5 3 Graphing Proportional Relationships Vocabulary linear equation

5 -3 Graphing Proportional Relationships Vocabulary linear equation linear function

5 -3 Graphing Proportional Relationships 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 6 Miles 1= 2 =3 = 3 2 8 4 6 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. 4 2 0 x 0 2 4 Hours

5 -3 Graphing Proportional Relationships Additional Example 1: Graphing Proportional Relationships Graph the linear function y = 4 x. Make a table. x y 0 0 1 4 2 8 3 12 Proportional relationships pass through (0, 0). Graph the ordered pairs (0, 0), (1, 4), (2, 8), (3, 12).

5 -3 Graphing Proportional Relationships y 12 Additional Example 1 Continued (3, 12) 10 (1, 4) Place each ordered pair on the coordinate grid and then connect the points with a line. (0, 0) 0 2 4 The graph is a straight line that passes through the origin. 8 (2, 8) 6 4 2 Check x 6 1 = 2= 3 4 8 12 8 10 The ordered pairs are proportional.

5 -3 Graphing Proportional Relationships Check It Out: Example 1 Graph y = 15 x. x 0 1 2 3 y 0 15 30 45 y 60 40 20 0 2 4 6 8 x

5 -3 Graphing Proportional Relationships 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.

5 -3 Graphing Proportional Relationships 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.

5 -3 Graphing Proportional Relationships Additional Example 2: Identify Proportional Relationships Tell whether the function is a proportional relationship. Then graph the function. A. y = – 2 x Make a table. x y – 1 2 0 0 1 – 2 2 – 4 3 – 6 – 1 = 2 = 3 – 2 – 4 – 6 2 The ordered pairs are proportional and the graph passes through (0, 0). y = – 2 x is a proportional relationship.

5 -3 Graphing Proportional Relationships Check It Out: Example 2 Tell whether y = 10 x – 1 is a proportional relationship. Then graph the function. x y 0 – 1 1 9 2 19 3 29 4 39 The ordered pairs are not proportional, and the graph does not pass through (0, 0). y = 10 x – 1 is not a proportional relationship.

5 -3 Graphing Proportional Relationships 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 = 15 x. Yes, the graph goes through the origin

5 -3 Graphing Proportional Relationships Additional Example 3 Continued Make a function table. Include a column for the rule. Input Rule Output x 15(x) y 0 15(0) 0 1 15(1) 15 2 15(2) 30 3 15(3) 45 Multiply the input by 15.

5 -3 Graphing Proportional Relationships Additional Example 3 Continued Graph the ordered pairs (0, 0), (1, 15), (2, 30), and (3, 45) from your table. Connect the points with a line. y Centimeters Check Use the ordered pairs 100 (1, 15), (2, 30), and 80 (3, 45) to see if the 60 relationship is proportional. 40 1 = 2 = 3 45 15 30 20 The ordered pairs are proportional and the graph 0 passes through (0, 0). y = 15 x is a proportional relationship. 2 4 8 Years 10 12 x

5 -3 Graphing Proportional Relationships Check It Out: Example 3 The outside temperature is increasing at the rate of 6 °F per hour. When Reid begins measuring the temperature, it is 52 °F. Write a linear function that describes the outside temperature over time. Graph the relationship. Is this a proportional relationship? Justify your answer. y = 6 x + 52, where x is the number of hours and y is the temperature. The ordered pairs are not proportional and the graph does not pass through (0, 0). y = 6 x + 52 is not a proportional relationship.

5 -3 Graphing Proportional Relationships Check it Out: Example 3 Continued Temperature 100 80 60 40 0 2 4 6 Hours 8

5 -3 Graphing Proportional Relationships Lesson Quiz: Part I Tell whether each function is a proportional relationship. Then graph the function. 1. y = 3 x – 4 no 2. y = –x yes 3. y = 2 x yes y = –x y = 3 x – 4 y = 2 x

5 -3 Graphing Proportional Relationships 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 – 12 x; no, the graph does not go through the origin.