Precalculus

Chapter 2: Polynomial and Rational Functions

2.1:Power Functions and Monomials

Before we define a polynomial function, we shall discuss some basic functions.
One of most important building blocks of functions is power functions. These are an important family of functions in their own right. The identity function, square function, cubic function, square-root function, and reciprocal function are five of the twelve basic functions that we discussed in Chapter 1. These belong to the family of power functions.

Definition: Any function that can be written in the form of $f(x) = k\times x^n$, where $n$ and $k$ are nonzero constants, is called a power functions. The constant $n$ is called the power of the function. Power functions are very useful in modeling a variety of practical application problems. We shall discuss these in the last module of this chapter.

Example 1:

For the power functions $(a) 2x^3, \quad (b) 3x^{-4}, \quad (c) 8 \sqrt{x^{\frac{3}{2}}}$, state the power of each.

Solution: $(a) 3; \quad (b) -4; \quad (c) \frac{3}{2}$

A power function is also called a monomial or a single term , when the power $n$ is a positive integer. In this case, the power $n$ is called the degree of the monomial and the constant $k$ is called the coefficient of the monomial.

Example 2:

Determine whether each of the following power functions is a monomial or not: $(a) 2x^3; \quad (b) -4x^{-2}; \quad (c) 5x^{\frac{4}{7}}$

Solution: $(a)$ Yes. The power is the positive integer $3$; $(b)$ No. the power is $-2$, which is not a positive integer; $(c)$ No. The power is $\frac{4}{7}$, which is not a positive integer.

Example 3:

Determine the degree and coefficient of each of the following monomials: $(a) -3x^4; \quad (b) 2x^8; \quad (c) \frac{2}{9}x; \quad (d) 6$

Solution: $(a)$ Degree: $4$, Coefficient: $-3$; $(b)$ Degree: $8$, Coefficient: $2$; $(c)$ Degree: $1$, Coefficient: $\frac{2}{9}$; $(d)$ Degree: $0$, Coefficient: $6$.

Monomial functions are either odd or even functions. They are odd, if the degree $n$ is odd and even when $n$ is even. We already know that odd functions are symmetric about the origin and even functions are symmetric about the $y$-axis. Because of this property of symmetry, we can easily generalize the end-behavior of monomial functions. Graphs of all monomial functions pass through origin.

Graphs of monomial functions:

End behavior of the graph of a function tells us whether the graph goes upwards or downwards when $x$ goes towards $\pm \infty$. As $x \rightarrow \pm \infty$, When the graph goes up, we say that $f(x)$ approaches $\infty$ or $y \rightarrow \infty$, and when the graph goes down, we say that $f(x)$ approaches $-\infty$ or $y \rightarrow -\infty$.

Linear monomial functions: Monomials of degree 1 are of the form $f(x) = mx$. Graphs of these are straight lines passing through origin. If the coefficient $m$ is positive, then the line is a rising line. This means, if $m \gt 0$, as $x \rightarrow -\infty, \quad y \rightarrow -\infty$ and as $x \rightarrow \infty, \quad y \rightarrow \infty$. If the coefficient $m$ is negative, then the graph is a falling line. This means that, as $x \rightarrow -\infty \quad y \rightarrow \infty$ and as $x \rightarrow \infty, \quad y \rightarrow -\infty$. Graphs of linear monomials do not turn in their directions. This means that there are no turning points. In the examples of these two types of graphs shown below, the blue graph that of $f(x) = 2x$ and, the brown graph is that of $f(x) = -3x$.
Quadratic monomial functions: These are monomials of degree 2 and are of the form $f(x) = ax^2, \quad a \neq 0$. Graphs of these functions are parabolic ( U shaped) where the curve opens up when the coefficient $a$ is positive and opens down when the coefficient is negative. This means that, when $a \gt 0$, as $x \rightarrow -\infty, \quad y \rightarrow \infty$. When $a \lt 0$, as $x \rightarrow -\infty, \quad y \rightarrow \infty$ and as $x \rightarrow \infty, \quad y \rightarrow -\infty$. Graphs of these monomials have one turning point. In the example graph shown below, the blue curve is the graph os $f(x) = 2x^2$ and the pink curve is that of $f(x) = -3x^2$. Both graphs turn direction at the origin.
Cubic monomial functions: These monomials of degree 3 are of the form $f(x) = ax^3$. The end behavior of the graphs of these monomials resemble the end behaviour of linear monomials. This means that, when $a \gt 0$, as $x \rightarrow -\infty, \quad y \rightarrow -\infty$ and as $x \rightarrow \infty, \quad y \rightarrow \infty$. If the coefficient $a$ is negative, $x \rightarrow -\infty \quad y \rightarrow \infty$ and as $x \rightarrow \infty, \quad y \rightarrow -\infty$. Graphs of these monomials have two turning points, both coinciding at the origin. The graph below shows these two types of cubic monomials. The blue curve is the graph of $f (x) = 2x^3$ and the pink curve is that of $f(x) = -3x^3$.

