5.2 Using Intercepts

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1 Name Class Date 5.2 Using Intercepts Essential Question: How can ou identif and use intercepts in linear relationships? Resource Locker Eplore Identifing Intercepts Miners are eploring 9 feet underground. The miners ascend in an elevator at a constant rate over a period of 3 minutes until the reach the surface. In the coordinate grid, the horizontal ais represents the time in minutes from when the miners start ascending, and the vertical ais represents the miners elevation relative to the surface in feet. What point represents the miners elevation at the beginning of the ascent? Plot this point. What point represents the miners elevation at the end of the ascent? Plot this point. Elevation below surface (ft) Time (min) Houghton Mifflin Harcourt Publishing Compan Image Credits: Lowell Georgia/Corbis Connect the points with a line segment. What is the point where the graph crosses the -ais? Reflect the -ais? 1. Discussion The point where the graph intersects the -ais represents the beginning of the miners ascent. Will the point where a graph intersects the -ais alwas be the lowest point on a linear graph? Eplain. Module Lesson 2

2 Eplain 1 Determining Intercepts of Linear Equations The graph in the Eplore intersected the aes at (, 9) and (3, ). The -intercept of a graph is the -coordinate of the point where the graph intersects the -ais. The -coordinate of this point is alwas. The -intercept of the graph in the Eplore is 9. The -intercept of a graph is the -coordinate of the point where the graph intersects the -ais. The -coordinate of this point is alwas. The -intercept of the graph in the Eplore is 3. Eample 1 Find the - and -intercepts. 3-2 = 6 To find the -intercept, replace with and solve for. 3-2 () = 6 3 = 6 = 2 The -intercept is = 6 To find the -intercept, replace with and solve for ( ) = 6 To find the -intercept, replace with and solve for. 3 () - 2 = 6-2 = 6 = -3 The -intercept is -3. To find the -intercept, replace with and solve for. -5 ( ) + 6 = 6-5 = 6 6 = 6 = The -intercept is. = The -intercept is. Reflect 2. If the point (5, ) is on a graph, is (5, ) the -intercept of the graph? Eplain. Your Turn Find the - and -intercepts = = 2 Houghton Mifflin Harcourt Publishing Compan Module Lesson 2

3 Eplain 2 Interpreting Intercepts of Linear Equations You can use intercepts to interpret a situation that is modeled b a linear function. Houghton Mifflin Harcourt Publishing Compan Eample 2 Find and interpret the - and -intercepts for each situation. The Sandia Peak Tramwa in Albuquerque, New Meico, travels a distance of about 5 meters to the top of Sandia Peak. Its speed is 3 meters per minute. The function ƒ () = 5-3 gives the tram s distance in meters from the top of the peak after minutes. To find the -intercept, replace ƒ() with and solve for. ƒ() = 5-3 = 5-3 = 15 It takes 15 minutes to reach the peak. To find the -intercept, replace with and find ƒ (). ƒ() = 5-3 ƒ () = 5-3() = 5 The distance from the peak when it starts is 5 m. A hot air balloon is 75 meters above the ground and begins to descend at a constant rate of 25 meters per minute. The function ƒ() = represents the height of the hot air balloon after minutes. To find the -intercept, replace ƒ() with and solve. ƒ () = = = It takes to reach the ground. To find the -intercept, replace with and find ƒ (). ƒ () = Reflect ƒ () = ( ) = 75 The height above ground when it starts is. Distance from peak (m) Height (m) f() Sandia Peak Tramwa Time (min) Height of Hot Air Balloon f() Time (min) 5. Critique Reasoning A classmate sas that the graph shows the path of the tram. Do ou agree? Your Turn 6. The temperature in an eperiment is increased at a constant rate over a period of time until the temperature reaches C. The equation = 5-7 gives the temperature in degrees Celsius hours 2 after the eperiment begins. Find and interpret the - and -intercepts. Module Lesson 2

4 Eplain 3 Graphing Linear Equations Using Intercepts You can use the - and -intercepts to graph a linear equation. Eample 3 Use intercepts to graph the line described b each equation. 1_ 2 = 3-3_ Write the equation in standard form. 3 _ + 1_ 2 = 3 Find the intercepts intercept: 3_ + 1_ 2 () = 3 3_ = 3 = -intercept: 3_ () + 1_ 2 = 3 1_ 2 = 3 = 6 Graph the line b plotting the points (, ) and (, 6) and drawing a line through them. 1 = Write the equation in standard form. = 1 Find the intercepts. -intercept: -intercept: ( ) = 1-12 = 1 = Graph the line b plotting the points and drawing through them. Your Turn 7. Use intercepts to graph 3 = = 1 1 = 1 = and Houghton Mifflin Harcourt Publishing Compan Module 5 21 Lesson 2

