Variation in Mathematics

Questions and Answers

What is the mathematical form of a direct variation relationship?

$y = kx^2$

$y = kx$ (correct)

$y = k + x$

$y = rac{k}{x}$

Which characteristic is true for direct variation?

The graph is a hyperbola.

If $x$ decreases, $y$ also decreases.

If $x$ increases, $y$ also increases. (correct)

The relationship has no constant of variation.

Which equation represents an inverse variation?

$y = kx$

$y = rac{k}{x}$ (correct)

$y = kx^2$

$y = k + x$

If $k = 8$ in an inverse variation, what is the value of $y$ when $x = 2$?

<p>$16$</p> Signup and view all the answers

What type of graph represents an inverse variation?

<p>A hyperbola.</p> Signup and view all the answers

Study Notes

Variation

Direct Variation

Definition: A relationship where one variable is a constant multiple of another.
Mathematical Form: ( y = kx )
- Where ( k ) is the constant of variation.
Characteristics:
- If ( x ) increases, ( y ) increases (and vice versa).
- The graph is a straight line passing through the origin (0,0).
Example:
- If ( k = 3 ), then ( y = 3x ). For ( x = 2 ), ( y = 6 ).

Inverse Variation

Definition: A relationship where one variable increases as the other decreases.
Mathematical Form: ( y = \frac{k}{x} )
- Where ( k ) is the constant of variation.
Characteristics:
- If ( x ) increases, ( y ) decreases (and vice versa).
- The graph is a hyperbola.
Example:
- If ( k = 12 ), then ( y = \frac{12}{x} ). For ( x = 3 ), ( y = 4 ).

Direct Variation

A relationship exists where one variable is directly proportional to another.
Represented mathematically as ( y = kx ), with ( k ) as the constant of variation.
If ( x ) increases, ( y ) also increases, maintaining a consistent ratio.
The graph of a direct variation is a straight line that intersects the origin (0,0).
Example of direct variation: If ( k = 3 ), then the equation becomes ( y = 3x ). For an input of ( x = 2 ), the output is ( y = 6 ).

Inverse Variation

A relationship exists where one variable increases while the other decreases.
Mathematically expressed as ( y = \frac{k}{x} ), with ( k ) being the constant of variation.
Inverse variation means that as ( x ) increases, ( y ) correspondingly decreases, and vice versa.
The graphical representation of inverse variation forms a hyperbola.
Example of inverse variation: If ( k = 12 ), then the equation becomes ( y = \frac{12}{x} ). For an input of ( x = 3 ), the output is ( y = 4 ).

Description

Explore the concepts of direct and inverse variation in mathematics. This quiz covers definitions, mathematical forms, characteristics, and examples of both types of variation. Test your understanding of how these relationships between variables work.