Inverse Variation

 
What is Inverse Variation?

Inverse variation describes a relationship where as one variable increases, the other decreases by the same factor. Similarly, if one decreases, the other increases proportionally. It can also be described as inverse proportion.

Key Characteristics of Inverse Variation:

• The product of the two quantities remains constant.
• There is a constant of proportionalityy (denoted as $k$).
• The equation follows the form:  $$y = \frac{k}{x}$$
• The graph of an inverse variation relationship is a curve that never touches the axes.

 

Example:

If a journey takes 4 hours at a speed of $60 \text{km/h}$, then:

• Doubling the speed to $120 \text{km/h}$ halves the time to 2 hours.
• Halving the speed to $30 \text{km/h}$ doubles the time to 8 hours.

 

Using Inverse Variation with Powers and Roots

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

Common Inverse Proportional Relationships:

• $y$ is inversely proportional to $x^2$ : $y = \frac{k}{x^2}$ 
• $y$ is inversely proportional to $\sqrt{x}$ : $y = \frac{k}{\sqrt{x}}$
• $y$ is inversely proportional to $x^3$ : $y = \frac{k}{x^3}$
• $y$ is inversely proportional to $\sqrt[3]{x}$ : \(y = \frac{k}{\sqrt[3]{x}\)

 

Example:

If the intensity of light is inversely proportional to the square of the distance from the source, then:

$$I = \frac{k}{d^2}$$

If $I = 20$ when $d = 2$, then:

$$20 = \frac{k}{2^2} \quad \Rightarrow 20 = \frac{k}{4} \quad \Rightarrow k = 80$$

 
So the equation is $I = \frac{80}{d^2}$.

 

Finding the Equation Between Two Inversely Proportional Variables

To find the equation for an inverse variation problem, follow these steps:

Step 1: Write the General Formula

Identify the relationship and set up an equation with .

• If $y$ is inversely proportional to $x$: $y = \frac{k}{x}$
• If $y$ is inversely proportional to $x^2$: $y = \frac{k}{x^2}$

 

Step 2: Sub in values

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

 

Step 3: Substitute $k$ back into original equation

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 inversely proportional to $x$ .

When $x = 6$, $y = 5$.

Find $y$ when $x = 2$.

Step 1: Set Up the Equation

$$y = \frac{k}{x}$$

 

Step 2: Find $k$

$$5 = \frac{k}{6}$$

 $$k = 5 \times 6 = 30$$

 

Step 3: Rewrite the Equation

$$y = \frac{30}{x}$$

 

Step 4: Find $y$ when $x = 2$ 

$$y = \frac{30}{2} = 15$$

Final Answer: When $x = 2$, then $y = 15$.

 

 

 

 

Graphing Inverse Proportions

• The graph of $y = \frac{k}{x}$ is a curved hyperbola.
• The graph never touches the axes because  never reaches zero.
• For relationships involving powers, the graph will have a different curved shape.

Example: If $y = \frac{50}{x^2}$, the graph will be a steep curve approaching the axes.