Sketching Polynomial Functions

Let's begin by determining the degree of each function. It is determined by the exponent with the highest value. The polynomial function $p(x)=0.6x^3+2x^2+4$ is of degree $3,$ $q(x)=x+4$ is of degree $1$ and $h(x)=0.2x^2+x+4$ is of degree $2.$ Now we need to link each graph to its corresponding function by thinking about what the functions' graphs should look in order for them to match.

Graph A

Graph A has the following end behavior. $$\begin{array}{c}\text{As }x\to \text{-}\infty ,f(x)\to \text{-}\infty \\ \text{and}\\ \text{As }x\to +\infty ,f(x)\to \infty \end{array}$$ That is consistent with a function with odd degree. We have two polynomial functions with odd degree, $p(x)$ and $q(x).$ Since the graph is not a linear function, then $q(x)$ is not a match. Therefore, we know that the graph must be that of the function $p(x)=0.6x^3+2x^2+4.$

Graph B

Now there are two functions left, $h(x)$ and $q(x).$ Since the graph B is a parabola, the function has a degree of $2.$ Therefore, the function $h(x)=0.2x^2+x+4$ is the function for this graph.

Graph C

Since the blue graph is a line it represents a linear function. Thus, we can conclude that it is the graph of $q(x)=x+4.$