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3.7 Rational functions  (Page 7/16)

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Find the vertical and horizontal asymptotes of the function:

f ( x ) = ( 2 x − 1 ) ( 2 x + 1 ) ( x − 2 ) ( x + 3 )

Vertical asymptotes at   x = 2   and   x = – 3 ;   horizontal asymptote at   y = 4.

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Intercepts of rational functions

A rational function    will have a y -intercept when the input is zero, if the function is defined at zero. A rational function will not have a y -intercept if the function is not defined at zero.

Likewise, a rational function will have x -intercepts at the inputs that cause the output to be zero. Since a fraction is only equal to zero when the numerator is zero, x -intercepts can only occur when the numerator of the rational function is equal to zero.

Finding the intercepts of a rational function

Find the intercepts of   f ( x ) = ( x − 2 ) ( x + 3 ) ( x − 1 ) ( x + 2 ) ( x − 5 ) .

We can find the y -intercept by evaluating the function at zero

f ( 0 ) = ( 0 − 2 ) ( 0 + 3 ) ( 0 − 1 ) ( 0 + 2 ) ( 0 − 5 )          = − 6 10          = − 3 5           = − 0.6

The x -intercepts will occur when the function is equal to zero:

0 = ( x − 2 ) ( x + 3 ) ( x − 1 ) ( x + 2 ) ( x − 5 ) This is zero when the numerator is zero . 0 = ( x − 2 ) ( x + 3 ) x = 2 ,   − 3

The y -intercept is   ( 0 , –0.6 ) ,   the x -intercepts are   ( 2 , 0 )   and   ( –3 , 0 ) .   See [link] .

Graph of f(x)=(x-2)(x+3)/(x-1)(x+2)(x-5) with its vertical asymptotes at x=-2, x=1, and x=5, its horizontal asymptote at y=0, and its intercepts at (-3, 0), (0, -0.6), and (2, 0).
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Given the reciprocal squared function that is shifted right 3 units and down 4 units, write this as a rational function. Then, find the x - and y -intercepts and the horizontal and vertical asymptotes.

For the transformed reciprocal squared function, we find the rational form.   f ( x ) = 1 ( x − 3 ) 2 − 4 = 1 − 4 ( x − 3 ) 2 ( x − 3 ) 2 = 1 − 4 ( x 2 − 6 x + 9 ) ( x − 3 ) ( x − 3 ) = − 4 x 2 + 24 x − 35 x 2 − 6 x + 9

Because the numerator is the same degree as the denominator we know that as   x → ± ∞ ,   f ( x ) → − 4 ;   so   y = – 4   is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is   x = 3 ,   because as   x → 3 , f ( x ) → ∞ .   We then set the numerator equal to 0 and find the x -intercepts are at   ( 2.5 , 0 )   and   ( 3.5 , 0 ) .   Finally, we evaluate the function at 0 and find the y -intercept to be at   ( 0 , − 35 9 ) .

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Graphing rational functions

In [link] , we see that the numerator of a rational function reveals the x -intercepts of the graph, whereas the denominator reveals the vertical asymptotes of the graph. As with polynomials, factors of the numerator may have integer powers greater than one. Fortunately, the effect on the shape of the graph at those intercepts is the same as we saw with polynomials.

The vertical asymptotes associated with the factors of the denominator will mirror one of the two toolkit reciprocal functions. When the degree of the factor in the denominator is odd, the distinguishing characteristic is that on one side of the vertical asymptote the graph heads towards positive infinity, and on the other side the graph heads towards negative infinity. See [link] .

Graph of y=1/x with its vertical asymptote at x=0.

When the degree of the factor in the denominator is even, the distinguishing characteristic is that the graph either heads toward positive infinity on both sides of the vertical asymptote or heads toward negative infinity on both sides. See [link] .

Graph of y=1/x^2 with its vertical asymptote at x=0.

For example, the graph of   f ( x ) = ( x + 1 ) 2 ( x − 3 ) ( x + 3 ) 2 ( x − 2 )   is shown in [link] .

Graph of f(x)=(x+1)^2(x-3)/(x+3)^2(x-2) with its vertical asymptotes at x=-3 and x=2, its horizontal asymptote at y=1, and its intercepts at (-1, 0), (0, 1/6), and (3, 0).
  • At the x -intercept   x = − 1   corresponding to the   ( x + 1 ) 2   factor of the numerator, the graph bounces, consistent with the quadratic nature of the factor.
  • At the x -intercept   x = 3   corresponding to the   ( x − 3 )   factor of the numerator, the graph passes through the axis as we would expect from a linear factor.
  • At the vertical asymptote   x = − 3   corresponding to the   ( x + 3 ) 2   factor of the denominator, the graph heads towards positive infinity on both sides of the asymptote, consistent with the behavior of the function   f ( x ) = 1 x 2 .
  • At the vertical asymptote   x = 2 ,   corresponding to the   ( x − 2 )   factor of the denominator, the graph heads towards positive infinity on the left side of the asymptote and towards negative infinity on the right side, consistent with the behavior of the function   f ( x ) = 1 x .

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
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Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
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Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
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Haryormhidey Reply
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ALIYU
field is a region of space under the influence of some physical properties
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determine the slope giving that 3y+ 2x-14=0
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Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
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Adah
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Nassze Reply
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Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
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Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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