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Polynomial Functions
- Thread starter willd
- Start date
Hello, willd!
The degree of the polynomial tells us (sort of) how many <u>directions</u> the curve takes.
You might <u>expect</u> the cubic: . f(x) = x<sup>3</sup> - 4x<sup>2</sup> + 1 .to look like this:
. . . . . ___ . . . . . . /
. . . . / . . . \ . . . . ./ . . . . up, down, up ... three directions
. . . / . . . . . \___/
. . /
I say "expect" because not all cubics are shaped like that.
. . Some, like f(x) = x<sup>3</sup>, are "smoother" and lack the "humps".
But certainly a first-degree polynomial (line) has one direction,
and a second-degree polynomial (parabola) has two directions.
Here's a real sloppy rule for "eyeballing" a polynomial function.What general characteristics for the graph of any polynomial
function can be found by looking at its equation?
The degree of the polynomial tells us (sort of) how many <u>directions</u> the curve takes.
You might <u>expect</u> the cubic: . f(x) = x<sup>3</sup> - 4x<sup>2</sup> + 1 .to look like this:
. . . . . ___ . . . . . . /
. . . . / . . . \ . . . . ./ . . . . up, down, up ... three directions
. . . / . . . . . \___/
. . /
I say "expect" because not all cubics are shaped like that.
. . Some, like f(x) = x<sup>3</sup>, are "smoother" and lack the "humps".
But certainly a first-degree polynomial (line) has one direction,
and a second-degree polynomial (parabola) has two directions.
Quote:
What general characteristics for the graph of any polynomial
function can be found by looking at its equation?
Here's a real sloppy rule for "eyeballing" a polynomial function.
The degree of the polynomial tells us (sort of) how many directions the curve takes.
You might expect the cubic: . f(x) = x3 - 4x2 + 1 .to look like this:
. . . . . ___ . . . . . . /
. . . . / . . . \ . . . . ./ . . . . up, down, up ... three directions
. . . / . . . . . \___/
. . /
I say "expect" because not all cubics are shaped like that.
. . Some, like f(x) = x3, are "smoother" and lack the "humps".
But certainly a first-degree polynomial (line) has one direction,
and a second-degree polynomial (parabola) has two directions.
Besides those things that Soroban mentioned (based on the degree of the polynomial function), which are the quickest and easiest ideas to apply, there are a variety of other "inspections" one can make:
If you are familiar with Descartes' theorems, et al, you can determine the possible combinations of real and complex roots (the number of real roots is the number of times the graph touches or crosses the x-axis). You can determine possible values of real roots. You can determine upper and lower bounds on those real roots.
If you determine that a function is odd, even, or neither, this reveals symmetries of the graph (or the lack thereof).
Your question is a bit vague since we don't know exactly what you're covering right now.