Identify the quadrant that contains each point. 1.(6, –4) 2. (5, 3) Warm Up Identify the quadrant that contains each point. 1.(6, –4) 2. (5, 3) 3. (–5, –2) IV I III

Vocabulary linear equation linear function

The table shows how far a kayak travels down a river if the kayak is moving at a rate of 2 miles per hour. Notice for all ordered pairs in the table for every 1 hour increase in time, the miles traveled increases by 2. These ordered pairs are in proportion. y 2 4 6 x Miles Hours 1 2 2 4 3 6 3 8 = = = If the ordered pairs are in proportion, then the data represents a proportional relationship. When you graph a proportional relationship, the result is a line that passes through the origin.

Additional Example 1: Graphing Proportional Relationships Graph the linear function y = 4x. Make a table. x 1 2 3 y 4 8 12 Proportional relationships pass through (0, 0). Graph the ordered pairs (0, 0), (1, 4), (2, 8), (3, 12).

Additional Example 1 Continued y 12 (3, 12) 10 Place each ordered pair on the coordinate grid and then connect the points with a line. 8 (2, 8) 6 4 (1, 4) The graph is a straight line that passes through the origin. 2 (0, 0) x 2 4 6 8 10 Check 1 4 2 8 3 12 = = The ordered pairs are proportional.

x 1 2 3 y 2 4 6 Check It Out: Example 1 Graph the linear function y = 2x. Make a table. x 1 2 3 y 2 4 6 Proportional relationships pass through (0, 0). Graph the ordered pairs (0, 0), (1, 2), (2, 4), (3, 6).

The graph is a straight line that passes through the origin. Check It Out: Example 1 y 10 Place each ordered pair on the coordinate grid and then connect the points with a line. 8 6 (3, 6) 4 (2, 4) The graph is a straight line that passes through the origin. 2 (1, 2) (0, 0) x 2 4 6 8 10 Check 1 2 2 1 3 6 = = The ordered pairs are proportional.

A linear equation is an equation whose graph is a line 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 nonvertical line.

Some relationships are linear but not proportional Some relationships are linear but not proportional. If the ordered pairs in a linear function are not all proportional then it is not a proportional relationship. These non-proportional relationships do not pass through the origin on a graph.

Additional Example 2A: Identify Proportional Relationships Tell whether the function is a proportional relationship. Then graph the function. A. y = –2x Make a table. x –1 1 2 3 y 2 –2 –4 –6 –1 2 1 –2 2 –4 3 –6 = = = The ordered pairs are proportional and the graph passes through (0, 0). y = –2x is a proportional relationship.

Additional Example 2B: Identify Proportional Relationships Tell whether the function is a proportional relationship. Then graph the function. B. y = x – 5 Make a table. x –1 1 2 3 y –6 –5 –4 –3 –2 –1 –6 1 –4 2 –3 3 –2 ≠ ≠ ≠ The ordered pairs are not proportional and the graph does not pass through (0, 0). y = x – 5 is not a proportional relationship.

Check It Out: Example 2A Tell whether the function is a proportional relationship. Then graph the function. A. y = –3x Make a table. x –1 1 2 3 y 3 –3 –6 –9 –1 3 1 –3 2 –6 3 –9 = = = The ordered pairs are proportional and the graph passes through (0, 0). y = –3x is a proportional relationship.

Check It Out: Example 2B Tell whether the function is a proportional relationship. Then graph the function. B. y = x – 4 Make a table. x –1 1 2 3 y –5 –4 –3 –2 –4 –1 –5 1 –3 2 –2 3 –1 ≠ ≠ ≠ The ordered pairs are not proportional and the graph does not pass through (0, 0). y = x – 4 is not a proportional relationship.

Additional Example 3: Earth Science Application The fastest-moving tectonic plates on Earth move apart at a rate of 15 centimeters per year. Write a linear function that describes the movement of the plates over time. Graph the relationship. Is this a proportional relationship? Justify your answer. Let x represent the input, which is the time in years. Let y represent the output, which is the distance in centimeters the plates move apart. distance in cm = 15 cm/yr  time in years y = 15  x The function is y = 15x. Yes, the graph goes through the origin

Additional Example 2 Continued Make a function table. Include a column for the rule. Input Rule Output x 15(x) y Multiply the input by 15. 15(0) 1 15(1) 15 2 15(2) 30 3 15(3) 45

Additional Example 2 Continued Graph the ordered pairs (0, 0), (1, 15), (2, 30), and (3, 45) from your table. Connect the points with a line. Check y 100 80 60 40 20 2 4 8 10 12 Use the ordered pairs (1, 15), (2, 30), and (3, 45) to see if the relationship is proportional. Centimeters 1 15 2 30 3 45 = = The ordered pairs are proportional and the graph passes through (0, 0). y = 15x is a proportional relationship. x Years

Check It Out: Example 2 Dogs are considered to age 7 years for each human year. Write a linear function that describes the age of the dog over time. Graph the relationship. Is this a proportional relationship? Justify your answer. Let x represent the input, which is the age in human years. Let y represent the output, which is the age of the dog in dog years. Age in dog years = 7 dog yr/human yr  time in human years y = 7  x The function is y = 7x.

Check It Out: Example 2 Continued Make a function table. Include a column for the rule. Input Rule Output x 7(x) y Multiply the input by 7. 7(0) 1 7(1) 7 2 7(2) 14 3 7(3) 21

Check it Out: Example 2 Continued Graph the ordered pairs (0, 0), (1, 7), (2, 14), and (3, 21) from your table. Connect the points with a line. y 80 60 40 20 2 4 8 10 Dog years x Human years