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3.05 The hyperbola and inverse variation

Lesson

Inverse variation

Inverse proportion, or inverse variation, means that as one amount increases the other amount decreases. 

Mathematically, we write this as:

$y$y $\propto$ $\frac{1}{x}$1x
 

For example, speed and travel time vary inversely because the faster you go, the shorter your travel time.

 

General Equation for Amounts that are Inversely Proportional

We express these inverse variation relationships generally in the form

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

where $k$k is the constant of proportionality and $x$x and $y$y are any variables

It is also possible to write the equation in the form:

$xy=k$xy=k or $x=\frac{k}{y}$x=ky

and it is clear that $k$k can be any number other than $0$0.

The hyperbola

The graph of an inverse relationship in the $xy$xy-plane is called a hyperbola. Let's see what inverse variation looks like in a table of values.

This table shows the relationship $y=\frac{1}{x}$y=1x:

$x$x $-4$4 $-2$2 $-1$1 $-0.5$0.5 $-0.25$0.25 $0.25$0.25 $0.5$0.5 $1$1 $2$2 $4$4
$y$y $-0.25$0.25 $-0.5$0.5 $-1$1 $-2$2 $-4$4 $4$4 $2$2 $1$1 $0.5$0.5 $0.25$0.25

Notice that:

  • If $x$x is positive, $y$y is positive
  • If $x$x is negative, $y$y is negative
  • As $x$x gets further from zero (in either direction), $y$y gets closer to zero

Here are some hyperbolas with equations of the form $y=\frac{k}{x}$y=kx (or $xy=k$xy=k).

Notice the following features:

  • Each hyperbola has two parts and they lie in opposite quadrants.
  • They all approach the line $y=0$y=0 (the $x$x-axis), and they also approach the line $x=0$x=0 (the $y$y-axis), but they never cross these lines. This is because the equation $y=\frac{k}{x}$y=kx is not defined for $x=0$x=0 or $y=0$y=0. We call the lines $x=0$x=0 and $y=0$y=0 asymptotes.
  • They are symmetrical about the lines $y=x$y=x and $y=-x$y=x, and they have rotational symmetry about the origin.
  • We can see that every $x$x and $y$y coordinate on a hyperbola multiply to give the value of $k$k in their specific equation, as is suggested by the general form $xy=k$xy=k. (This gives us an easy method of finding an equation of a hyperbola from a sketch.)

Practice Questions

Question 1

Which of the following equations represent inverse variation between $x$x and $y$y?

Select all correct answers.

  1. $y=\frac{7}{x}$y=7x

    A

    $y=6x+8$y=6x+8

    B

    $y=-\frac{9}{x}$y=9x

    C

    $y=\frac{8}{x^2}$y=8x2

    D

    $y=2x^2-7x-4$y=2x27x4

    E

    $y=3-x$y=3x

    F

Question 2

In the table of values below, $m$m is proportional to $\frac{1}{p}$1p.

$p$p $2$2 $4$4 $5$5 $x$x
$m$m $140$140 $y$y $56$56 $40$40
  1. Determine the constant of proportionality, $k$k.

  2. Using the solution to part (a), find the unknowns in the table:

    $x$x$=$=$\editable{}$

    $y$y$=$=$\editable{}$

Question 3

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

Loading Graph...

  1. For positive values of $x$x, as $x$x increases $y$y approaches what value?

    $0$0

    A

    $1$1

    B

    $-\infty$

    C

    $\infty$

    D
  2. As $x$x takes small positive values approaching $0$0, what value does $y$y approach?

    $\infty$

    A

    $0$0

    B

    $-\infty$

    C

    $\pi$π

    D
  3. What are the values that $x$x and $y$y cannot take?

    $x$x$=$=$\editable{}$

    $y$y$=$=$\editable{}$

  4. The graph is symmetrical across two lines of symmetry. State the equations of these two lines.

    $y=\editable{},y=\editable{}$y=,y=

Question 4

Consider the hyperbola that has been graphed.

Loading Graph...

  1. Fill in the gap to complete the statement.

    Every point $\left(x,y\right)$(x,y) on the hyperbola is such that $xy$xy$=$=$\editable{}$.

  2. Considering that the relationship between $x$x and $y$y can be expressed as $xy=6$xy=6, which of the following is true?

    If $x$x increases, $y$y must increase.

    A

    If $x$x increases, $y$y must decrease.

    B
  3. Which of the following relationships can be modelled by a function of the form $xy=a$xy=a?

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

    A

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

    B

    The relationship between height and weight.

    C

 

Outcomes

MA11-1

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

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