partial and direct variation

PARTIAL AND DIRECT VARIATION

Recall: If a relationship is linear, its graph forms a straight line.

Recall: If a relationship is linear, its graph forms a straight line.

We can further divide linear relationships into two categories:

1. direct linear relationships

2. partial linear relationships

DIRECT PARTIAL when the independent

variable is zero, the dependent variable is also zero

both sides of the equation have only one term

ex: x = 0 y = 2xy = 2(0)y = 0

DIRECT PARTIAL when the independent

variable is zero, the dependent variable is also zero

both sides of the equation have only one term

ex: x = 0 y = 2xy = 2(0)y = 0

when the independent variable is zero, the dependent variable is not zero

one side of the equation has two terms

ex: x = 0 y = 2x + 5y = 2(0) + 5y = 5

DIRECT PARTIAL value of the

dependent variable is based solely on the value of the independent variable

passes through the origin (0,0)

DIRECT PARTIAL value of the

dependent variable is based solely on the value of the independent variable

passes through the origin (0,0)

value of the dependent variable is based on both the independent variable and a constant

does not pass through the origin

Which of the following lines show:

Partial Variation?

Direct Variation?

A

B

C

Which of the following lines show:

Partial Variation?B

Direct Variation?A,C

A

B

C

1. y = 3x2. C = 200 + 4d3. a = – 70 + b 4. h = 5t

Which of the equations are partial linear relationships?

Which of the equations are direct linear relationships?

1. y = 3x2. C = 200 + 4d3. a = – 70 + b 4. h = 5t

Which of the equations are partial linear relationships?2,3

Which of the equations are direct linear relationships?1,4

We can’t use x and y intercepts alone to graph a direct linear relationship.

ex: y = 3x x-intercept: y = 0 0 = 3x0 = x (0,0)

y-intercept: x = 0 y = 3(0)y = 0 (0,0)

The x and y intercepts are the same, and we need two points to graph a line.

CALCULATING STEEPNESS WITH RATE TRIANGLES

We can also compare linear relationships based on their steepness. We determine the steepness of a line using a rate triangle.

hypotenuseheigh

t

base

Using any two points on the line as the hypotenuse, draw a right angled triangle.

The rate is a measure of steepness where:

rate = height base

6

3

Using any two points on the line as the hypotenuse, draw a right angled triangle.

The rate is a measure of steepness where:

rate = height base = 2

6

3

Determine the rate/steepness of lines A, B, and C.

A

B

C

Determine the rate/steepness of lines A, B, and C.

A

B

C

25

5

Determine the rate/steepness of lines A, B, and C.

A

B

C5

5

Determine the rate/steepness of lines A, B, and C.

A

B

C10

5

Determine the rate/steepness of lines A, B, and C.

A. r = 25 = 5 B. r = 2 = 1 C. r = 10 = 2

5 2 5

INTERPOLATION AND EXTRAPOLATION

We can use the fact that a relationship is linear to identify other data points without performing calculations.

interpolation: finding another data point that exists between two points you already know

We can use the fact that a relationship is linear to identify other data points without performing calculations.

interpolation: finding another data point that exists between two points you already know

extrapolation: finding another point that exists beyond the points you already know (extend the line)

INTERPOLATION EXTRAPOLATION