If You Know The Equation Of A Proportional Relationship, How Can You Draw The Graph Of The Equation?

By Michael Avidon, math editor

Graphs: For Students

More in this series
Lesson i.
Lesson 2.
If you lot want to assist your kid, use our version for guardians.
Spanish version of this lesson.

Performance Expectations (CCSS)

This lesson covers the following parts of vii.RP.A.ii:
Recognize and represent proportional relationships between quantities.
a. Determine whether 2 quantities are in a proportional relationship, due east.chiliad., past … graphing on a coordinate plane and observing whether the graph is a direct line through the origin.
b. Place the constant of proportionality (unit rate) in … graphs … of proportional relationships.
d. Explain what a bespeak (x, y) on the graph of a proportional relationship means in terms of the situation, with special attending to the points (0, 0) and (i, r) where r is the unit rate.

In the previous ii lessons (Lesson ane and 2), proportional relationships were divers and were represented past tables, exact descriptions, and equations. In this lesson y'all volition learn how to represent proportional relationships equally graphs and to recognize graphs as representing proportional relationships. (If yous would rather have this printed out, see a printable version.)

Graphs of proportional relationships

You lot know that for each quart of a liquid in that location are two pints. The ratio of pints to quarts is always ii. This is a proportional relationship, and 2 is the abiding of proportionality.

x	y
0	0
1	2
two	4
?	6
iv	?

If x = the number of quarts of water in a container, and y = the number of pints of water in that container, so this relationship is represented by the table to the right:

Each row, or ordered pair of numbers (10, y), can be represented by a bespeak in the coordinate airplane, as shown at the left.
Proportions relations
Notice that points lie on a ray and that the ray starts at the origin. In general, the following is true.
The graph of a proportional human relationship is a line through the origin or a ray whose endpoint is the origin.

The real-earth examples volition generally have nonnegative values for both variables. In such a instance, the graph will be a ray in the first quadrant. Just in other cases, the graph could exist a line. Why practise the points lie on a line? Examine the ray beneath.

Proportional relationsips

For every increment of 1 in the value of x, the increase in y is the same amount. Phone call this corporeality k. So if the ray starts at (0, 0), information technology will and so pass through (1, chiliad), (two, 2k), (3, 3k), and so forth. These points satisfy the equation y = kx. This represents a proportional relationship, where g is the constant of proportionality.
Conversely, every proportional relationship is represented by an equation of the grade y = kx, and hence by a ray (or if negative values are permitted, a line). Regardless of the value of k, the ordered pair (0, 0) satisfies this equation, so the ray or line must pass through the origin.

Instance 1

Represent the cupcake equation, C = 3.5n, from the previous lesson by a graph.
Solution: Make a table of values that satisfy the equation.
Proportional relationships

Cupcakes
Number	Toll
0	$0
1	$3.50
2	$seven.00
3	$10.50
4	$
5	$

These are points on the graph. Describe a ray through the points.
Note that but the bulleted points on this graph correspond to bodily cupcakes, because they do non sell fractions of cupcakes. In the graph for quarts and pints, all points on the ray would represent existent values.

Equations from graphs

If you lot are given the graph of a proportional relationship, y'all can make up one's mind its equation.

Example ii

The graph for the number of toys produced in a factory over the course of several hours is shown. What do the points (0, 0) and (3, ninety) represent? What is the constant of proportionality and the equation representing the graph?
Solution: Proportional relationships
The graph is a ray starting at the origin, so it represents a proportional human relationship.
The bespeak represents the fact that when no fourth dimension has passed (0 hours), no toys take been produced (0 toys). The signal represents the fact that when __ hours have passed, ____ toys accept been produced. The unit rate, or abiding of proportionality, is ^xc ⁄ ₃ ________toys per hour. This could be computed using the coordinates of whatsoever point on the graph (other than (0, 0)), considering the ratio ^y ⁄ _x is abiding (definition of proportional relationship). The equation for this graph is y = 30x.

This graph contains the bespeak (1, xxx). This represents the fact that subsequently 1 hour, thirty toys have been produced. These coordinates straight show the unit rate. In general, the point (1, r) on the graph of a proportional relationship shows that the unit charge per unit is r.

Determining if a human relationship is proportional

If the graph of a relationship is a line or a ray through the origin, and then it is proportional. If it is a line or ray that does not pass through the origin, then it is not proportional. Also, if it is non linear, and then information technology is not proportional.

Example 3

Which graphs stand for proportional relationships?

Solution:
All 3 graphs pass through the origin (that is, the point (__,__)). Graph C is also a line. So it represents a proportional relationship.
Graph A is composed of line segments, but it is not a ray or a line. Graph B is a curve. So neither of these is a proportional relationship.

You can start with a table or a exact description and produce a graph. So you can decide if information technology is proportional from the graph.

Case four

Does the table represent a proportional relationship?
Solution:

x	y
2	2
4	3
6	iv
8	5

Proportional relationships graph After graphing the points, you tin can come across that they prevarication on a line. Describe the line and extend it to the y-centrality. You lot can run into that the y-intercept is not the __________. Therefore, it is not a proportional relationship.

Example 5

A plumber charges $sixty for the first hour of work, and $40 for every addition hour of piece of work. Is the relationship betwixt total cost and number of hours proportional?
Solution:
The first hour of work is represented by the point (1, 60). For each increase in the value of ten by 1, the value of y increases by 40. This same increase will put the points on a line. However, as you can run into from the graph, the line does not pass through the origin (the y-intercept is the betoken (__,____)). Then the relationship is not proportional.

Exercises for lesson iii

Draw the graphs of the given equations.
Proportional relationships graph
1. y = 3x, where 10 = the number of lbs. of ground beef, and y = the price in $
ii. y = 0.5x, with no restrictions on the variables

For each table, describe the graph and determine if the relationship is proportional.

x	y
2	2.five
iv	5
8	10
10	12.5

iv.

10	y
1	1
2	3
three	5
4	7

For each description, describe the graph and determine if the relationship is proportional.
5. A tutor charges $l for ane 60 minutes, $90 for two hours, and $120 for three hours.

half-dozen. Asparagus sells for $2.50 per pound.

7. For each relationship in exercises 3 through 6 that was proportional, find the equation.

8. Proportional relationships graph
The graph to the right shows the relationship betwixt the fourth dimension, 10, in minutes someone jogs on a treadmill and the distance, y, in miles they run. Explain the meaning of the points (iv, 0.4), (1, 0.one) and (0, 0) in this context.

Does each graph below represent a proportional relationship?
ix. Proportional relationships graph

ten. Propotion graph eleven. Proportions graph

12. Proportions graph

13. Challenge Problem:

A line passes through the point (a, b), where both coordinates are positive. If the line represents a proportional relationship, what is its equation? Explain.

REVIEW THE Entire Series: Lesson 1 | Lesson 2 | Overview and answers to each question