It turns out that we can also consider functions that are made up of higher powers of \(x\text{,}\) like \(x^3, x^4, x^5\text{,}\) and so on. In general, we call this a polynomial function. So, linear and quadratic functions are specific types of polynomial functions.
Let’s explore polynomial functions. We’ll start with the most basic types, which are just a single power of \(x\text{.}\) We can graph polynomial functions using Desmos.
Exploration of power functions. Consider the graphs of \(y = x\text{,}\)\(y = x^2\text{,}\)\(y = x^3\text{,}\)\(y = x^4\text{,}\)\(y = x^5\text{,}\)\(y = x^6\text{.}\) Observe the pattern in the shapes.
Power functions with negatives. Consider \(y = -x\text{,}\)\(y = -x^2\text{,}\)\(y = -x^3\text{,}\)\(y = -x^4\text{,}\)\(y = -x^5\text{,}\)\(y = -x^6\text{.}\) These are basically the previous graphs except “upside down”. In other words, a reflection over the \(x\)-axis of the previous graphs.
\(x^3\) functions with more terms. Consider adding terms with smaller exponents. Take a function that starts with \(x^3\text{,}\) and add some terms like \(x^2\text{,}\)\(x\text{,}\) or a constant.
Observe that the first 3 all have the same S-shape, that starts in the bottom left and ends in the top right. The last 2 also share similar shapes, starting in the top left and ending in the bottom right. Observe that the negative sign in front basically reflects it upside down. The precise behavior of the function and how it goes up and down is much more complicated, however, and can’t be determined just by looking at the equation.
In fact, you have seen this pattern before with parabolas (quadratic functions). Quadratic functions with various numbers. Recall that the number in front of \(x^2\) (called the leading coefficient) determines if it opens up or opens down. In particular, if \(a \gt 0\) it opens up, and if \(a \lt 0\) it opens down.
Basically, a polynomial function is made up of powers of \(x\text{,}\) combined with numbers through multiplication, addition, and subtraction. Sometimes, we refer to them simply as polynomials.
The domain of a polynomial is all real numbers, or \(x \in \mathbb{R}\) (or, \((-\infty,\infty)\)). This is because it only uses the basic operations of addition, subtraction, and multiplication of numbers, and these operations are always defined.
\(f(x) = \frac{1}{5}(x+3)(x+1)(x-2)(x-3)\) is a polynomial, intuitively because if you were to expand out all of the brackets, all of the powers of \(x\) would be whole numbers.
You can always just say “degree 3 polynomial” or “degree 4 polynomial”, but you should be aware of these terms. Typically, the higher the degree of the polynomial, the more complicated it is.
As we’ve seen, the graph of a polynomial function can go either up or down on the left and right side, depending on its equation. This is called it’s end behavior.
Graphically, end behavior is what the function does to the “far right” or “far left” side of the graph. The graph of a polynomial function may increase in some places and decrease in others, however it will eventually tend towards either going “up” or “down”.
The end behavior of a polynomial function is determined by the leading term. Intuitively, this is because when \(x\) is large (either large positive, or large negative), the leading term dominates the other terms, in the sense that it’s size is much bigger than the others. All other terms become insignificant in size.
For odd-degree polynomials, the sign of the leading coefficient determines the overall trend, if it overall has “positive” slope (like \(\swarrow \nearrow\)) or “negative” slope (like \(\nwarrow \searrow\)), (similar to a line).
\begin{align*}
y = mx + b \amp \qquad \longrightarrow \qquad \text{$y$-int} = (0,b)\\
y = ax^2 + bx + c \amp \qquad \longrightarrow \qquad \text{$y$-int} = (0,c)
\end{align*}
This is because to find the \(y\)-intercept, set \(x = 0\text{,}\) and solve for \(y\text{.}\) Doing this makes all of the \(x\) terms vanish, leaving only the constant term left. In general, the \(y\)-intercept of a polynomial function is the constant term.
Subsection9.1.5x-Intercepts (Zeros) of a Polynomial
For \(x\)-intercepts, first note that the terms zero, root, and \(x\)-intercept are all somewhat used interchangably. Zero and root are more truly synonyms, whereas \(x\)-intercept typically refers to the point on the graph (where it touches the \(x\)-axis).
We will explore various techniques to solve equations like this. Most of the examples will be simple, with numbers chosen very carefully so that the problem is solvable. In general, solving this equation by hand and exactly is very difficult, and sometimes actually impossible (and only possible approximately).