The behavior of the graph as $x \rightarrow -\infty$ is called the left-end behavior and its behavior as $x \rightarrow \infty$ is called the righ-end behavior.
Quartic monomials: These monomials of degree 4 are of the form $f(x) = ax^4$. The end behaviour of the graphs are similar to the end behaviour of quadratic monomials.Graphs of these monomials hae 3 turning points, all coinciding at the origin. In the example graph shown below, the blue curve is the graph of $f(x) = 2x^4$ and the pink curve is that of $f(x) = -3x^4$.

Based on the above illustrations, we conclude the following patterns of the graphs of monomial functions:

Graphs of odd degree monomials have end behaviour similar to linear monomials. These are symmetric about the origin.
Graphs of even degree monomials end behaviour similar to quadratic monomials. These are symmetric about the $y$- axis.
If the monomial is of degree $n$, the graph has $(n-1)$ turning points.

Example 4:

State the power of each of the following power functions. If any of these are monomials, then state the degree and coefficient of these monomials:
$(a) f(x) = -\frac{1}{2}x^3;\quad (b) g(x) = -3x^{-3}; \quad (c) h(x) = \frac{4}{x}; \quad (d) p(x) = -1.5x^5$

Solution: $(a)$. Monomial of degree $5$ and coefficient $\frac{1}{2}$; $(b)$. Power is $-3$ and coefficient is $3$. $(c)$. Power is $-1$ and coefficient is $4$. $(d)$. Monomial of degree $3$ and coefficient is $-1.5$.

Example 5:

For each of the following monomials, (a) state the degree and coefficient; (b) state whether the function is even or odd; (c) state the type of symmetry; (d) describe the end behaviour; and (e) number of turning points:
$(i) \quad f(x) = 8x^{11}$
$(ii) \quad f(x) = - \frac{2}{3}x^6$;
$(iii) \quad g(t) = -3t^5$

Solution:$(i)$ degree 11, coefficient 8; odd function; symmetric about origin; as $x \rightarrow -\infty, \quad y \rightarrow -\infty$ and as $x \rightarrow \infty, \quad y \rightarrow \infty$; 10 turning points.
$(ii)$ degree 6, coefficient $-\frac{2}{3}$; even function; symmetric about $y$-axis; as $x \rightarrow -\infty, \quad y \rightarrow \infty$ and as $x \rightarrow \infty, \quad y \rightarrow -\infty$; 5 turning points.
$(iii)$ degree 5, coefficient $-3$; odd function; symmetric about origin; as $x \rightarrow -\infty \quad y \rightarrow \infty$ and as $x \rightarrow \infty, \quad y \rightarrow -\infty$; 4 turning points.

Practice Problems

For each of the problems from #1 - #5, state the power of the power function and identify whether it is a monomial.Justify the reason for your decision.

$-4$. Not a monomial. Power is not a positive integer.

$6$. Yes. It is a monomial because the power is a positive integer.

$1$.Yes. It is a monomial because the power is a positive integer.

$-2$. No.It is not a monomial because the power is not a positive integer.

$5$. Yes. It is a monomial because the power is a positive integer.

For each of the monomials in problems #6 - #10, state the degree, coefficient, even or odd, type of symmetry, end behaviour, and number of turning points.

degree 8; coefficient $\frac{1}{2}$; even; symmetric about $y$- axis; goes up at both ends; 7 turning points.

degree 9; coefficient $-2$; odd; symmetric about origin; goes up as $x \rightarrow -\infty$ and goes down as $x \rightarrow \infty$; 8 turning points.

degree 7; coefficient $\frac{2}{5}$; odd; symmetric about origin; goes down as $x \rightarrow -\infty$ and goes up as $x \rightarrow \infty$; 6 turning points.

degree 14; coefficient $-5$; even; symmetric about $y$-axis; goes down at both ends; 13 turning points.

degree 5; coefficient $-2$; odd; symmetric about origin; goes up as $x \rightarrow -\infty$ and goes down as $x \rightarrow \infty$; 4 turning points.