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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.

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

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!)

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. 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.