Lesson 7-4

Example 1     Use a Graph to Find a Constant Rate of Change
GRAVEL     Find the constant rate of change for the cost of -inch gravel in the graph shown.  Describe what the rate means.      

Choose any two points on the line and find the rate of change between them.  We will use the points at (2, 25) and (6, 75).

(2, 25)  2 tons, cost $25
(6, 75)  6 tons, cost $75

dollars 
tons

rate of change =  

The cost goes from $25 to $75. 
The weight goes from 2 tons to 6 tons. 

	= 	
		= 		Simplify.
		= $12.50/ton		Express this as a unit rate.
The rate of change $12.50/ton means that the cost of gravel increases at a rate of $12.50 per ton.

Real-World EXAMPLE   
Example 2     Use Graphs to Find Proportional Linear Relationships
BOOKS     The cost of books, as the number of books increases, is shown in the graph.  Determine if there is a proportional linear relationship between the cost and the number of books.

To determine if the quantities are proportional, find for points on the graph.  Four points on the graph are shown.  

= 10	= 9		= 8.67	= 8.5

Since the ratios are not the same, the cost is not proportional to the number of books.

Real-World EXAMPLE
Example 3	Use Direct Variation to Solve Problems
Time
(days)	Height
(in.)	Rate of changes (days/in.)	
x	y	k = 	
4	1	0.25	
6	1.5	0.25	
8	2	0.25	
10	2.5	0.25	
GARDENING     The height of a tomato seedling as it grows is recorded in the table.
			
a.  Write an equation that relates time and height.

Step 1	Find the value of k using the equation  	y = kx.  Choose any point in the table.  
Then solve for k.

      y 	= kx		Direct variation
      1	= k(4)		Replace y with 1 and x with 4.
 0.25	= k		Simplify.

Step 2	Use k to write an equation.
	      y 	= kx		Direct variation
    	      y 	= 0.25x		Replace k with 0.25.

b.  Predict how many days it will take for the seedling to be 8 inches tall.

                        y 	= 0.25x		Write the direct variation equation.
                              8	= 0.25x		Replace y with 8.
                            32	= x		Divide each side by 0.25.

It will take about 32 days for the seedling to grow to a height of 8 inches.