 # 5.10 Hyperbolas

Lesson

Graphs of equations of the form $y=\frac{a}{x-h}+k$y=axh+k (where $a$a, and $h$h and $k$k are any number are called hyperbolas. The hyperbola defined by $y=\frac{1}{x}$y=1x

### Features of hyperbolas

Hyperbolas can have either $0$0 or $1$1 $x$x-intercepts. This is the point on the graph which touches the $x$x-axis. We can find this by setting $y=0$y=0 and finding the value of $x$x. If the $x$x-value is undefined, there is no $x$x-intercept. For example, there is no $x$x-intercept of $y=\frac{1}{x}$y=1x.

Similarlty, hyperbolas can have either $0$0 or $1$1 $y$y-intercepts. This is the point on the graph which touches the $y$y-axis. We can find this by setting $x=0$x=0 and finding the value of $y$y. If the $y$y-value is undefined, there is no $y$y-intercept. For example, there is no $y$y-intercept of $y=\frac{1}{x}$y=1x.

Hyperbolas have a vertical asymptote which is the vertical line which the graph approaches but does not touch. For example, the vertical asymptote of $y=\frac{1}{x}$y=1x is $x=0$x=0.

Hyperbolas also have a horizontal asymptote which is the horizontal line which the graph approaches but does not touch. For example, the horizontal asymptote of $$is y=0y=0 ### Transformations of hyperbolas A hyperbola can be vertically translated by increasing or decreasing the yy-values by a constant number. So to translate y=\frac{1}{x}y=1x up by kk units gives us y=\frac{1}{x}+ky=1x+k. Vertically translating up by 22 (y=\frac{1}{x}+2y=1x+2) and down by 22 (y=\frac{1}{x}-2y=1x2) Similarly, a hyperbola can be horizontally translated by increasing or decreasing the xx-values by a constant number. However, the xx-value together with the translation must both be in the denominator. That is, to translate y=\frac{1}{x}y=1x to the left by hh units we get y=\frac{1}{x+h}y=1x+h. Horizontally translating left by 22 (y=\frac{1}{x+2}y=1x+2) and right by 22 (y=\frac{1}{x-2}y=1x2) A hyperbola can be scaled by changing the value of the numerator. So to expand the hyperbola y=\frac{1}{x}y=1x by a scale factor of aa we get y=\frac{a}{x}y=ax. We can compress a hyperbola by dividing by the scale factor instead. Note that for hyperbolas, vertically scaling is equivalent to horizontally scaling. Expanding by a scale factor of 22 (y=\frac{2}{x}y=2x) and compressing by a scale factor of 22 (y=\frac{1}{2x}y=12x). We can reflect a hyperbola about either axis by taking the negative of the yy-values. So to reflect y=\frac{1}{x}y=1x about the xx-axis gives us y=-\frac{1}{x}y=1x. Note that for hyperbolas, vertically reflecting is equivalent to horizontally reflecting. Reflecting the hyperbola about the yy-axis ($$)

Summary

The graph of an equation of the form $y=\frac{a}{x+h}+k$y=ax+h+k is a hyperbola.

Hyperbolas can have $0$0 or $1$1 $x$x-intercepts and can have $0$0 or $1$1$y$y-intercepts, depending on the solutions to the equation.

Hyperbola have a vertical asymptote which is the vertical line that the graph approaches but does not intersect and a horizontal asymptote which is the horizontal line that the graph approaches but does not intersect.

Hyperbolas can be transformed in the following ways (starting with the hyperbola defined by $y=\frac{1}{x}$y=1x):

• Vertically translated by $k$k units: $y=\frac{1}{x}+k$y=1x+k
• Horizontally translated by $h$h units: $y=\frac{1}{x-h}$y=1xh
• Scaled by a scale factor of $a$a: $y=\frac{a}{x}$y=ax
• Reflected about the $y$y-axis: $y=-\frac{1}{x}$y=1x

#### Practice questions

##### Question 1

Consider the function $y=\frac{1}{x}$y=1x which is defined for all real values of $x$x except $0$0.

1. Complete the following table of values.

 $x$x $-2$−2 $-1$−1 $-\frac{1}{2}$−12​ $-\frac{1}{4}$−14​ $\frac{1}{4}$14​ $\frac{1}{2}$12​ $1$1 $2$2 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Plot the points in the table of values.

3. Hence draw the curve.

4. In which quadrants does the graph lie?

$3$3

A

$2$2

B

$1$1

C

$4$4

D

$3$3

A

$2$2

B

$1$1

C

$4$4

D
##### Question 2

Consider the function $y=-\frac{1}{2x}$y=12x

1. Complete the following table of values.

 $x$x $y$y $-3$−3 $-2$−2 $-1$−1 $1$1 $2$2 $3$3 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
2. Sketch the graph.

3. In which quadrants does the graph lie?

$2$2

A

$1$1

B

$3$3

C

$4$4

D

$2$2

A

$1$1

B

$3$3

C

$4$4

D
##### Question 3

Consider the equation $f\left(x\right)=\frac{3}{x}$f(x)=3x.

1. Sketch a graph of the function on the axes below:

2. Which of the following statements about the symmetry of the graph is true?

The graph is symmetric about the $x$x-axis

A

The graph is symmetric about the $y$y-axis.

B

The graph has no symmetry.

C

The graph is rotationally symmetric about the origin.

D

The graph is symmetric about the $x$x-axis

A

The graph is symmetric about the $y$y-axis.

B

The graph has no symmetry.

C

The graph is rotationally symmetric about the origin.

D
3. Find an expression for $f\left(-x\right)$f(x):

### Outcomes

#### VCMNA359 (10a)

Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations.