Graphing Proportional Relationships

Transcription

1 .3.3 Graphing Proportional Relationships equation = m? How can ou describe the graph of the ACTIVITY: Identifing Proportional Relationships Work with a partner. Tell whether and are in a proportional relationship. Eplain our reasoning. a. Earnings (dollars) Mone Hours worked b. Height (meters) Helicopter Time (seconds) Graphing Equations In this lesson, ou will write and graph proportional relationships. c. e. Cost (dollars) 8 6 Tickets Number of tickets Laps, 3 Time (seconds), d. Cost (dollars) f. Cups of Sugar, Cups of Flour, ACTIVITY: Analzing Proportional Relationships Pizzas Number of pizzas 3 Work with a partner. Use onl the proportional relationships in Activit to do the following. Find the slope of the line. Find the value of for the ordered pair (, ). What do ou notice? What does the value of represent? 58 Chapter Graphing and Writing Linear Equations

2 3 ACTIVITY: Deriving an Equation Work with a partner. Let (, ) represent an point on the graph of a proportional relationship. (, ) (, m) (, ) Math Practice View as Components What part of the graph can ou use to find the side lengths? a. Eplain wh the two triangles are similar. b. Because the triangles are similar, the corresponding side lengths are proportional. Use the vertical and horizontal side lengths to complete the steps below. = m = m Simplif. Ratios of side lengths = m Multiplication Propert of Equalit What does the final equation represent? c. Use our result in part (b) to write an equation that represents each proportional relationship in Activit.. IN YOUR OWN WORDS How can ou describe the graph of the equation = m? How does the value of m affect the graph of the equation? 5. Give a real-life eample of two quantities that are in a proportional relationship. Write an equation that represents the relationship and sketch its graph. Use what ou learned about proportional relationships to complete Eercises 3 6 on page 6. Section.3 Graphing Proportional Relationships 59

3 .3 Lesson Lesson Tutorials Direct Variation Stud Tip In the direct variation equation = m, m represents the constant of proportionalit, the slope, and the unit rate. Words When two quantities and are proportional, the relationship can be represented b the direct variation equation = m, where m is the constant of proportionalit. Graph The graph of = m is a line with a slope of m that passes through the origin. (, ) (, m) EXAMPLE Graphing a Proportional Relationship Cost (dollars) 5 Internet Plan 3 (, ) (, ) 3 5 Data used (gigabtes) 6 The cost (in dollars) for gigabtes of data on an Internet plan is represented b =. Graph the equation and interpret the slope. The equation shows that the slope m is. So, the graph passes through (, ) and (, ). Plot the points and draw a line through the points. Because negative values of do not make sense in this contet, graph in the first quadrant onl. The slope indicates that the unit cost is $ per gigabte. Stud Tip EXAMPLE In Eample, the slope indicates that the weight of an object on Titan is one-seventh its weight on Earth. Writing and Using a Direct Variation Equation The weight of an object on Titan, one of Saturn s moons, is proportional to the weight of the object on Earth. An object that weighs 5 pounds on Earth would weigh 5 pounds on Titan. a. Write an equation that represents the situation. Use the point (5, 5) to find the slope of the line. = m Direct variation equation 5 = m(5) Substitute 5 for and 5 for. 7 = m Simplif. So, an equation that represents the situation is = 7. b. How much would a chunk of ice that weighs 3.5 pounds on Titan weigh on Earth? 3.5 = 7 Substitute 3.5 for..5 = Multipl each side b 7. So, the chunk of ice would weigh.5 pounds on Earth. 6 Chapter Graphing and Writing Linear Equations

4 Eercises 7 8. WHAT IF? In Eample, the cost is represented b =. Graph the equation and interpret the slope.. In Eample, how much would a spacecraft that weighs 35 kilograms on Earth weigh on Titan? Distance (meters) 8 6 (, ) EXAMPLE Two-Person Lift (6, ) Time (seconds) 3 Comparing Proportional Relationships The distance (in meters) that a four-person ski lift travels in seconds is represented b the equation =.5. The graph shows the distance that a two-person ski lift travels. a. Which ski lift is faster? Interpret each slope as a unit rate. Four-Person Lift =.5 The slope is.5. The four-person lift travels.5 meters per second. Two-Person Lift change in slope = change in = 8 = The two-person lift travels meters per second. So, the four-person lift is faster than the two-person lift. b. Graph the equation that represents the four-person lift in the same coordinate plane as the two-person lift. Compare the steepness of the graphs. What does this mean in the contet of the problem? The graph that represents the four-person lift is steeper than the graph that represents the two-person lift. So, the four-person lift is faster. Distance (meters) 8 6 Ski Lift four-person two-person Time (seconds) Eercise 9 3. The table shows the distance (in meters) that a T-bar ski lift travels in seconds. Compare its speed to the ski lifts in Eample 3. (seconds) 3 (meters) Section.3 Graphing Proportional Relationships 6

5 .3 Eercises Help with Homework. VOCABULARY What point is on the graph of ever direct variation equation?. REASONING Does the equation = + 3 represent a proportional relationship? Eplain. 9+(-6)=3 3+(-3)= +(-9)= 9+(-)= Tell whether and are in a proportional relationship. Eplain our reasoning. If so, write an equation that represents the relationship TICKETS The amount (in dollars) that ou raise b selling fundraiser tickets is represented b the equation = 5. Graph the equation and interpret the slope. 8. KAYAK The cost (in dollars) to rent a kaak is proportional to the number of hours that ou rent the kaak. It costs $7 to rent the kaak for 3 hours. a. Write an equation that represents the situation. b. Interpret the slope. c. How much does it cost to rent the kaak for 5 hours? Distance (miles) Car 3 5 Gasoline (gallons) 3 9. MILEAGE The distance (in miles) that a truck travels on gallons of gasoline is represented b the equation = 8. The graph shows the distance that a car travels. a. Which vehicle gets better gas mileage? Eplain how ou found our answer. b. How much farther can the vehicle ou chose in part (a) travel than the other vehicle on 8 gallons of gasoline? 6 Chapter Graphing and Writing Linear Equations

6 . BIOLOGY Toenails grow about 3 millimeters per ear. The table shows fingernail growth. a. Do fingernails or toenails grow faster? Eplain. Weeks 3 Fingernail Growth (millimeters) b. In the same coordinate plane, graph equations that represent the growth rates of toenails and fingernails. Compare the steepness of the graphs. What does this mean in the contet of the problem?. REASONING The quantities and are in a proportional relationship. What do ou know about the ratio of to for an point (, ) on the line?. PROBLEM SOLVING The graph relates the temperature change (in degrees Fahrenheit) to the altitude change (in thousands of feet). a. Is the relationship proportional? Eplain. b. Write an equation of the line. Interpret the slope. c. You are at the bottom of a mountain where the temperature is 7 F. The top of the mountain is 55 feet above ou. What is the temperature at the top of the mountain? Temperature ( F) Altitude Change Altitude (thousands of feet) 3. Consider the distance equation d = rt, where d is the distance (in feet), r is the rate (in feet per second), and t is the time (in seconds). a. You run 6 feet per second. Are distance and time proportional? Eplain. Graph the equation. b. You run for 5 seconds. Are distance and rate proportional? Eplain. Graph the equation. c. You run 3 feet. Are rate and time proportional? Eplain. Graph the equation. d. One of these situations represents inverse variation. Which one is it? Wh do ou think it is called inverse variation? Graph the linear equation. (Section.). = 5. = = MULTIPLE CHOICE What is the value of? (Section 3.3) A B 35 C 35 D Section.3 Graphing Proportional Relationships 63