M151Q Precalculus · Unit 3: Quadratics, polynomials, triangles · on Exam 3
MFG 7.1–7.2 Polynomial functions and their graphs
Study notes for Polynomial functions and their graphs in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in StudyCat.
Big ideas
- A polynomial $a_nx^n + \dots + a_0$ has degree $n$ and leading coefficient $a_n$. End behavior depends only on the leading term: even degree, both ends go the same way (up if $a_n>0$); odd degree, opposite ways (like $x^3$ if $a_n>0$).
- Zeros come from factors: $(x - r)^k$ gives a zero at $r$ of multiplicity $k$. Odd multiplicity crosses the axis; even multiplicity touches and turns. A degree-$n$ polynomial has at most $n$ real zeros and at most $n-1$ turning points.
- To sketch: find zeros and multiplicities, the $y$-intercept, end behavior; then connect smoothly. To write a formula from a graph: use the zeros for factors and one extra point to find the constant $a$.
- Products and powers of binomials: expand carefully; $(x-2)^3 \ne x^3 - 8$.
Key formulas
End behavior
$$\text{leading term } a_nx^n \text{ decides: } n \text{ even} \Rightarrow \text{same ends};\ n \text{ odd} \Rightarrow \text{opposite ends}$$
Zeros and multiplicity
$$(x-r)^k:\ k \text{ odd crosses},\ k \text{ even bounces}$$
From a graph
$$y = a(x-r_1)(x-r_2)\cdots,\quad \text{use one more point to find } a$$
Worked example
Write a polynomial with zeros $-2$ (multiplicity 1) and $3$ (multiplicity 2) whose $y$-intercept is $-18$.
Show the solution
$y = a(x+2)(x-3)^2$. At $x = 0$: $a(2)(9) = -18 \Rightarrow a = -1$. $y = -(x+2)(x-3)^2$, degree 3, falls to the right, touches at 3, crosses at $-2$.
Common mistakes
- Deciding end behavior from the constant term or a middle term.
- Drawing a crossing at an even-multiplicity zero.
Exam tipSketch the end behavior arrows first, then place the zeros; the picture usually fills itself in.
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