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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 $y=0$y=0

 

Transformations of hyperbolas

A hyperbola can be vertically translated by increasing or decreasing the $y$y-values by a constant number. So to translate $y=\frac{1}{x}$y=1x up by $k$k units gives us $y=\frac{1}{x}+k$y=1x+k.

Vertically translating up by $2$2 ($y=\frac{1}{x}+2$y=1x+2) and down by $2$2 ($y=\frac{1}{x}-2$y=1x2)

Similarly, a hyperbola can be horizontally translated by increasing or decreasing the $x$x-values by a constant number. However, the $x$x-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 $h$h units we get $y=\frac{1}{x+h}$y=1x+h.

Horizontally translating left by $2$2 ($y=\frac{1}{x+2}$y=1x+2) and right by $2$2 ($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 $a$a 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 $2$2 ($y=\frac{2}{x}$y=2x) and compressing by a scale factor of $2$2 ($y=\frac{1}{2x}$y=12x).

We can reflect a hyperbola about either axis by taking the negative of the $y$y-values. So to reflect $y=\frac{1}{x}$y=1x about the $x$x-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 $y$y-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.

    Loading Graph...

  3. Hence draw the curve.

    Loading Graph...

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

    Loading 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:

    Loading Graph...

  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.

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