The graph shown at right is called a hyperbola and is given by the equation y= 1.x
Power functions are functions of the form f (ff x) = xn, n∈ R. The value of the power, n, determines the type of function. We saw earlier that when n= 1, f (ff x) = x and the function is linear. When xx n= 2, f (ff x) = x2 and the function is quadratic. When n= 3, f (ff x) = x3 and the function is cubic. When n= 4, f (ff x) = x4 and the function is quartic.
The power function that produces the graph of a hyperbola has a value of n=−1. Thus, the function f xf xf x( )= 1 can also be expressed as x the power function f (ff x) = x−1.
The graph exhibits asymptotic behaviour. That is, as x becomes very large, the graph xx approaches the x-axis, but never touches it, and as x becomes very small (approaches 0), the xx graph approaches the y-axis, but never touches it. So the line x = 0 (the y-axis) is a vertical asymptote and the line y = 0 (the x-axis) is the horizontal asymptote. Both the domain and the range of the function are all real numbers, except 0; that is, R\{0}.
The graph of y
= 1 can be subject to a number of transformations.x Consider y a
x b c
= − + or y = a(x − b)−1+ c
Dilation
The value a is a dilation factor. It dilates the graph from the x-axis.
Refl ection
If a is negative, the graph of the basic hyperbola is refl ected in the x-axis. If x is replaced with xx −x, the graph of the basic hyperbola is refl ected in the y-axis.
2C
x = 0 y = 0
y
0 x
x = 0 y = 0
y
0 x
a= 1 a= 2 a=1–2
y=a–x
For example, the graphs of y are refl ections of each other across the y-axis.
Translation
Horizontal translation
The value b translates the graph b units horizontally, that is, parallel to the x-axis. If b > 0, the graph is translated to the right, and if b< 0, the graph is translated to the left. For example, the graph with equation y
=x−1
3 is a basic hyperbola translated
3 units to the right. This graph has a vertical asymptote of x= 3 and domain R\{3} (and a horizontal asymptote y = 0). If a basic hyperbola is translated 3 units to the left, it becomes y= x
+ 1
3, with a vertical asymptote of x=−3 and domain R\{−3}. Hence, the equation of the vertical asymptote is x = b and the domain is R\{b}. The horizontal asymptote and the range remain the same, x= 0 and R \ {0}, respectively.
Vertical translation
The value c translates the graph c units vertically, that is, parallel to the y-axis. If c > 0, the graph is translated upward, and if c< 0, the graph is translated c units downward. The graph
with equation y
= +x
= +1 3 is a basic hyperbola translated 3 units up. This graph has a horizontal asymptote of y= 3 and a range of R\{3} (and a vertical asymptote x = 0). If a basic hyperbola isxx translated 3 units down, it becomes y
= −x
= −1 3, with a horizontal asymptote of y =−3 and a range of R\{−3} (and a vertical asymptote x = 0). Hence the equation of the horizontal asymptote is xx y = c and the range is R\{c}.
Always draw the asymptote as a dotted line and label it with its equation (for example, y= 3) at the end of the asymptote. Ensure that the graph continues to approach the asymptote getting closer but not touching or crossing the asymptote or bouncing away from the asymptote.
Combination of transformations
The graph of y a
x b c
= − + shows the combination of these transformations.
x = b
Finally, if the coeffi cient of x is a number other than 1, to obtain the value of xx h the equation should be rearranged fi rst. For example,
y= x4 = x eBookplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplus eBoo eBookkkplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplusplus
Digital doc Spreadsheet 051 The hyperbola
WORKED EXAMPLE 7
State the changes that should be made to the graph of y
==== 1 in order to obtain the graph ofx y= −
+4 x 2 − 1.
THINK WRITE
1 Write the general equation of the hyperbola. y a
x b c
= x bx b+
2 Identify the value of a. a=−4
3 State the changes to y= 1, caused by a.x The graph of y= 1 is dilated by the x factor of 4 from the x-axis and refl ected in the x-axis.
4 Identify the value of b. b=−2
5 State the effect of b on the graph. The graph is translated 2 units to the left.
6 Identify the value of c. c =−1
7 State the changes to the graph, caused by c. The graph is translated 1 unit down.
WORKED EXAMPLE 8 For the graph of y
==== x2 3 2
− + , state:
a the equations of the asymptotes b the domain c the range.
