By Michael Avidon, math editor
Graphs: For Students
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Performance Expectations (CCSS)
This lesson covers the following parts of 7.RP.A.2:
Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by … graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in … graphs … of proportional relationships.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
In the previous two lessons (Lesson 1 and 2), proportional relationships were defined and were represented by tables, verbal descriptions, and equations. In this lesson you will learn how to represent proportional relationships as graphs and to recognize graphs as representing proportional relationships. (If you would rather have this printed out, see a printable version.)
Graphs of proportional relationships
You know that for each quart of a liquid there are two pints. The ratio of pints to quarts is always 2. This is a proportional relationship, and 2 is the constant of proportionality.
If x = the number of quarts of water in a container, and y = the number of pints of water in that container, then this relationship is represented by the table to the right:
Each row, or ordered pair of numbers (x, y), can be represented by a point in the coordinate plane, as shown at the left.
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 relationship is a line through the origin or a ray whose endpoint is the origin.
The real-world examples will generally have nonnegative values for both variables. In such a case, the graph will be a ray in the first quadrant. But in other cases, the graph could be a line. Why do the points lie on a line? Examine the ray below.
For every increase of 1 in the value of x, the increase in y is the same amount. Call this amount k. So if the ray starts at (0, 0), it will then pass through (1, k), (2, 2k), (3, 3k), and so forth. These points satisfy the equation y = kx. This represents a proportional relationship, where k is the constant of proportionality.
Conversely, every proportional relationship is represented by an equation of the form 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.
Represent the cupcake equation, C = 3.5n, from the previous lesson by a graph.
Solution: Make a table of values that satisfy the equation.
These are points on the graph. Draw a ray through the points.
Note that only the bulleted points on this graph correspond to actual cupcakes, because they do not sell fractions of cupcakes. In the graph for quarts and pints, all points on the ray would represent real values.
Equations from graphs
If you are given the graph of a proportional relationship, you can determine its equation.
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, 90) represent? What is the constant of proportionality and the equation representing the graph?
The graph is a ray starting at the origin, so it represents a proportional relationship.
The point represents the fact that when no time has passed (0 hours), no toys have been produced (0 toys). The point represents the fact that when __ hours have passed, ____ toys have been produced. The unit rate, or constant of proportionality, is 90 ⁄ 3 ________toys per hour. This could be computed using the coordinates of any point on the graph (other than (0, 0)), because the ratio y ⁄ x is constant (definition of proportional relationship). The equation for this graph is y = 30x.
This graph contains the point (1, 30). This represents the fact that after 1 hour, 30 toys have been produced. These coordinates directly show the unit rate. In general, the point (1, r) on the graph of a proportional relationship shows that the unit rate is r.
Determining if a relationship is proportional
If the graph of a relationship is a line or a ray through the origin, 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 not linear, then it is not proportional.
Which graphs represent proportional relationships?
All three 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 verbal description and produce a graph. Then you can determine if it is proportional from the graph.
Does the table represent a proportional relationship?
After graphing the points, you can see that they lie on a line. Draw the line and extend it to the y-axis. You can see that the y-intercept is not the __________. Therefore, it is not a proportional relationship.
A plumber charges $60 for the first hour of work, and $40 for every addition hour of work. Is the relationship between total cost and number of hours proportional?
The first hour of work is represented by the point (1, 60). For each increase in the value of x by 1, the value of y increases by 40. This same increase will put the points on a line. However, as you can see from the graph, the line does not pass through the origin (the y-intercept is the point (__,____)). So the relationship is not proportional.
Exercises for lesson 3
Draw the graphs of the given equations.
1. y = 3x, where x = the number of lbs. of ground beef, and y = the price in $
2. y = 0.5x, with no restrictions on the variables
For each table, draw the graph and determine if the relationship is proportional.
For each description, draw the graph and determine if the relationship is proportional.
5. A tutor charges $50 for one hour, $90 for two hours, and $120 for three hours.
6. Asparagus sells for $2.50 per pound.
7. For each relationship in exercises 3 through 6 that was proportional, find the equation.
The graph to the right shows the relationship between the time, x, in minutes someone jogs on a treadmill and the distance, y, in miles they run. Explain the meaning of the points (4, 0.4), (1, 0.1) and (0, 0) in this context.
Does each graph below represent a proportional relationship?
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.