# 3.7 Rational functions  (Page 7/16)

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Find the vertical and horizontal asymptotes of the function:

$f\left(x\right)=\frac{\left(2x-1\right)\left(2x+1\right)}{\left(x-2\right)\left(x+3\right)}$

Vertical asymptotes at $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=–3;\text{\hspace{0.17em}}$ horizontal asymptote at $\text{\hspace{0.17em}}y=4.$

## 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 $\text{\hspace{0.17em}}f\left(x\right)=\frac{\left(x-2\right)\left(x+3\right)}{\left(x-1\right)\left(x+2\right)\left(x-5\right)}.$

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

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

The y -intercept is $\text{\hspace{0.17em}}\left(0,–0.6\right),\text{\hspace{0.17em}}$ the x -intercepts are $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(–3,0\right).\text{\hspace{0.17em}}$ See [link] .

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. $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{\left(x-3\right)}^{2}}-4=\frac{1-4{\left(x-3\right)}^{2}}{{\left(x-3\right)}^{2}}=\frac{1-4\left({x}^{2}-6x+9\right)}{\left(x-3\right)\left(x-3\right)}=\frac{-4{x}^{2}+24x-35}{{x}^{2}-6x+9}$

Because the numerator is the same degree as the denominator we know that as is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ because as $\text{\hspace{0.17em}}x\to 3,f\left(x\right)\to \infty .\text{\hspace{0.17em}}$ We then set the numerator equal to 0 and find the x -intercepts are at $\text{\hspace{0.17em}}\left(2.5,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3.5,0\right).\text{\hspace{0.17em}}$ Finally, we evaluate the function at 0 and find the y -intercept to be at $\text{\hspace{0.17em}}\left(0,\frac{-35}{9}\right).$

## 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] .

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] .

For example, the graph of $\text{\hspace{0.17em}}f\left(x\right)=\frac{{\left(x+1\right)}^{2}\left(x-3\right)}{{\left(x+3\right)}^{2}\left(x-2\right)}\text{\hspace{0.17em}}$ is shown in [link] .

• At the x -intercept $\text{\hspace{0.17em}}x=-1\text{\hspace{0.17em}}$ corresponding to the $\text{\hspace{0.17em}}{\left(x+1\right)}^{2}\text{\hspace{0.17em}}$ factor of the numerator, the graph bounces, consistent with the quadratic nature of the factor.
• At the x -intercept $\text{\hspace{0.17em}}x=3\text{\hspace{0.17em}}$ corresponding to the $\text{\hspace{0.17em}}\left(x-3\right)\text{\hspace{0.17em}}$ factor of the numerator, the graph passes through the axis as we would expect from a linear factor.
• At the vertical asymptote $\text{\hspace{0.17em}}x=-3\text{\hspace{0.17em}}$ corresponding to the $\text{\hspace{0.17em}}{\left(x+3\right)}^{2}\text{\hspace{0.17em}}$ factor of the denominator, the graph heads towards positive infinity on both sides of the asymptote, consistent with the behavior of the function $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{{x}^{2}}.$
• At the vertical asymptote $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ corresponding to the $\text{\hspace{0.17em}}\left(x-2\right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{x}.$

So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?