Preview Warm Up California Standards Lesson Presentation

A rocket is fired into the air. Warm Up Interpret the graph. A rocket is fired into the air. y Rocket ‘s Altitude x Time Possible answer: The rocket’s speed increases until gravity gradually slows the rocket and causes it to fall to the ground.

AF3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph. California Standards

Vocabulary linear equation linear function

Recall that the solution of an equation with one variable is the value of the variable that makes the equation true. The solutions of an equation with two variables are the ordered pairs that make the equation true. When these ordered pairs form a line, the equation is called a linear equation. A function described by a linear equation is a linear function. To graph a linear function, plot some solutions of the related linear equation, then draw a line through them. The line represents all of the ordered pair solutions of the equation.

For example, the function that relates distance d, rate r, and time t is described by the linear equation d = rt. This graph shows solutions of this equation when r = 2 feet per second. Time (s) Distance (ft)

Additional Example 1: Graphing Linear Functions Graph the linear function y = 4x – 1. Make a table. Input Rule Output Ordered Pair x 4x – 1 y (x, y) Substitute positive, negative, and zero values for x. 4(0) – 1 –1 (0, –1) 1 4(1) – 1 3 (1, 3) –1 4(–1) – 1 –5 (–1, –5)

Not all linear equations describe functions Not all linear equations describe functions. The graphs of some linear equations are vertical lines, which do not pass the vertical line test. Helpful Hint

Additional Example 1 Continued Graph the linear function y = 4x - 1. x y –2 –4 2 4 (1, 3) Plot each ordered pair on the coordinate grid and then connect the points with a line. (0, –1) (–1, –5)

Graph the linear function y = 3x + 1. Check It Out! Example 1 Graph the linear function y = 3x + 1. Make a table. Input Rule Output Ordered Pair x 3x + 1 y (x, y) Substitute positive, negative, and zero values for x. 3(0) + 1 1 (0, 1) 1 3(1) + 1 4 (1, 4) –1 3(–1) + 1 –2 (–1, –2)

Check It Out! Example 1 Continued Graph the linear function y = 3x + 1. x y (1, 4) 4 Plot each ordered pair on the coordinate grid. Then connect the points with a line. 2 (0, 1) –4 –2 2 4 –2 (–1, –2) –4

Additional Example 2: Earth Science Application 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. Write a linear function that describes the movement of the plates over time. Then make a graph to show the movement over 4 years. Let x represent the input and y represent the output. The function is y = 15x + 30, where x is the number of years and y is the total distance apart the two plates are.

Additional Example 2 Continued Input Rule Output x 15(x) + 30 y Multiply the input by 15 and then add 30. 15(0) + 30 30 2 15(2) + 30 60 4 15(4) + 30 90

Additional Example 2 Continued y 100 80 60 40 20 2 4 8 10 12 Graph the ordered pairs (0, 30), (2, 60), and (4, 90) from your table. Connect the points with a line. Distance (cm) x Years

Check It Out! Example 2 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. Let x represent the input and y represent the output. The function is y = 7x + 21, where x is the number of years from now and y is the total age of the dog in human years.

Check It Out! Example 2 Continued Input Rule Output x 7(x) + 21 y Multiply the input by 7 and then add 21. 7(0) + 21 21 2 7(2) + 21 35 4 7(4) + 21 49

Check It Out! Example 2 Continued y 80 60 40 20 2 4 8 10 Graph the ordered pairs (0, 21), (2, 35), and (4, 49) from your table. Connect the points with a line. Age in Human Years x Years

Graph the linear functions. 1. y = 3x – 4 2. y = –x + 4 3. y = 2 Lesson Quiz: Part I Graph the linear functions. 1. y = 3x – 4 2. y = –x + 4 3. y = 2 y = –x +4 y = 2 y = 3x – 4

Lesson Quiz: Part II 4. The temperature of a liquid is decreasing at a rate of 12°F per hour. Susan begins measuring the liquid at 200°F. Write a linear function that describes the change in temperature over time. Then make a graph to show the temperature over 5 hours. y 200 180 160 140 120 1 2 3 4 5 y = 200 – 12x