Rational Function (Domain, x & y - Intercepts, Zeros, Vertical and Horizontal Asymptotes and Hole)
Summary
TLDRThis tutorial video teaches viewers how to analyze rational functions by determining their domain, x-intercepts, y-intercept, zeros, vertical and horizontal asymptotes, and holes. The instructor uses step-by-step examples to illustrate the process of finding these characteristics, starting with defining the domain as all real numbers except the zeros of the denominator. The video then demonstrates how to find x and y-intercepts by setting y to zero and x to zero respectively, and explains how to identify zeros and asymptotes based on the degrees of the numerator and denominator. The concept of holes, or removable discontinuities, is also covered, showing how they occur when there's a common factor in the numerator and denominator. The tutorial is designed to provide a comprehensive understanding of rational functions.
Takeaways
- 📘 A rational function is defined as the ratio of two polynomial functions.
- 🚫 The domain of a function includes all real numbers except the zeros of the denominator.
- ✖️ To find the x-intercept, set y to zero and solve for x, ensuring x is within the domain.
- 🔍 The y-intercept is found by setting x to zero and solving for y.
- 🔢 The zeros of a rational function are the values of x that make the function equal to zero, considering the domain restrictions.
- 📉 Vertical asymptotes occur at values of x that make the denominator zero, but are not included in the domain.
- 📈 The horizontal asymptote is determined by comparing the degrees of the numerator and the denominator, and can be none if the degree of the numerator is greater.
- 🕳 A hole, or removable discontinuity, is present when there is a common factor in the numerator and denominator, leading to a discontinuity at the value of x that makes the common factor zero.
- 🔄 The process involves factoring the numerator and denominator to identify intercepts, asymptotes, zeros, and holes.
- 📚 The tutorial provides a systematic approach to analyzing rational functions, emphasizing the importance of domain considerations.
Q & A
What is a rational function?
-A rational function is any function that can be written as the ratio of two polynomial functions, with the numerator and the denominator being polynomials.
How do you find the domain of a rational function?
-The domain of a rational function is the set of all real numbers except for the values that make the denominator zero.
What is an x-intercept and how do you find it?
-An x-intercept is a point where the graph of the function crosses the x-axis, which means the y-value is zero. To find it, set y to zero and solve the equation for x.
What is a y-intercept and how is it determined?
-A y-intercept is the point where the graph of the function crosses the y-axis, which occurs when x is zero. Determine it by substituting x with zero in the function and solving for y.
What are the zeros of a rational function?
-The zeros of a rational function are the values of x that make the function equal to zero. They are typically found by setting the function to zero and solving for x.
How do you determine if a rational function has a vertical asymptote?
-A rational function has a vertical asymptote at values of x that make the denominator zero but are not included in the domain of the function.
What is a horizontal asymptote and how is it found?
-A horizontal asymptote is a horizontal line that the graph of the function approaches as x approaches infinity or negative infinity. It is found by comparing the degrees of the numerator and the denominator.
What is meant by a 'hole' in a rational function?
-A 'hole' in a rational function refers to a removable discontinuity, which is a point where the function is not defined due to a common factor in the numerator and denominator that can be canceled out.
How do you find the coordinates of a 'hole' in a rational function?
-To find the coordinates of a 'hole', identify the common factor in the numerator and denominator, set it equal to zero to find the x-value, and then substitute this x-value back into the function to find the y-value.
What is the significance of the degrees of the numerator and denominator in determining the horizontal asymptote?
-The degrees of the numerator and denominator determine the horizontal asymptote as follows: if the degree of the numerator is less than the degree of the denominator, the asymptote is y=0; if they are equal, the asymptote is the ratio of the leading coefficients; if the degree of the numerator is greater, there is no horizontal asymptote.
Outlines
📘 Introduction to Rational Functions
This paragraph introduces the concept of rational functions, defined as the ratio of two polynomial functions. The tutorial aims to teach viewers how to find various aspects of a rational function, including its domain, x-intercepts, y-intercepts, zeros, vertical and horizontal asymptotes, and holes. The first example provided is a function f(x) = (x^2 - x - 6) / (x - 3), and the process begins with determining the domain by setting the denominator equal to zero and solving for x, which results in x = 3. The domain is all real numbers except x ≠ 3.
