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Functions whose function rules contain polynomials exhibit certain characteristics that that make it possible to sketch the functions by hand.

A polynomial function is a function whose rule contains a polynomial. For example, $p(x)=x^2-17 \quad \text{and} \quad q(x)=3x^4-x+9,$

are both polynomial functions. Unless explicitly stated, the domain of a polynomial function is all real numbers.The end behavior of a polynomial function is determined by its degree and the sign of its leading coefficient, that is, the coefficient of the highest degree monomial. As $x$ approaches to infinity, a positive leading coefficient makes the graph extend upward, and a negative one makes the graph extend downward. For even-degree polynomials, the left-end behaves the same way as the right-end. For odd-degree polynomials, they have opposite behavior.

In the figure, the $3$rd and $6$th degree polynomials must have negative leading coefficients, as their right-ends extend downward. The rest have right-ends that extend upward, so they have positive leading coefficients. Notice that the higher degree polynomial functions have more turns. While a higher degree doesn't necessarily imply more turns, it increases the number of turns the graph could have.

Four graphs of polynomial functions are shown in the image.

Observe the end behavior of the graphs to match them with the following function rules. $\begin{aligned} f(x) &= x^3 & t(x) &= \text{-} x^3+x^2 \\ g(x) &= x^2-4 \ \ & h(x) &= \text{-} x^4+3x^2 \end{aligned}$

From the end behavior of the graphs, we can determine whether they have a positive or negative leading coefficient and if the function is of even or odd degree. By looking at the right-end of the graphs, we can find the sign of the leading coefficient. If it extends upward, it's positive, and if it extends downward, it's negative.

The graphs I and II have a positive leading coefficient. This can be compared with the two function rules that have a positive leading coefficient, $f(x)=x^3 \quad \text{and} \quad g(x)=x^2-4.$ To determine which rule matches with I and II, we'll examine the degree of $f(x)$ and $g(x).$ The degree is even if the ends have the same behavior and odd if it's opposite.

By this, graph I must be of even degree and graph II of odd degree. Thus, the graphs can be matched with the following function rules: $\text{I: }g(x)=x^2-4,\quad\text{II: }f(x)=x^3.$ Now we have two graphs left to match with the function rules $t(x)=\text{-} x^3+x^2 \quad \text{and} \quad h(x)=\text{-} x^4+3x^2.$ They both have a negative leading coefficient, and the same is true for the graphs III and IV.

Because the left- and right-end have different behavior for graph III, it is of odd degree. Therefore, it can be matched with $t(x),$ since it also is of an odd degree, $\text{III: } t(x)=\text{-} x^3+x^2.$ Finally, the graph IV is of even degree, as its ends extend in the same direction. This satisfies the last function rule, $h(x).$ We have now matched all graphs with their corresponding function rule: $\begin{aligned} \text{I: }g(x)&=x^2-4 & \text{II: }f(x)&=x^3 \\ \text{III: }t(x)&=\text{-} x^3+x^2 & \text{IV: }h(x)&=\text{-} x^4+3x^2. \end{aligned}$

Using some of the characteristics of a polynomial function, such as its zeros and end behavior, a rough sketch of its graph can be drawn. For instance, the graph of a polynomial function with the following characteristics can be sketched.

Its zeros are $x=\text{-}2,$ $x=1$ and $x=4.$ As $x$ approaches positive infinity, the graph of the function extends upward, and as $x$ approaches negative infinity, it extends downward.

Plot the zeros

Sketch the end behavior

Next, mark the graph's end behavior in the coordinate plane. Note that the ends must be sketched to the left of the leftmost zero and to the right of the rightmost zero, or the graph will not represent a function. The example's right-end extends upward and its left-end extends downward.

Sketch the graph

Lastly, the points and the ends can be connected by a smooth curve. Note that the curve must change from increasing to decreasing or vice versa at least once between two zeros.

The graph above fulfills the criteria for the function. However, there are an infinite number of other polynomials that also do so. Another example is sketched below.

Thus, the information given was not enough to specify a single function.

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