Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

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

Problem of the Day Graph the ordered pairs form the table. What letter do the points form? V

Sunshine State Standards MA.7.A.1.4 Graph proportional relationships…

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.

Check It Out: Example 1 Graph y = 15x. x 1 2 3 y 15 30 45 40 20 60 2 4 1 2 3 y 15 30 45 y 40 20 x 60 2 4 6 8

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 2: 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.

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

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 3 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 3 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 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 = 6x + 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 = 6x + 52 is not a proportional relationship.

Check it Out: Example 3 Continued 100 80 60 40 2 4 6 8 Temperature Hours

Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

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

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 – 12x; no, the graph does not go through the origin.

Lesson Quiz for Student Response Systems 1. Tell whether the linear function y = 2x is a proportional relationship. A. yes B. no

Lesson Quiz for Student Response Systems 2. Tell whether the graph of the given linear function is a proportional relationship. A. yes B. no

Lesson Quiz for Student Response Systems 3. Larry has 150 cents in his piggy bank. He puts 20 cents into it everyday. Identify a linear function that describes the amount in the piggy bank over time. Is this a proportional relationship? A. y = 20x; yes B. y = –20x; yes C. y = 150 + 20x; no D. y = 150 – 20x; no