topic badge

3.05 The hyperbola and inverse variation

Worksheet
Inverse variation
1

If y varies inversely with x, write an equation that uses k as the constant of variation.

2

Determine whether each of the following is an example of a direct variation or an inverse variation:

a

The variation relating the distance between two locations on a map and the actual distance between the two locations.

b

The variation relating the number of workers hired to build a house and the time required to build the house.

3

If r varies inversely with a, write an equation that uses k as the constant of variation.

4

Determine whether the following equations represent an inverse relationship between x and y:

a

x = 1 + y^{3}

b

x = \dfrac {8}{y^{2}}

c

y = 6 x + 8

d

x y = - 7

e

x = \dfrac {2}{y}

f

x y = 5 x

5

Consider the equation s = \dfrac {375}{t}.

a

State the constant of proportionality.

b

Find the exact value of s when t = 6.

c

Find the exact value of s when t = 12.

6

State whether the following tables represent an inversely proportional relationship between x and y.

a
x1234
y31.510.75
b
x1234
y3618129
c
x15610
y375108300
d
x1234
y4567
7

m is proportional to \dfrac {1}{p}. Consider the values in the table which represents the relationship.

p467x
m63y3628
a

Determine the constant of proportionality, k.

b

Find the values of x and y.

8

Find the equation relating t and s for each of the following tables of values:

a
s1234
t48241612
b
s36912
t\dfrac{2}{9}\dfrac{1}{18}\dfrac{2}{81}\dfrac{1}{72}
9

Use technology to graph the inverse relationship y = \dfrac {6}{x}.

a

How many x-intercepts does the graph have?

b

How many y-intercepts does the graph have?

Key features of a hyperbola
10

Consider the graph of y = \dfrac {2}{x}.

a

For x > 0, as x increases, what does y approach?

b

For x > 0, as x approaches 0, what does y approach?

c

State the equation of the vertical asymptote.

d

State the equation of the horizontal asymptote.

e

The graph has two axes of symmetry. State their equations.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
11

The equation y = - \dfrac {12}{x} represents an inverse relationship between x and y.

a

When x = 3, what is the value of y?

b

If x is a positive value, is the corresponding y value positive or negative?

c

If x is a negative value, is the corresponding y value positive or negative?

d

In which quadrants does the graph of y = \dfrac {- 12}{x} lie?

12

Consider the function y = \dfrac {2}{x}.

a

Complete the table of values.

x-2-1-\dfrac{1}{2}-\dfrac{1}{10}-\dfrac{1}{100}\dfrac{1}{100}\dfrac{1}{10}\dfrac{1}{2}1
y
b

For what value of x is the function undefined?

c

Rewrite the equation to make x the subject.

d

For what value of y is the function undefined?

e

Below is the graph of y = \dfrac {2}{x}.

Complete the following statement:

As x approaches from the right, the function value approaches \infty. As x approaches from the left, the function value approaches -\infty.

As x approaches \infty and -\infty, y approaches . This is called the limiting value of the function.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
13

Consider the function y = - \dfrac {5}{x}.

a

For what value of x is the function undefined?

b

As x approaches 0 from the positive side, what does y approach?

c

As x approaches 0 from the negative side, what does y approach?

d

As x approaches \infty, what does y approach?

e

As x approaches -\infty, what does y approach?

14

Consider the hyperbola shown:

Complete the statement:

"Every point \left(x, y\right) on the hyperbola is such that x y = ⬚."

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
15

Determine whether the following relationships can be modelled by a function of the form x y = a?

a

The relationship between the number of people working on a job and how long it will take to complete the job.

b

The relationship between the number of sales and the amount of revenue.

c

The relationship between height and weight.

16

Consider the function graphed below:

a

What is the equation of the vertical asymptote?

b

What is the equation of the horizontal asymptote?

c

State the domain of the function.

d

State the range of the function.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
17

For functions of the form f \left( x \right) = \dfrac {k}{x}, where k is a constant, the domain is always \left(-\infty, 0\right) \cup \left(0, \infty\right). What does this mean for the function when x = 0?

18

Consider the function y = \dfrac {1}{x}.

a

For what value of x is the function undefined?

b

What is the domain of y = \dfrac {1}{x}?

c

For what value of y is the function underfined?

d

What is the range of y = \dfrac {1}{x}?

