Algebra 2 / Unit 3: Polynomial Functions
3.6 Graphing Polynomial Functions
Notes
Use the zeros, their multiplicities and the end behavior to sketch.
- A zero with odd multiplicity crosses the $x$-axis. A zero with even multiplicity touches the axis and turns around.
- A degree-$n$ polynomial has at most $n - 1$ turning points.
- The $y$-intercept is $f(0)$.
- Example: $f(x) = (x + 1)^2(x - 3)$ touches at $x = -1$, crosses at $x = 3$, and has $y$-intercept $f(0) = -3$.
Practice
How does the graph of $f(x) = (x - 1)^2(x + 4)$ behave at $x = 1$?
The factor $(x - 1)$ has even multiplicity 2.
What is the greatest number of turning points the graph of a degree-5 polynomial can have?
At most $n - 1 = 4$.
Which polynomial has a zero at $-2$ with multiplicity 1, a zero at 3 with multiplicity 2, and a positive leading coefficient?
The zero $-2$ comes from $x + 2$, and the squared factor $(x - 3)^2$ gives multiplicity 2.
Ready for more? Take the Unit 3 test.