Graphs of polynomial functions

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Polynomials are some of our favorite functions.

What do the graphs look like?

We understand the graphs of polynomials of degrees $1$ and $2$ very well.

Polynomial functions with degree $1$ are referred to as linear polynomials. This is due to the fact that such a function can be written as $f(x) = mx + b$ . The graph of such a function is a straight line with slope $m$ and $y$ -intercept at $(0,b)$ .

Quadratic functions, written as $f(x) = ax^2 + bx + c$ with $a \ne 0$ , have parabolas as their graphs.

Sketch the graph of the function $f(x) = \frac {1}{2} x^2 + 2x + 1$ .

To plot the parabola, we first complete the square to write our function in the form $f(x) = a(x-h)^2 +k$ . $\begin{align*} f(x) &= \frac{1}{2} x^2 + 2x + 1 \\ &= \frac{1}{2} \left( x^2+ 4x \right) + 1\\ &= \frac{1}{2}\left( x^2 + 4x + 4\right) + 1 - \frac{1}{2} \cdot 4\\ &= \frac{1}{2} \left( x+2 \right)^2 -1 \end{align*}$

The parabola we are looking for has vertex at $(-2, -1)$ , opens upward (since $1/2 > 0$ ), and is wider than the standard $y=x^2$ parabola.

In terms of graph transformations, we can think of this as the graph of $g(x) = x^2$ , which has been shifted horizontally $2$ units to the left, vertically compressed by a factor of $\frac {1}{2}$ , then shifted vertically $1$ unit downward.

Use the graph of $f(x) = 2x^2 - 4x + 3$ to estimate the value of $\lim _{x\to 2} f(x)$ . $\begin {prompt} \lim _{x\to 2} f(x) = \answer {3} \end {prompt}$

For polynomials of higher degree, we will have to take everything else we have been talking about and put it together. We need to understand the zeroes of the polynomial and its end behavior. The zeroes will correspond to $x$ -intercepts, and the end behavior will tell us what happens out past those intercepts.

One upshot of the Fundamental Theorem of Algebra (in terms of graphing) is that when we plot a polynomial of degree $n$ , its graph will cross the $x$ -axis at most $n$ times. Each crossing corresponds to a real root of that polynomial. (Complex roots do not give crossings!)

Sketch a rough graph of the function $f(x) = 2(x-1)^2(x+2)$ .

We start with the end behavior of this function. It is a degree $3$ polynomial with leading coefficient $2$ , so $\begin{align*} \text{As} \quad x \to \infty \, , \quad & f(x) \to \infty \\ \text{As} \quad x \to -\infty \, , \quad & f(x) \to -\infty \end{align*}$

We also see from the factorization that $x=1$ is a zero of multiplicity $2$ , and $x=-2$ is a zero of multiplicity $1$ . That means our graph will have $x$ -intercepts at those locations.

At these $x$ -intercepts, will the graph pass through the $x$ -axis like a line does, or will it touch the axis and turn back around, like a parabola does at it’s vertex? To answer this, look at a sign-chart for $f$ .

Notice that around $x=-2$ , only the $x+2$ factor changes signs. That means $f(x)$ will change signs, so the graph will pass through the $x$ -axis at $x=-2$ . At $x=1$ , however, the $(x-1)^2$ factor does NOT change signs (due to the even exponent). That means $f(x)$ will turn around at that $x$ -intercept. Putting this together, we have the following graph.

The function $f(x) = -4x^4 - 24x^3-32x^2+24x+36$ has zeroes at $x=-3$ , $x=1$ , and $x=-1$ . At which of those $x$ -intercepts does the graph of the function PASS THROUGH the $x$ -axis? (Don’t use a graphing calculator)

At $x=-3$ . At $x=1$ . At $x=-1$ . At none of them.

Our discussion in that example showed that, given an $x$ -intercept, we can determine if the graph passes through the $x$ -axis or just turns around there, by examining the multiplicity of that zero.

Suppose the polynomial function $f$ has a zero at $x=a$ of multiplicity $n$ . If $n$ is odd, then the graph of $f$ will pass through $x$ -axis at the point $(a,0)$ . If $n$ is odd, then the graph of $f$ will intersect the $x$ -axis at the point $(a,0)$ , but not pass through the $x$ -axis.

More specifically: The graph of $f$ will resemble, at least locally near the point $(a,0)$ , of the graph of $\pm (x-a)^n$ .

This last statement means that if $x=4$ is a zero of $f$ with multiplicity $3$ , then the graph of $f$ will pass through the $x$ -axis at $(4,0)$ , but it will flatten out the same way that the graph of $y=x^3$ flattens out around the origin.

Here we see the the graphs of four polynomial functions.

For each of the curves, determine if the polynomial has even or odd degree, and if the leading coefficient (the one next to the highest power of $x$ ) of the polynomial is positive or negative.

Curve $A$ is defined by an evenodd degree polynomial with a positivenegative leading term.
Curve $B$ is defined by an evenodd degree polynomial with a positivenegative leading term.
Curve $C$ is defined by an evenodd degree polynomial with a positivenegative leading term.
Curve $D$ is defined by an evenodd degree polynomial with a positivenegative leading term.