Inverse Variation and Reciprocal Function

Key Concepts

• Identify Inverse Variation.
• Use an Inverse Variation Model.
• Graph the Reciprocal Function.
• Graph the Translations for the Reciprocal Function.

Inverse Variation  

Identify Inverse Variation  

How do you determine if a relationship represents an inverse variation? 

1. Does the table of values represent an inverse variation? 

An inverse variation is a relation between two variables such that one variable increases, the other decreases proportionally. For the table to represent an inverse variation, the product of x and y must be constant. Find the product, xy, for each column in the table.  

2. Does the table of values represent an inverse variation? 

Find the product.  

Concept  

When a relation between x and y is an inverse variation, we say that x varies inversely as y. Inverse variation is modeled by the equation y = k/x or with an equivalent form x = k/y  or xy = k, when k ≠ 0. The variable k represents the constant of variation, the number that relates the two variables in an inverse variation.  

In this table, the constant of variation is 24.  

Use an Inverse Variation Model  

On a guitar, the string length, s, varies inversely with the frequency, f, of its vibrations.  

The frequency of a 26-inch E-string is 329.63 cycles per second. What is the frequency when the string length is 13 inches?  

Solution:  

s = k/f————————–Write the equation for an inverse variation. 

26 = k/329.63—————-Substitute 26 and 329.63 for s and f.  

8570.38 = k ——————-Multiply by 329.63 to solve for k.  

After solving for k, write an equation for an inverse variation.  

s = 8570.38/f—————–Substitute 8570.38 for k in the equation.  

13 = 8570.38/f—————Substitute 13 for s in the equation.  

f = 659.26 ———————Solve for f.  

So, the frequency of the 13-inch string is 659.26 cycles per second.  

Reciprocal Function  

Graph the Reciprocal Function  

How do you graph the reciprocal function, y = 1/x? 

The reciprocal function maps every non-zero real number to its reciprocal.  

Step 1: Consider the domain and range of the function. 

Step 2: Graph the function.   

Step 3:  

Observe the graph of y = 1/x as it approaches positive infinity and negative infinity.  

An asymptote is a line that a graph approaches. Asymptotes guide the end behaviour of a function.  

As x approaches infinity, f(x) approaches 0. The same is true as x-values approach negative infinity, so the line y = 0 is a horizontal asymptote.  

Step 4:  

Observe the graph of y = 1/x as x approaches 0 for positive and negative x-values.  

For positive values of x, as x approaches 0, f(x) approaches positive infinity. 

For negative values of x, as x approaches 0, f(x) approaches negative infinity. The domain of the function excludes 0, so the graph will never touch the line x = 0. The line x = 0 is a vertical asymptote.   

Graph the Translations of the Reciprocal Function  

Graph g(x) = (1/x – 3)+ 2. What are the equations of the asymptotes? What are the domain and the range? 

Start with the graph of the parent function,

f(x) = 1/x

.  

Recall that adding h to x in the definition of f translates the graph of f horizontally. Adding k to f(x) translates the graph of f  vertically.  

The function (1/x – 3) + 2 is a transformation of the parent function f   that shifts the graph of f  horizontally by h units and then shifts the graph of f  vertically by k units.   

The graph of g(x) = (1/x – 3) + 2  is a translation of the graph of the parent function 3 units right and 2 units up. 

The line x = 3 is a vertical asymptote. The line y = 2 is a horizontal asymptote.  

The domain is {x ∣ x ≠ 3}.  

The range is {y ∣ y ≠ 2}.  

Questions  

Question 1 

Determine if this table of values represents an inverse variation. 

Solution: 

Since the product xy is constant, this table of values represents an inverse variation.   

Question 2 

The amount of time it takes for an ice cube to melt varies inversely to the air temperature, degrees. At 20° Celsius, the ice will melt in 20 minutes. How long will it take the ice to melt if the temperature is 30° Celsius? 

Solution:  

Let t be the time it takes for an ice cube to melt and T be the air temperature in degrees.  

t = k/T (Inverse Variation)  

For T = 20 degrees, t = 20 minutes.  

20 = k/20… k = 400  

Equation: t = 400/T

If T = 30 degrees, t = 400/30 = 13.33 minutes.  

Question 3 

Graph g(x) = (1/x+2) – 4. What are the equations of the asymptotes? What are the domain and the range? 

Solution:  

Start with the graph of the parent function, g(x) = 1/x

. 

g(x) = (1/x+2) – 4 

Adding 2 to x shifts the graph horizontally to the left by 2 units and subtracting 4 from the output shifts the graph down by 4 units.   

Below is the graph of the function: g(x) = (1/x+2) – 4 

As we can see in the graph, vertical asymptote is x = -2 and horizontal asymptote is y = -4.  

The domain is {x ∣ x ≠ -2}.  

The range is {y ∣ y ≠ -4}.  

Key Concepts Covered  

Exercise

Graph the following functions. What are the asymptotes? What are the domain and the range?  

1. f(x) = 1/x
2. f(x) = (1/(x+5)) – 4
3. f(x) = (1/(x-1)) + 2
4. f(x) = (1/(x-6)) – 3
5. f(x) = (1/(x+1)) – 1
6. f(x) = (1/(x+10)) – 9
7. f(x) = (1/(x-5.5)) + 6
8. f(x) = (1/(x+4)) – 6
9. f(x) = (1/(x+3.2)) – 3.4
10. f(x) = (1/(x-9)) – 9