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# 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.

1. 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?

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 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:

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}.

## 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

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