📐 Finding X-Intercepts and Y-Intercepts
The tutorial continues with the process of finding the x-intercepts by setting y to zero and solving the equation. The function is factored to simplify the process, resulting in (x - 3)(x + 2) / (x - 3). It's noted that x = 3 is not included in the domain, so the x-intercept is at x = -2, y = 0. The y-intercept is found by setting x to zero, leading to y = 2. The zeros of the function are the same as the x-intercepts, but since x = 3 is not in the domain, it's not considered a zero. The concept of horizontal asymptotes is introduced, and it's determined that there is no horizontal asymptote for this function because the degree of the numerator is greater than the degree of the denominator.
🔍 Analyzing Rational Functions for Asymptotes and Holes
The tutorial moves on to discuss vertical asymptotes, which are the values of x that are not in the domain. For the given function, x = 3 is a vertical asymptote. The concept of holes is introduced, which are removable discontinuities caused by common factors in the numerator and denominator. Since there is a common factor of (x - 3), x = 3 is identified as a hole, and by substituting x = 3 into the remaining factor (x + 2), the y-coordinate of the hole is found to be 5, making the hole's coordinates (3, 5).
📉 Exploring More Rational Functions and Their Characteristics
A second example is presented, f(x) = 1 / (x - 1), to illustrate finding the domain, x-intercepts, y-intercepts, and asymptotes without common factors. The domain is all real numbers except x ≠ 1. There is no x-intercept because the equation 0 = 1 / (x - 1) has no solution for x. The y-intercept is found by setting x to zero, resulting in y = -1. Since there are no common factors, there are no zeros or holes. The vertical asymptote is at x = 1, and the horizontal asymptote is y = 0 because the degree of the numerator is less than the degree of the denominator.
📚 Comprehensive Review of Rational Functions
The final example, f(x) = (x^2 - 4) / (x^2 - 9), is used to demonstrate finding the domain, x-intercepts, y-intercepts, zeros, vertical asymptotes, and horizontal asymptotes. The domain excludes x = ±3. The x-intercepts are found to be x = -2 and x = 2. The y-intercept is calculated as y = 4/9. The zeros are the same as the x-intercepts. Vertical asymptotes are at x = ±3, and the horizontal asymptote is y = 1, as the degrees of the numerator and denominator are equal. The tutorial concludes with a summary of the methods to analyze rational functions.
Mindmap
Keywords
💡Rational Function
💡Domain
💡X-intercept
💡Y-intercept
💡Zeros
💡Vertical Asymptote
💡Horizontal Asymptote
💡Whole
💡Factoring
💡Degree of a Polynomial
Highlights
Definition of a rational function as the ratio of two polynomial functions.
Explanation of finding the domain of a rational function by excluding the zeros of the denominator.
Method to determine the x-intercept by setting y to zero and solving the equation.
Procedure for finding the y-intercept by substituting x with zero in the function.
Technique to identify the zeros of a rational function, which are the same as the x-intercepts.
Criterion for determining the presence of a horizontal asymptote based on the degrees of the numerator and denominator.
Identification of vertical asymptotes as the values that make the denominator zero, excluding them from the domain.
Concept of a hole or removable discontinuity in a rational function where a common factor exists in both the numerator and denominator.
Tutorial on solving for the domain, x-intercept, y-intercept, zeros, vertical, and horizontal asymptotes, and holes for a given rational function.
Example problem demonstrating the calculation of the y-intercept resulting in a value of 2.
Example illustrating that there is no x-intercept when the function results in an undefined value.
Explanation of how to find the vertical asymptote by identifying the values that are excluded from the domain.
Guide on calculating the horizontal asymptote by comparing the degrees of the numerator and the denominator.
Process for determining the hole in a rational function by factoring and identifying common factors.
Second example problem walkthrough for a different rational function, emphasizing the steps to find various characteristics.
Final summary of the tutorial's content, reinforcing the understanding of rational functions and their key features.
