Direct Variation

[nycmathguy]

Section 1.10

In your own words define the different types of variations.

Direct Variation:
Inverse Variation:
Combined Variation:
Joint variation:

 

[MathLover1]

the different types of variations are:

Direct Variation:
Inverse or Indirect Variation:
Inverse Variation:
Combined Variation:
Joint variation:
Partial Variation


Direct Variation, where one variable is a constant multiple of another.

Inverse or Indirect Variation, where when one of the variables increases, the other one decreases (their product is constant).

Joint Variation, where more than two variables are related directly.

Combined Variation, which involves a combination of direct or joint variation, and indirect variation.

Partial Variation, where two variables are related by a formula, such as the formula for a straight line (with a non-zero y-intercept)

Here are some examples of direct and inverse variation:

Direct: The number of dollars I make varies directly (or you can say varies proportionally) with how much I work (k is positive).
Direct: The length of the side a square varies directly with the perimeter of the square.
Inverse: The number of people I invite to my bowling party varies inversely with the number of games they might get to play (or you can say is proportional to the inverse of).
Inverse: The temperature in my house varies indirectly (same as inversely) with the amount of time the air conditioning is running.
Inverse: My GPA may vary directly inversely with the number of hours I watch TV.
Partial (Direct): The total cost of my phone bill consists of a fixed cost per month, and also a charge per minute.

Direct or Proportional Variation:
y=kx or y1/x1=y2/x2 ->k is the ratio of y to x (which is the same as the slope or rate)
Special Case:
Direct Square variation: y=kx^2

Inverse or Indirect Variation:

y=k/x or xy=k

Joint Variation:
Like direct variation, but involves more than one variable.
Example: y=kxz^2 ..........( y varies jointly with x and the square of z. )

Combined Variation:

Involves a combination of direct variation or joint variation, and indirect variation.
Example: y=(kxw)/z^2..........( y varies jointly as x and w and inversely as the square of z)

Partial Variation:

Two variables are related by the sum of two or more variables (one of which may be a constant).
Example: y=k1x+k2...........(y is partly constant and partly varies directly with x)

 

[nycmathguy]

the different types of variations are:

Direct Variation:
Inverse or Indirect Variation:
Inverse Variation:
Combined Variation:
Joint variation:
Partial Variation


Direct Variation, where one variable is a constant multiple of another.

Inverse or Indirect Variation, where when one of the variables increases, the other one decreases (their product is constant).

Joint Variation, where more than two variables are related directly.

Combined Variation, which involves a combination of direct or joint variation, and indirect variation.

Partial Variation, where two variables are related by a formula, such as the formula for a straight line (with a non-zero y-intercept)

Here are some examples of direct and inverse variation:

Direct: The number of dollars I make varies directly (or you can say varies proportionally) with how much I work (k is positive).
Direct: The length of the side a square varies directly with the perimeter of the square.
Inverse: The number of people I invite to my bowling party varies inversely with the number of games they might get to play (or you can say is proportional to the inverse of).
Inverse: The temperature in my house varies indirectly (same as inversely) with the amount of time the air conditioning is running.
Inverse: My GPA may vary directly inversely with the number of hours I watch TV.
Partial (Direct): The total cost of my phone bill consists of a fixed cost per month, and also a charge per minute.

Direct or Proportional Variation:
y=kx or y1/x1=y2/x2 ->k is the ratio of y to x (which is the same as the slope or rate)
Special Case:
Direct Square variation: y=kx^2

Inverse or Indirect Variation:

y=k/x or xy=k

Joint Variation:
Like direct variation, but involves more than one variable.
Example: y=kxz^2 ..........( y varies jointly with x and the square of z. )

Combined Variation:

Involves a combination of direct variation or joint variation, and indirect variation.
Example: y=(kxw)/z^2..........( y varies jointly as x and w and inversely as the square of z)

Partial Variation:

Two variables are related by the sum of two or more variables (one of which may be a constant).
Example: y=k1x+k2...........(y is partly constant and partly varies directly with x)

This is a book load of information. Cool.

 

[nycmathguy]

This is a book load of information. Cool.

Why is k called the constant of proportionality?

 

[MathLover1]

Why is k called the constant of proportionality?

k= (some number ) determines how the two variables are related to one another

The value of this constant (some number which does not change) is called the coefficient of proportionality. The constant of proportionality is also known as unit rate.

 

[nycmathguy]

k= (some number ) determines how the two variables are related to one another

The value of this constant (some number which does not change) is called the coefficient of proportionality. The constant of proportionality is also known as unit rate.

Interesting. This is going to be a good section. Be advised, I am more interested in the set up in terms of applications. If the set up is right, the wrong answer can be avoided. By the way, Section 1.10 is the last section before we move on to Chapter 2, Section 2.1. There is a test at the end of each chapter but I will skip all tests and move on in the textbook. As you know, this is just for fun.