Pre Algebra Slope and Direct Variation

Slope and Direct Variation

The relationship between the number of pounds of apples squeezed, x, and the ounces of apple juice made, y, is a linear function. The table shows some solutions to the function. Is the function a direct variation?

Input (x)	Output (y)
3	9
5	15
9	27
10	30

• The ratio of the two quantities is [3/1]
• The slope is always [3/1]
• Since the ratio and the slope are equal, the function is a direct variation

Yes

A store sells apples for $ 1.75 per pound. Graph the relationship between the pounds of apples purchased, x, and the total cost, y. Find the slope of the line and explain what it represents.

• y = 1.75x
• Make a table to find the solutions:

• Input (x)	Output (y)
  0	0
  1	1.75

• Use two points to graph the line on a coordinate plane:
• 
• Find the slope:
• m = [(y₁ − y₂)/(x₁ − x₂)], (0,0) and (1,1.75)
• m = [(1.75 − 0)/(1 − 0)] =

Slope = 1.75
The slope represents the increase in cost for every increase of 1 pound of apples purchased. That is, for every pound of apples bought, the cost increases by $ 1.75.

Is the linear function in this table a direct variation?

Input (x)	Output (y)
0	0
1	3.5
2	7
3	10.5

• The ratio of the two quantities is [3.5/1]
• The slope is always [3.5/1]

Yes

Is the linear function in this table a direct variation?

Input (x)	Output (y)
2	6
3	9
4	11
5	16

• [6/2] = 3
• [9/3] = 3
• [11/4] ≠ 3
• [16/5] ≠ 3
• The ratios aren't the same

No

Graph the direct variation y = − 2x and find the slope of the line.

• Make a table of x- and y - coordinates before graphing the equation.

• Input (x)	Output (y)
  0	0
  1	-2

• 
• m = [(y₁ − y₂)/(x₁ − x₂)], (0,0) and (1, − 2)
• m = [( − 2 − 0)/(1 − 0)] =
• [( − 2)/1] =

Slope = − 2

Suppose you eat 3 times each day. Graph the relationship between the number of days passed, x, and the cumulative number of meals you've eaten, y. Find the slope of the line and explain what it represents.

• y = 3x

• Input (x)	Output (y)
  0	0
  1	3
  2	6

• 
• m = [(y₁ − y₂)/(x₁ − x₂)], (0,0) and (1,3)
• m = [(3 − 0)/(1 − 0)] =
• [3/1] =

Slope = 3
The slope represents the increase of cumulative meals eaten for every increase of a day passing. That is, for every day that goes by, 3 more meals are eaten.

Is the function in this table a direct variation?

Input (x)	Output (y)
1	5
2	6
3	7
4	8

• [5/1] = 5
• [6/2] = 3
• [7/3]
• [8/4] = 2
• The ratio of quantities are not the same.

No

Graph y = − 0.5x and find the slope of the line.

• Make a table of x- and y- coordinates before graphing the equation.

• Input (x)	Output (y)
  0	0
  2	1

• 
• m = [(y₁ − y₂)/(x₁ − x₂)], (0,0) and (2,1)
• m = [(1 − 0)/(2 − 0)] =

Slope = [1/2]

Graph the equation y = − 3x and find the slope of the line.

• Make a table of x- and y- coordinates before graphing the equation.

• Input (x)	Output (y)
  0	0
  1	-3

• 
• m = [(y₁ − y₂)/(x₁ − x₂)], (0,0) and (1, − 3)
• m = [( − 3 − 0)/(1 − 0)] =
• [( − 3)/1] =

Slope = − 3

Graph y = [1/3]x and find the slope of the line.

• Make a table of x- and y- coordinates before graphing the equation.

• Input (x)	Output (y)
  0	0
  3	1

• 
• m = [(y₁ − y₂)/(x₁ − x₂)], (0,0) and (3,1)
• m = [(1 − 0)/(3 − 0)] =

Slope =[1/3]

Slope and Direct Variation

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

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• Intro 0:00
• What You'll Learn and Why 0:05
• Topics Overview
• Vocabulary 0:33
• Direct Variation
• Identifying a Direct Variation 0:47
• Steps in Identifying a Direct Variation
• Slope and Direct Variation 3:01
• Example: Slope and Direct Variation
• Extra Example 1: Direct Variation 6:16
• Extra Example 2: Direct Variation 7:14
• Extra Example 3: Graphing Direct Variation 8:10
• Extra Example 4: Slope and Direct Variation 11:23