Direct Variation

 
What is Direct Variation?

Direct variation describes a consistent relationship between two variables. This means that as one variable increases, the other increases by the same factor. Similarly, if one decreases, the other decreases proportionally. It can also be described as direct proportion

Key Characteristics of Direct Variation:

• The ratio between the two quantities remains constant.
• There is a constant of proportionality (denoted as $k$).
• The equation follows the form: $$y = kx$$
• The graph of a directly proportional relationship is a straight line through the origin with gradient $k$.

 

Example:

If a worker earns $\pounds 15$ per hour, then:

• 2 hours = £30
• 5 hours = £75
• The ratio of hours:earnings always remains the same.

 

Using Direct Variation with Powers and Roots

Sometimes, direct variation problems involve powers or roots of a variable. In these cases, the relationship follows different equations.

Common Proportional Relationships:

• $y$ is directly proportional to $x^2$: $y = kx^2$
• $y$ is directly proportional to $\sqrt{x}$: $y = k\sqrt{x}$
• $y$ is directly proportional to $x^3$: $y = kx^3$
• $y$ is directly proportional to $\sqrt[3]{x}$: $y = k\sqrt[3]{x}$

 

Example:

If the area of a circle is directly proportional to the square of its radius, then:

$$A = kr^2$$

If $A = 50$ when $r = 5$, then:

$$50 = k(5^2) \Rightarrow 50 = 25k \Rightarrow k = 2$$

So the equation is $A = 2r^2$.

 
Finding the Equation Between Two Directly Proportional Variables

To find the equation for a direct variation problem, follow these steps:

Step 1: Write the General Formula

Identify the relationship and set up an equation with $k$.

If $y$ is proportional to $x$: $y = kx$
If $y$ is proportional to $x^2$: $y = kx^2$

Step 2: Find $k$

Substitute given values into the equation and solve for $k$.

 

Step 3: Rewrite the Equation with $k$

Once $k$ is found, substitute it back into the equation.

 

Step 4: Use the Equation to Find Other Values

Use the final equation to calculate other values as needed.

 

Example:

It is known that $y$ is directly proportional to $x^2$.

When $x = 3$, $y = 18$.
Find $y$ when $x = 4$.

Step 1: Set Up the Equation

$$y = kx^2$$

Step 2: Find $k$

$$18 = k(3^2)$$

$$18 = 9k \Rightarrow k = 2$$

Step 3: Rewrite the Equation

$$y = 2x^2$$

Step 4: Find $y$ when $x = 4$

$$y = 2(4^2)$$

$$y = 2(16) = 32$$

Final Answer: When $x = 4$, then $y = 32$.

 
Graphing Direct Proportions

The graph of $y = kx$ is a straight line through the origin.

The gradient = $k$.

For relationships involving powers, the graph will have a curved shape.

Example: If $y = 2x^2$, the graph will be a parabola passing through the origin.