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

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 Linear Functions Rocket Speed Time y x

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 Linear Functions

Learn to identify and graph linear equations. Course Linear Functions

Vocabulary linear equation linear function Insert Lesson Title Here Course Linear Functions

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 x Miles Hours 0 Course Linear Functions

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 Linear Functions

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 Linear Functions

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 – –2 –4 (0, –1) (1, 3) (–1, –5) Course Linear Functions

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 Linear Functions

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 –2 –4 (0, –1) (–2, –1) Course Linear Functions

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) –2 (0, 1) (1, 4) (–1, –2) Try This: Example 1A Course Linear Functions

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 – –2 –4 (0, 1) (1, 4) (–1, –2) Course Linear Functions

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) (0, 1) (3, 1) (–2, 1) The equation y = 1 is the same equation as y = 0x + 1. Try This: Example 1B Course Linear Functions

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 –2 –4 (0, 1)(–2, 1) Course Linear Functions

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 Linear Functions

Additional Example 2 Continued InputRuleOutput 15(x) + 30 y = 15x + 30 x (0) (2) (4) + 30 y x Course Linear Functions

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 Linear Functions

Try This: Example 2 Insert Lesson Title Here x y = 7x + 21 InputRuleOutput 7(x) + 21 x (0) (2) (4) + 21 y Course Linear Functions

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

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 Linear Functions