19

Patricia determined the range of the function y = \dfrac {2}{x} is \left(-\infty, 0\right) \cup \left(0, \infty\right).

By filling in the gaps below, complete her reasoning.

Looking at y = \dfrac {2}{x}, notice that the numerator is non-zero, and so \dfrac {2}{x} can never be equal to .

Another approach is to rearrange the equation and make x the subject. Then we get x = ⬚ . In this form we can see that the denominator cannot be equal to .

20

Consider the functions y = \dfrac {4}{x} and y = \dfrac {2}{x}.

a

When x = 2, what is the value of y if y = \dfrac {4}{x}?

b

When x = 2, what is the value of y if y = \dfrac {2}{x}?

c

Which graph lies further away from the axes?

d

For hyperbolas of the form y = \dfrac {k}{x}, as the k value increases, what happens to the graph?

Graphs of hyperbolas
21

Consider the inverse variation equation y = \dfrac {6}{x}.

a

Complete the following tables of values:

x\dfrac{1}{4}\dfrac{1}{2}124
y
b

Plot the points from the table of values on a cartesian plane.

22

Ursula wants to sketch the graph of y = \dfrac {7}{x}, but knows that it will look similar to many other hyperbolas.

What can she do to the graph to show that it is the hyperbola y = \dfrac {7}{x}, rather than any other hyperbola of the form y = \dfrac {k}{x}?

23

For each of the following functions:

i

Complete a table of values of the form:

x-2-1-\dfrac{1}{2}\dfrac{1}{2}12
y
ii

Sketch the graph of the function.

iii

State which quadrants the function lies in.

a
y = \dfrac {2}{x}
b
y = - \dfrac {1}{x}
c
y = - \dfrac {1}{4 x}
Applications
24

The relationship between the current, C, (in amperes) and resistance, R, (in ohms) in an electrical circuit is given by: C = \dfrac {240}{R}where the voltage provided to the circuit is 240 V.

a

Graph this function for 0 \leq R \leq 50.

b

What happens to the current as the resistance increases?

25

The rent, electricity, telephone bill and other expenses for a flat costs a total of \$490 per week. These expenses are shared equally between the tenants of the flat.

a

How much will each occupant pay in dollars if the flat is shared by two people?

b

Let the number of occupants be x, and the cost paid by each occupant be y.

Write a formula that relates the two pronumerals.

c

Use this equation to fill in the following table of values.

x123456
y
d

Sketch the curve from your equation.

e

What type of relationship exists between x and y?

26

Consider the hyperbola that has been graphed. Points A \left(4, 2\right), B, C \left( - 4 , k\right) and D form the vertices of a rectangle.

a

Find the value of k.

b

Hence find the area of the rectangle with vertices ABCD.

-8
-6
-4
-2
2
4
6
8
x
-6
-4
-2
2
4
6
y
27

Boyle's law describes the relationship between pressure and volume of a gas of fixed mass under constant temperature. The pressure for a particular gas can be found using P = \dfrac {6000}{V} where P has units \text{kg/cm}^2 and V has units \text{cm}^3.

a

Graph the relationship P = \dfrac {6000}{V} for 0 \leq V \leq 2000.

b

What is the pressure if the volume is 1\text{ cm}^3?

c

What happens to the pressure as the volume increases?

28

The time it takes a commuter to travel 100\text{ km} depends on how fast they are going. We can write this using the equation t = \dfrac{100}{S} where S is the speed in \text{ km/h} and t is the time taken in hours.

a

Graph the relationship t = \dfrac {100}{S}.

b

What is the time taken if the speed travelled is 10\text{ km/h}?

c

What is the time taken if the speed travelled is 50\text{ km/h}?

d

If we want the travel time to decrease, what must happen to the speed of travel?

29

The time, t, taken by a typist to type up a document is inversely proportional to his typing speed, s. That is, the quicker the typing speed, the less time it will take. If it takes a typist 20 minutes to type a particular document, typing at a speed of 61 words per minute:

a

Find the constant of variation k.

b

How long (in minutes) will it take a typist with a typing speed of 30.5 words per minute to type up the document?

Sign up to access Worksheet
Get full access to our content with a Mathspace account

Outcomes

MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

What is Mathspace

About Mathspace