THINK WRITE
a 1 Write the general equation of the hyperbola. a y a x b c
=x bx b+
2 Identify the values of b and c and hence write the equations of the asymptotes:
Horizontal asymptote: y = c Vertical asymptote: x = b
b= 3, c = 2
Horizontal asymptote: y = 2 Vertical asymptote: x = 3 b State the domain of the hyperbola: R\{b}. b Domain: R\{3}
c State the range of the hyperbola: R\{c}. c Range: R\{2}
Sketching the graph of the hyperbola by hand can be easily done by following these steps:
1. Find the position of the asymptotes.
2. Find the values of the intercepts with the axes.
3. Decide whether the hyperbola is positive or negative.
4. On the set of axes draw the asymptotes (using dotted lines) and mark the intercepts with the axes.
5. Treating the asymptotes as the new set of axes, sketch either the positive or negative hyperbola, making sure it passes through the intercepts that have been previously marked.
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Worked example 9
Sketch the graph of y
= x+2 −
2 4 , clearly showing the intercepts with the axes and the position of the asymptotes.
THINK WRITE/DRAW
1 Compare the given equation with y a
x b c
= x bx b+ and state the values of a, b and c.
a = 2, b =−2, c=−4
2 Write a short statement about the effects of a, b and c on the graph of y
=1x .
The graph of y
=1x
is dilated by the factor of 2 from the x-axis, translated 2 units to the left and 4 units down.
3 Write the equations of the asymptotes.
The horizontal asymptote is at y= c.
The vertical asymptote is at x= b.
Asymptotes: x =−2; y =−4
4 Find the value of the y-intercept by
letting x= 0. y-intercept: x = 0
5 Find the value of the x-intercept by
making y = 0. x-intercept: y = 0
6 To sketch the graph:
x = −2 a Draw the set of axes and label them.
b Use dotted lines to draw the asymptotes. The asymptotes are x=−2 and y=−4.
c Mark the intercepts with the axes. The intercepts are y=−3 and x= − 32.
d Treating the asymptotes as your new set of axes, sketch the graph of the hyperbola (as a is positive, the graph is not refl ected); make sure the upper branch passes through the x- and y-intercepts previously marked.
The next example shows how to fi nd the equation of the hyperbola from its graph.
WORKED EXAMPLE 10
Find the equation of the graph shown. y
0 2 4 x 3 6
THINK WRITE
1 Write the general equation of the hyperbola. y a
x b c
= x bx b+
2 From the graph, identify the values of b and c Remember that the equation of the horizontal asymptote is y = c and of the vertical asymptote is x = b.
b= 2, c = 3
3 Substitute the values of b and c into the formula. y a
= x
− +
2 3
4 Substitute the coordinates of any of the 2 known points of intersection with the axes into the formula (say, x-intercept).
Substitute (4, 0):
0=4 24 24 2a +3
5 Solve for a. 0
2 3
2 3
6
= +
= +
=
=
−
−
a a a
6 Substitute the value of a into y a
= x
− +
2 3. y
= x
− +
−6
2 3
7 Transpose (optional). y
= −x 3 −
= −
= − 6
2
The graph of
1. y
=x1 is called a hyperbola.
The graph of
2. y a
x b c
=x bx b+ is the graph of the basic hyperbola, dilated by the factor of a from the x-axis, translated b units horizontally (to the right if b> 0, or to the left if b < 0) and c units vertically (up if c> 0, or down if c < 0). If a < 0, the graph is refl ected in the x-axis. The equations of the asymptotes are:
x = b and y = c. The domain of the function is R\{b} and its range is R\{c}.
x = b
y = c y
x
y c
0
=x − ba + REMEMBER
The power function (the hyperbola)
1 WE 7 State the changes that should be made to the graph of State the changes that should be made to the graph of y
=1 in order to obtain the x graph of each of the following.
a y
2 Which of the following transformations were applied to the graph of Which of the following transformations were applied to the graph of y
=x1 to obtain each of the graphs shown below?
i translation to the right ii translation to the left
iii translation up iv translation down
v refl ection in the x-axis
a y
i the equations of the asymptotes ii the domain iii the range.
a y
4 For each of the following graphs, state:
i the equations of the asymptotes ii the domain
6 WE9 Sketch each of the following, clearly showing the position of the asymptotes and the intercepts with the axes. Check your answers, using a CAS calculator.
a y
C The equation of the horizontal asymptote is y =−3.
D The equation of the vertical asymptote is x = 2.
E None of the above.
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Digital doc Spreadsheet 236 Function
grapher
9 WE10 Find the equation for each of the following hyperbolas, if they are of the form
, sketch each of the following, labelling the asymptotes and the intercepts with the axes.
a f (ff x+ 2) b f (ff x) − 1 c −−−fff x) ( − 2 d f (1 ff − x) + 2 e −−−fff x ( − 1) − 1 f 1 − f (ff x− 2) 11 Sketch the graph of yx − 3x + 1 = 0, and
state its domain and range. (Hint: First transpose the equation to make y the subject.)