1. 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

2. Vocabulary
linear equation
linear function

3. 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.

4. 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).

5. Additional Example 1 Continuedy 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.

6. 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).

7. 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.

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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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

15. 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

16. 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
= =
The ordered pairs are proportional and the graph passes through (0, 0). y = 15x is a proportional relationship. x
Years

17. 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.

18. 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

19. 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