kx. The constant of proportionality k tells you how steep the graph of the relationship is. The greater the absolute value of k, the steeper the line. EXAMPLE The graph shows the relationship between time in minutes and the number of miles Damon runs. Write an equation for this relationship. STEP 1 STEP STEP 3 Choose a point on the graph and tell what the point represents. The point (5,.5) represents the distance (.5 miles) that Damon runs in 5 minutes. What is the constant of proportionality? Because distance =.5 mi time of proportionality is 1. 5 min = 1, the constant Write an equation in the form y = kx. y = 1 x Distance (mi) Distance (mi) Time (min) Reflect. Communicate Mathematical Ideas What does the point (0, 0) on the graph represent? 3. What If? Esther runs faster than Damon. Suppose you drew a graph representing the relationship between time in minutes and distance run for Esther. How would the graph compare to the one for Damon? YUR TURN. The graph shows the relationship between the distance a bicyclist travels and the time in hours. a. What does the point (, 0) represent? A, 7..C Math n the Spot The points appear to form a line through the origin so the relationship is proportional. Math Talk Mathematical Processes Give some examples of other situations that could be represented by proportional relationships. b. What is the constant of proportionality? c. Write an equation in the form y = kx for this relationship. Time (h) Personal Math Trainer nline Assessment and Intervention Lesson.3 75
4 Guided Practice Complete each table. Tell whether the relationship is a proportional relationship. Explain why or why not. (Explore Activity) 1. A student reads 5 pages per hour. Time (h) 3 5 Pages 55. A babysitter makes $7.50 per hour. Time (h) 5 Pages.50 0 Tell whether the relationship is a proportional relationship. Explain why or why not. (Explore Activity and Example 1) 3. Chores. Movie Rentals Number of chores Cost ($) Age (years) 1 0 Number of movies Write an equation of the form y = kx for the relationship shown in each graph. (Example )? 5.. Balloon height (ft) Time (s) Number of items ESSENTIAL QUESTIN CHECK-IN 7. How does a graph show a proportional relationship? Cost ($) 7 Unit
5 Name Class Date.3 Independent Practice 7..A, 7..C, 7..D For Exercises 1, the graph shows the relationship between time and distance run by two horses. Distance (mi) Time (min). Explain the meaning of the point (0, 0). B A Personal Math Trainer nline Assessment and Intervention 13. A bullet train can travel at 170 miles per hour. Will a graph representing distance in miles compared to time in hours show a proportional relationship? Explain. 1. Critical Thinking When would it be more useful to represent a proportional relationship with a graph rather than an equation? 9. How long does it take each horse to run a mile?. Multiple Representations Write an equation for the relationship between time and distance for each horse. 11. Draw Conclusions At the given rates, how far would each horse run in 1 minutes? 1. Analyze Relationships Draw a line on the graph representing a horse than runs faster than horses A and B. 15. Multiple Representations Bargain DVDs cost $5 each at Mega Movie. a. Graph the proportional relationship that gives the cost y in dollars of buying x bargain DVDs. Cost ($) DVDs b. Give an ordered pair on the graph and explain its meaning in the real world context. Lesson.3 77
6 The graph shows the relationship between distance and time as Glenda swims. 1. How far did Glenda swim in seconds? 17. Communicate Mathematical Ideas Is this a proportional relationship? Explain your reasoning. 1. Multiple Representations Write an equation that shows the relationship between time and distance. Distance (ft) Time (s) FCUS N HIGHER RDER THINKING 19. Make a Conjecture If you know that a relationship is proportional and are given one ordered pair other than (0, 0), how can you find another pair? Work Area The tables show the distance traveled by three cars. Car 1 Car Car 3 Time (h) Distance (mi) Time (h) Distance (mi) Time (h) Distance (mi) Communicate Mathematical Ideas Which car is not traveling at a constant speed? Explain your reasoning. 1. Make a Conjecture Car is traveling at twice the rate of speed of car. How will the table values for car compare to the table values for car? 7 Unit
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1. Write an inequality that describes the graph. 0 1. Use words to describe the line graph. 4. Create a line graph of the inequality. x > 4 4. Write an inequality that describes the graph. 0 5. Write an
GUIDELINES FOR COMPLETING THE ASSIGNMENT
RAHWAY HIGH SCHOOL MATHEMATICS DEPARTMENT Algebra 1 Summer Assignment packet Summer 2018 Due date: September 7th GUIDELINES FOR COMPLETING THE ASSIGNMENT This packet was created to help you succeed in
A theme park charges $12 entry to visitors. Find the money taken if 1296 people visit the park.
Write an Equation An equation is a term used to describe a collection of numbers and variables related through mathematical operators. An algebraic equation will contain letters that relate to real quantities
Algebra I EOC Packet #
1. Which inequality best describes the graph shown below? A y > x + 5 B y < x + 5 C y < x + 5 D y > x + 5 2. The table shows a set of values for x and y. x -3-2 1 3 6 y 7 5-1 -5-11 Which equation best
B; = to 2, or 3 : 2; The ratio means that there are 3 tulips for
Chapter 7. a. Old Mixture New Mixture The smallest possible container for these ratios has nuts. Almonds Both graphs begin at (0, 0) The graph for the new mixture is slightly steeper, so it has a greater