21.09 The hyperbola
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.)
Note: In this course you can use technology to graph hyperbolas, find stationary points, intercepts, and points of intersection.
Practice Questions
Question 1
Consider the graph of $y=\frac{2}{x}$y=2x.
A Cartesian plane has an $x$x-axis and $y$y-axis ranging from $-10$−10 to $10$10. Each axis has major tick marks at $2$2-unit intervals and minor tick marks at $1$1-unit intervals. A graph of the hyperbola $y=\frac{2}{x}$y=2x is plotted with two symmetrical branches. One branch lies in the first quadrant curving downward and rightward. And the other branch lies in the third quadrant curving downward and leftward.
For positive values of $x$x, as $x$x increases $y$y approaches what value?
$0$0
A$1$1
B$-\infty$−∞
C$\infty$∞
DAs $x$x takes small positive values approaching $0$0, what value does $y$y approach?
$\infty$∞
A$0$0
B$-\infty$−∞
C$\pi$π
DWhat are the values that $x$x and $y$y cannot take?
$x$x$=$=$\editable{}$
$y$y$=$=$\editable{}$
The graph is symmetrical across two lines of symmetry. State the equations of these two lines.
$y=\editable{},y=\editable{}$y=,y=
Question 2
Consider the hyperbola that has been graphed.
Fill in the gap to complete the statement.
Every point $\left(x,y\right)$(x,y) on the hyperbola is such that $xy$xy$=$=$\editable{}$.
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.
AIf $x$x increases, $y$y must decrease.
BWhich 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.
AThe relationship between the number of sales and the amount of revenue.
BThe relationship between height and weight.
C