1. 12-4 Linear Functions Course 2 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

2. Warm Up Interpret the graph. A rocket is fired into the air. The rocket’s speed increases until gravity gradually slows the rocket and causes it to fall to the ground. Course 2 12-4 Linear Functions Rocket Speed Time y x

3. Problem of the Day The mean of a, 31, 42, 65, and b is 51. The greatest number is 67 more than the least number. What are the missing numbers? 25 and 92 Course 2 12-4 Linear Functions

4. Learn to identify and graph linear equations. Course 2 12-4 Linear Functions

5. Vocabulary linear equation linear function Insert Lesson Title Here Course 2 12-4 Linear Functions

6. The graph at right shows how far an inner tube travels down a river if the current flows 2 miles per hour. The graph is linear because all the points fall on a line. It is part of the graph of a linear equation. A linear equation is an equation whose graph is a line. The solutions of a linear equation are the points that make up its graph. Linear equations and linear graphs can be different representations of linear functions. A linear function is a function whose graph is a non vertical line. y 0 24 2 4 6 x Miles Hours 0 Course 2 12-4 Linear Functions

7. You need to know only two points to draw the graph of a linear function. However, graphing a third point serves as a check. You can use a function table to find each ordered pair. Course 2 12-4 Linear Functions

8. Graph the linear function. Additional Example 1A: Graphing Linear Functions A. y = 4x – 1 Input RuleOutput Ordered Pair x4x – 1y(x, y) 0 1 –1 4(0) – 1 4(1) – 1 4(–1) – 1 –1 3 –5 (0, –1) (1, 3) (–1, –5) Course 2 12-4 Linear Functions

9. Additional Example 1A Continued Graph the linear function. A. y = 4x – 1 Place each ordered pair on the coordinate grid and then connect the points with a line. x y 0 –2 –4 24 2 4 –2 –4 (0, –1) (1, 3) (–1, –5) Course 2 12-4 Linear Functions

10. Graph each linear function. Additional Example 1B: Graphing Linear Functions B. y = –1 InputRule Output Ordered Pair x0x – 1y(x, y) 0 3 –2 0(0) – 1 0(3) – 1 0(–2) – 1 –1 (0, –1) (3, –1) (–2, –1) The equation y = –1 is the same equation as y = 0x – 1. Course 2 12-4 Linear Functions

11. Additional Example 1B Continued Graph the linear function. B. y = –1 Place each ordered pair on the coordinate grid and then connect the points with a line. (3, –1) x y 0 2 4 –2 –4 (0, –1) (–2, –1) Course 2 12-4 Linear Functions

12. Graph the linear function. A. y = 3x + 1 Input RuleOutput Ordered Pair x3x + 1y(x, y) 0 1 –1 3(0) + 1 3(1) + 1 3(–1) + 1 1 4 –2 (0, 1) (1, 4) (–1, –2) Try This: Example 1A Course 2 12-4 Linear Functions

13. Try This: Example 1A Continued Graph the linear function. A. y = 3x + 1 Place each ordered pair on the coordinate grid and then connect the points with a line. x y 0 –2 –4 24 2 4 –2 –4 (0, 1) (1, 4) (–1, –2) Course 2 12-4 Linear Functions

14. Graph each linear function. B. y = 1 InputRule Output Ordered Pair x0x + 1y(x, y) 0 3 –2 0(0) + 1 0(3) + 1 0(–2) + 1 1 1 1 (0, 1) (3, 1) (–2, 1) The equation y = 1 is the same equation as y = 0x + 1. Try This: Example 1B Course 2 12-4 Linear Functions

15. Try This: Example 1B Continued Graph the linear function. B. y = 1 Place each ordered pair on the coordinate grid and then connect the points with a line. (3, 1) x y 0 2 4 –2 –4 (0, 1)(–2, 1) Course 2 12-4 Linear Functions

16. The fastest-moving tectonic plates on Earth move apart at a rate of 15 centimeters per year. Scientists began studying two parts of these plates when they were 30 centimeters apart. How far apart will the two parts be after 4 years? Additional Example 2: Earth Science Application The function y = 15x + 30, where x is the number of years and y is the spread in centimeters. Course 2 12-4 Linear Functions

17. Additional Example 2 Continued InputRuleOutput 15(x) + 30 y = 15x + 30 x 0 2 4 15(0) + 30 15(2) + 30 15(4) + 30 y 30 60 90 x Course 2 12-4 Linear Functions

18. Try This: Example 2 Insert Lesson Title Here Dogs are considered to age 7 years for each human year. If a dog is 3 years old today, how old in human years will it be in 4 more years? Write a linear equation which would show this relationship. Then make a graph to show how the dog will age in human years over the next 4 years. The function y = 7x + 21, would describe this situation where x is the number of years, 21 is the current age and y would be the future age. Course 2 12-4 Linear Functions

19. Try This: Example 2 Insert Lesson Title Here x y = 7x + 21 InputRuleOutput 7(x) + 21 x 0 2 4 7(0) + 21 7(2) + 21 7(4) + 21 y 21 35 49 Course 2 12-4 Linear Functions

20. Lesson Quiz: Part 1 Graph the linear functions. 1. y = 3x – 4 2. y = –x + 4 3. y = 2 Insert Lesson Title Here y = 3x – 4 y = –x +4 y = 2 Course 2 12-4 Linear Functions

21. Lesson Quiz: Part 2 4. The temperature of a liquid is decreasing at a rate of 12°F per hour. Susan begins measuring the liquid at 200°F. What will the temperature be after 5 hours? 140°F Insert Lesson Title Here Course 2 12-4 Linear Functions