5 Elaborate. A line intersects the -ais at the point (a, b). Is a =? Is b =? Eplain. 9. What does a negative -intercept mean for a real-world application? 1. Essential Question Check-in How can ou find the -intercept of the graph of a linear equation using the equation? How is using the graph of a linear equation to find the intercepts like using the equation? Evaluate: Homework and Practice Identif and interpret the intercepts for each situation, plot the points on the graph, and connect the points with a line segment. Online Homework Hints and Help Etra Practice 1. An electronics manufacturer has 1 capacitors, and the same number of capacitors is needed for each circuit board made. The manufacturer uses the capacitors to make 35 circuit boards. 2. A dolphin is 2 feet underwater and ascends at a constant rate for 1 seconds until it reaches the surface. Houghton Mifflin Harcourt Publishing Compan Capacitors remaining Circuit boards made Elevation below sea level (ft) Time (sec) Module Lesson 2

6 Find the - and -intercepts = = 5. + = = _ 5 + 1_ 2 = _ + 5_ = 15 6 Interpret the intercepts for each situation. Use the intercepts to graph the function. 9. Biolog A lake was stocked with 35 trout. Each ear, the population decreases b 1. The population of trout in the lake after ears is represented b the function ƒ () = The air temperature is 6 C at sunrise and rises 3 C ever hour for several hours. The air temperature after hours is represented b the function ƒ () = 3-6. Population f() Temperature ( C) Trout Population Time (ears) f() Houghton Mifflin Harcourt Publishing Compan Time (hours) Module Lesson 2

7 11. The number of brake pads needed for a car is, and a manufacturing plant has brake pads. The number of brake pads remaining after brake pads have been installed on cars is ƒ () = Connor is running a 1-kilometer cross countr race. He runs 1 kilometer ever minutes. Connor s distance from the finish line after minutes is represented b the function ƒ () = 1-1. Use intercepts to graph the line described b each equation. Brake pads remaining Distance from finish line f() f() Number of cars Time (min) = = Houghton Mifflin Harcourt Publishing Compan 15. = 1_ = Module Lesson 2

8 17. 3_ 2 = _ 3 = 2-1_ Kim owes her friend $25 and plans to pa $35 per week. Write an equation of the function that shows the amount Kim owes after weeks. Then find and interpret the intercepts of the function. 2. Eplain the Error Arlo incorrectl found the -intercept of = 1. His work is shown = 1 9 () + 12 = 1 12 = 1 Eplain Arlo s error. = 12 The -intercept is Determine whether each point could represent an -intercept, -intercept, both, or neither. A. (, 5) B. (, ) C. (-7, ) D. (3, -) E. (19, ) 22. A bank emploee notices an abandoned checking account with a balance of $36. The bank charges an $ monthl fee for the account. a. Write and graph the equation that gives the balance ƒ () in dollars as a function of the number of months,. b. Find and interpret the - and -intercepts. Balance ($) f() Time (months) Houghton Mifflin Harcourt Publishing Compan Module 5 21 Lesson 2

9 23. Kathrn is walking on a treadmill at a constant pace for 3 minutes. She has programmed the treadmill for a 2-mile walk. The displa counts backward to show the distance remaining. a. Write and graph the equation that gives the distance ƒ () left in miles as a function of the number of minutes she has been walking. b. Find and interpret the - and -intercepts. Distance remaining (mi) f() Time (min) H.O.T. Focus on Higher Order Thinking 2. Represent Real-World Problems Write a real-world problem that could be modeled b a linear function whose -intercept is 6 and whose -intercept is 6. Houghton Mifflin Harcourt Publishing Compan Image Credits: Mango Productions/Corbis 25. Draw Conclusions For an linear equation A + B = C, what are the intercepts in terms of A, B, and C? 26. Multiple Representations Find the intercepts of 3 + = 12. Eplain how to use the intercepts to determine appropriate scales for the graph and then create a graph. Module Lesson 2

10 Lesson Performance Task A sail on a boat is in the shape of a right triangle. If the sail is superimposed on a coordinate plane, the point where the horizontal and vertical sides meet is (, ) and the sail is above and to the right of (, ). The equation of the line that represents the sail s hpotenuse in feet is 1 + = 2. a. Find and interpret the intercepts of the line and use them to graph the line. Then use the triangle formed b the -ais, -ais, and the line described b the above equation to find the area of the sail. b. Now find the area of a sail whose hpotenuse is described b the equation A + B = C, where A, B, and C are all positive. Height (ft) Length (ft) Houghton Mifflin Harcourt Publishing Compan Module 5 22 Lesson 2