Transcripts
in this tutorial video i will be
teaching you how to get
the domain
the x-intercepts
the y-intercepts the zeros
the vertical asymptote the horizontal
asymptote and the whole of rational
functions
let us first define what is a rational
function
a rational function is any function that
can be written as the ratio
so when we say ratio we have the
numerator and denominator of two
polynomial functions
now let's have our problem number one
let's say we have f of x is equal to x
squared minus x minus six all over x
minus three
we're going to find the domain the
x-intercept y-intercept zeros the
vertical asymptote horizontal asymptote
and the whole if you notice we have
guide in every item
so for the domain
so let's find first the domain
the domain is the set of all numbers
except the zeros of the denominator so
for the domain
let's say a
domain
find the zeros of our denominator
so our denominator here is
x minus three
so copy x minus three just equate to
zero
so solve for x so we have
x is equal to
opacity of three
so our domain is
x such that x is an element
of real numbers
exact
exactly
exact
3
or simply just get the value for x to
make our rational function undefined so
get our denominator then equate to 0.
letter b
x intercept
our guide to find the x intercept let y
be equal to zero
so f of x that stands for our y
so we have
y
f of x is our y
then we have
x squared minus
x minus six
all over
x minus three
let y be equal to zero so
y be zero
and we have now
x squared minus x minus six
all over
x minus three
cross multiply so we have zero times x
minus three that will become zero is
equal to
x squared minus x minus six
now let us factor x squared minus x
minus six
okay to get the factor
um get the product of negative negative
six the sum must be negative one
so factoring general trinomial so x and
x
factors of negative six
we have negative three and positive two
let's check
negative three times positive two that
is negative six
negative three plus two that is negative
one or simply negative x here
so
this function
is just equal to x minus three times x
plus two all over x minus three
okay let us write here go back in letter
b zero is equal to
just copy the factored form
x minus 3 and x plus 2.
now
equate to 0
so we have x minus 3 is equal to 0
and
x plus two is equal to zero
solve for x
so x is equal to
positive three
and
here
x is equal to negative two
so we have two x-intercepts we have
three zero
and
negative two
zero
but take a look in our
domain that our x must not be equal
to three
so
we're just going to consider negative
two
zero
so our x-intercept is negative two
zero
where did we get zero
that is the value of our y
so
our uh let's have a short review
our x-intercept is a point that means it
must contain our
[Music]
abscissa and ordinate so our x here is
negative two then y
zero
now letter c
y-intercept
y-intercept
for the y-intercept
let x be equal to zero
so we have
let x be equal to zero
so
y
is equal to
just substitute
our x to zero so we have
zero squared minus zero
change x to zero
minus six all over
zero minus three
now we have
zero square that is zero minus zero
that is zero minus six so negative six
all over zero minus three negative three
negative six divided by negative three
that is positive 2.
so our y intercept is
our x is 0 so we have 0
positive 2.
this will be our x intercept
okay or y is equal to two
next
letter d
now let's have the zeros of our rational
fraction
zeros
in our guide this is the same as our
x-intercept so letter b x-intercept we
have our solution here
and we're going to find the value for
our x
so x here is equal to negative 2 and
positive 3 but take a look but take a
look in our domain
our restricted value for x is three so
x is a restricted value for our zeros
x is equal to
negative
discuss here because this is a crucial
part
let's have
our first
condition
d here stands for the degree
if the degree of the numerator and
stands for the numerator if the degree
of the numerator is less than the degree
of the denominator
our y is equal to zero
if the degree
of the numerator is equal to the degree
of the denominator we're going to use
a over b
where a is our leading coefficient in
the numerator and b is the leading
coefficient in that denominator
and if the degree of the numerator is
greater than the degree of the
denominator we don't have a horizontal
asymptote
now studying our given
the degree
of our numerator
is greater than
the denominator
in our numerator earlier our quadratic
so our degree in the numerator is 2
and in our denominator that is a linear
function
so 1
so
the degree of the numerator is greater
than the degree of our denominator
so horizontal asymptote in this case we
don't have
none
there is no horizontal asymptote
now let's have
g
the whole
okay
hole is also known as the removable
discontinuities
so these are the values or the inputs
that causes our numerator and
denominator b zero
so in this case notice that
we have a common factor
x minus 3
x minus 3 in our numerator and in the
denominator so
these are the discontinuity of our
function so get the common factor
for the whole we have x minus 3 is equal
to 0
that is the common factor
so find the value for x so x is equal to
3
now we have
our abscissa or our x
to find our y
get the remaining factor in our
numerator which is x plus two
so x plus two
now
substitute the value of our x which is
