Title: Algebra 1
Author: Mandy McDaniel
Last saved by: MHE

Lesson 4-1

Example 1      Find a Rate of Change
Use the table to find the rate of change. Explain the meaning of the rate of change.

Number of
Movie Tickets

Total Cost
($)

x | y
3 | 22.50
6 | 45
9 | 67.50

Each time x increases by 3 tickets, y increases by $22.50. 
rate change = (change in y)/(change in x)
						= 
						= 
						=  or 
The rate of change is . This means it costs $7.50 per ticket.

Real-World EXAMPLE
Example 2     Find a Rate of Change
The graph shows the density of population	
for the state of Idaho in various years.
a.	Find the rates of change for 1930-1960 
and 1990-2000.
Use the formula for slope.
 = 
1930 – 1960:		= 	Substitute.
	=  or 0.09	Simplify.
Density increased by 2.7 people per square mile in a 30-year period for a rate of change of 0.09 people per square mile per year.

1990-2000:		= 	Substitute.
	=  or 0.34	Simplify.
Over this 10-year period, the density increased by 3.4 people per square mile, for a rate of change of 0.34 people per square mile per year.

b.	Explain the meaning of the slope in each case.
For 1930-1960, on average, the people per square mile increased by 0.09 from the year before.
For 1990-2000, on average, the people per square mile increased by 0.34 from the year before.

c.	How are the different rates of change shown on the graph?
There is a greater vertical change between 1990-2000 than for 1930-1960. Therefore, the section of the graph for 1990-2000 is steeper.

Example 3    Positive Slope
Find the slope of the line that passes through (31/2, 2) and (-1, 1/2).
Let (31/2, 2) = (x, y) and (-1, 1/2) = (x, y).	
m = 	rise/run
	= 	Substitute. 31/2= 7/2
	= 	Simplify.
	= 	Divide.
The slope is .

Example 4     Negative Slope
Find the slope of the line that passes through (1, 3.6) and (4, 1.6).
Let (1, 3.6) = (x, y) and (4, 1.6) = (x, y).	
m = 	rise/run
	= 	Substitute.
	= 	Simplify.
The slope is -.

Example 5     Zero Slope
Find the slope of the line that passes through (-3, 1/3) and (-4, 1/3).
Let (-3, 1/3) = (x, y) and (-4, 1/3) = (x, y).	
m = 	rise/run
	= 	Substitute.
	=  or 0	Simplify.

Example 6     Undefined Slope
Find the slope of the line that passes through (-3, 5.5) and (-3, 2.5).
Let (-3, 5.5) = (x, y) and (-3, 2.5) = (x, y).	
m = 	rise/run
	=  or 	Substitute.

Since division by zero is undefined, the slope is undefined.

Example 7    Find Coordinates Given Slope
Find the value of r so that the line through (-6, -3) and (-1, r) has a slope of 2.
m = 	       rise/run
	   2 	= 	Let (-6, -3) = (x, y) and (-1, r) = (x, y).
	  	= 	Subtract. 2 = 2/1.
  2(5) 	= 1(r + 3)	Find the cross products.
	 10 	= r + 3	Simplify.
10 –3	= r + 3 – 3 	Subtract 3 from each side.
	   7 	= r	Simplify.
So, the line goes through (-1, 7)