Unit D: Graphing

What is Partial Variation?

When a linear relation demonstrates partial variation.

  • the graph is a straight line
  • the graph does not include the ordered pair (0, 0)
  • the slope, or rate of change, is constant

In partial variation, there is a fixed value associated with the y-value. A fixed value is a number that is constant and does not change. The rate of change stays the same as in direct variation.

Compare Dave's and Krista's earnings in the table of values.

Dave's Earnings
(direct variation)
x
(Hours worked)
y
(Earnings in $)
0 0
1 16
2 32
3 48
4 64

Krista's Earnings
(partial variation)
x
(Hours worked)
y
(Earnings in $)
0 10
1 26
2 42
3 58
4 74


The rate of change for Dave's and Krista's earnings are both $16, which is the hourly pay.

Dave's Earnings
(direct variation)



Krista's Earnings
(partial variation)




Krista’s earnings are $10 higher than Dave’s. The $10 is called the fixed value, as Krista receives this no matter how many hours she works. The fixed value, in a partial variation relationship, is located on the y-axis of the graph. The fixed value that is represented by the point (0, 10) is called the y-intercept.

 
Direct Variation




In any direct variation relationship, the equation of the line is

y = mx,

where m is the slope, or rate of change.

The equation for the line representing Dave’s wages is

y = 16x,

where m = $16, which is the slope, or rate of change.
 
Partial Variation



In any partial variation relationship, the equation of the line is

y = mx + b,

where is the slope, or rate of change, and b is the y-intercept.

In partial variation, there is a fixed value for y when x = 0 . The rate of change stays the same as in direct variation.

The equation for the line representing Krista’s wages is

y = 16x + 10,

where m = $16, which is the slope, or rate of change, and b = $10 , which is the y-intercept (fixed value).

Identify the graph as direct variation, partial variation, or neither. If the graph represents either direct variation or partial variation, calculate the slope.





The graph represents direct variation. Since the graph passes through the point (0, 0) and is a straight line, it represents direct variation.

Let (0, 0) = (x1, y1) and (2, 0.4) = (x2, y2).

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The slope of the line is m = 0.2.
Identify the graph as direct variation, partial variation, or neither. If the graph represents either direct variation of partial variation, calculate the slope.





The graph does not pass through the point (0, 0) and is not a straight line. Therefore, it does not represent direct variation. The graph has a y-intercept of 4 since the curve intersects with the y-axis at point (0, 4). As the line is not straight, it does not represent partial variation. This graph is neither direct variation or partial variation.