three so we have three plus two
now we have three plus two we have five
now the coordinate of our whole
is
x is 3
then this will be the value of our y
[Music]
3 comma 5
that would be the discontinuities of our
rational function 3 comma 5 this is the
whole
again
if you have a common factor in the
numerator and denominator
get the common factor then equate to 0
then solve for x and after that
substitute to get our
organic
okay that is on how to find the domain
x-intercept y-intercept zeros vertical
asymptotes horizontal asymptote and
whole let's have our second example
now let's have our second example
f of x is equal to one over x minus one
let us find the domain
so letter a
domain
so get the denominator x minus one
equate to zero so x is equal to one
so our domain is
x such that x is an element of real
numbers
except
positive one
b
the x-intercept for the x-intercept let
y be equal to zero so x intercept
so
y
is equal to
one over x minus one again f of x is the
same as our y so
let y be equal to zero so we have zero
is equal to
one over
x minus one
cross multiply
so we have zero is equal to one notice
that we don't have a value for x
because zero times x minus one is zero
so that means we don't have
no
x
intercept
or simply none for the x-intercept
letter c
y-intercept
for the y intercept let x be equal to
zero
so we have
y is equal to one minus
x minus one
let x be zero so we have y is equal to
one over
zero minus one
that is
one over negative one
therefore y is equal to negative one
so our y intercept is
zero comma
negative one
that is our y y-intercept
and for the zeros
letter d
zeros
the same as the x-intercept so we don't
have
x-intercept so 0's
9
and for the vertical asymptote
letter e
these are the restricted values for x
vertical asymptotes
according to our domain
x such that x is an element of real
numbers except one
so vertical asymptote
x is equal to
one
letter f
horizontal asymptote
1 over x minus 1. so study our condition
if the degree of the numerator is less
than the degree of the denominator
our horizontal asymptote is y is equal
to zero
so
we have here a degree of one
in our denominator that means
the degree of the
denominator is greater than the degree
of the numerator
so we have our first condition
y is equal to zero
and lastly
to find the whole
we don't have a common factor
both in numerator and denominator
so
f of x is equal to x squared minus four
all over x squared minus nine
okay let us find first the factored form
of our given
in our numerator
that is
x plus 2
x minus 2
and in our denominator
we have
x plus three
and x minus three
different of two squares
now get the domain
domain
our denominator
is x squared minus nine that is
the same as quantity x plus three
times
x minus three so that is
x plus three
and x minus three equate to zero
now solve for x
we have x plus three is equal to zero
and x minus three is equal to zero
so we have x is equal to negative 3
and x is equal to positive 3.
for our domain
x such that x is an element of real
numbers
except
positive negative 2
and for the x-intercept letter b
let y be equal to zero
so
f of x that stands for our y so
zero
is equal to
x squared minus four all over
x squared minus nine
cross multiply
we have zero is equal to
x squared minus four
get the factored form of x squared minus
nine so that is zero is equal to
x squared minus four
factors are
x plus two
x minus two
now equate to zero
so
x plus two is equal to zero
the other one x minus two is equal to
zero
so our x-intercept we have two values we
have
negative 2 and
positive 2.
okay
that is our x-intercept
and
letter c y-intercept
okay
letter c
y intercept let x be equal to zero
so in this case
y is equal to
let x be zero so zero squared minus four
all over
zero squared minus nine
so that is negative four all over
negative nine
negative divided by negative
that is
four over nine
so y is equal to four over nine
so
we can write this as 0
4 over 9 so that you can easily plot the
points and for this one we have
x is negative 2 0
and
two zero
okay
now get the zeros letter d
zeros
for the zeros
same as the x intercept so letter b x
intercept
so our zeros x is equal to
positive negative two
next letter e
vertical asymptote
okay
for the vertical asymptote
according to our domain our restricted
values
are positive and negative three
so for the vertical asymptote we have x
is equal to
positive three
and x is equal to negative
[Applause]
those are
the vertical asymptotes
and letter f
horizontal
so examine our condition
condition one if the degree of the
numerator is less than the degree of the
debit the denominator
that is
y is equal to zero
so we study our given we have the same
degree
so our degree
of the numerator is equal to the degree
of the denominator so we're going to use
y is equal to
a over b
wherein
a is the leading coefficient in the
numerator in this case you have one
over
b is the leading coefficient in the
denominator
we have also one one divided by one
that is one
so our horizontal asymptote is y is
equal to
one and our whole
as you can see in our factor factored
form we don't have a common factor so we
don't have
i hope you understand the rational
function and how to get
the domain x intercept y intercept the
zeros vertical asymptotes horizontal and
the whole
thank you for watching senior pablo tv
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