Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.3 Quadratic Functions
The extension that immediately follows quadratic functions occurs by simply adding more terms in a similar way that we did from linear to quadratic. We could go through each function from cubic to quartic, quintic, and so on, but that would take far too long. Instead, generalizing the group of functions is more efficient and will develop a good enough foundation. The functions that extend linear and quadratics with more terms are called Polynomials.
Forms
Polynomial functions are functions of the form:
In words, a polynomial is the sum of terms, where each term can be written as a constant, an, multiplied by a variable raised to some positive, whole number exponent, xn. This is an extension of quadratic equations. That is to say, if we keep adding new terms with new exponents, then it is called a polynomial. Quadratic and Linear functions are both called polynomials as well. A polynomial can be categorized by the largest exponent in the equation. This is called the degree of the polynomial. For example, a quadratic is a 2nd degree polynomial while a linear equation is a 1st degree polynomial. f(x)=3x7+2 would be a 7th degree polynomial.
A Rational Function can be written as the quotient of two polynomials:
Since there is division by a polynomial, we can think of a rational function having negative whole number exponents as well. It could also be represented as:
Graphing
The graph of a polynomial depends on several factors:
The degree of the polynomial tells us if the ‘tails’ or ends of the graph go the same direction. If the degree is an even number, then the tails go the same direction (like a parabola). If the degree is an odd number, then the tails go opposite directions (like a linear equation or a cubic function).
The leading coefficient, an, tells us which direction the tails go. If an>0 then the positive tail (where x is a large, positive number) is going up. If an<0 then the positive tail is going down.
A factored form of p(x) tells us the x-intercepts. If x=a is a repeated factor, like (x-a)3, then the exponent tells us is if the graph ‘bounces’ off the x-intercept or goes through it. Odd exponents mean the graph extends through that intercept, like a linear equation, while an even exponent means the graph bounces off that intercept, like on y=x2.
a0 tells us the y-intercept.
Example 1: Draw a sketch of y=x(x+1)(x-3)(x+5)
Solution: It is helpful to first identify the general shape, and the intercepts. This has four terms, so it is an even polynomial degree polynomial. Additionally, all the x-terms are positive, so an>0. Finally, it has four x-intercepts: x=-5, -1, 0, 3 and a y-intercept can be found by multiplying out all of the numerical terms y-intercept =1(-3)(5)= -15.
We can begin by plotting the intercepts and then connecting them with a smooth curve:
The graph of a rational function also depends on several factors. It is helpful to first fully factor the function.
Holes occur at points where the function can equal 0/0. The easiest way to identify these spots is to look for common factors between the numerator and denominator of a function.
x-intercepts occur when only the numerator is 0, like 0/(some number). Use the factors of the numerator that don’t appear in the denominator to identify the x-intercepts. Similar to polynomials, the multiplicity of the intercept determines whether the graph ‘bounces off’ the intercept or goes through it.
y-intercept can be easily found by plugging in x=0.
Vertical asymptotes will occur when the function is divided by 0, but has a numerical value on top like (some number)/0. Use the factors of the denominator to help identify these points. Like x-intercepts, the multiplicity of the asymptote describes its behavior. An asymptote that occurs an even number of times will be horizontally symmetrical (think of y=1/x2 ) while an odd multiplicity results in an asymptote where the graphs goes down on one side and up on the other (think of y=1/x).
The behavior of the ‘tails’ of the graph can be determined by plugging in a very large negative number (for the left tail) and a very large positive number (for the right tail).
Example 2: Draw a sketch of 
Solution: The values on top of the fraction will help find the x-intercept, while the values on bottom will help with vertical asymptotes.
x-intercepts: x=-2, -1, 4
Asymptotes: x=-3, 1
The y-intercept can be found by plugging in 0, or evaluating only the number terms:
Now we can plot the points and connect the lines as smoothly as possible, without crossing an asymptote. If the line approaches an asymptote, the line should go up or down as much as possible without touching it.
Domain
Polynomial functions are just like linear and quadratic functions in terms of domain. Polynomials have no domain restrictions.
A rational function will have domain restrictions at any point where there is an asymptote or a hole. The easiest way to identify this is to look at the denominator. If the denominator can equal 0, then the x-value for that output is not part of the domain.
Example 3: Identify the domain of 
Solution: The domain restrictions occur when the denominator equals 0. So we take (x-1)(x+3)≠0 and solve.
So, the domain can be written as x∈(-∞,-3)∪(-3,1)∪(1,∞) or x∈
\{-3,1}
Range
The range of a polynomial function will depend on the degree of the polynomial. For odd polynomials, the range is y∈. For even polynomials, the minimum or maximum would need to be calculated in a method similar to that of quadratics if possible. Another strategy would be to graph the function and use that to help identify the range of the polynomial.
For rational functions, the range will depend on the asymptotes closest to the tails and the tails themselves. It might be most helpful to graph the function and use it to identify the range.
Applications
Review
Polynomial and Rational Functions can be described using variables and whole number exponents.
Polynomials have no domain restrictions.
Rationals have domain restrictions anywhere the denominator equals 0.
The range of a Polynomial is determined by the type of exponent (even or odd).
The range of a Rational Function is defined similarly to Polynomial, but needs to take into account the tails of the function.
Rational Functions may have asymptotes or holes.
Equations
