Polynomials and Rational Functions
4.1 Power Functions and Polynomials
Power Function Basics
Power functions are useful benchmark graphs for larger polynomial behavior.
Definition: Power Function
A power function has the form
$$f(x)=kx^p,$$
where $k$ and $p$ are constants.
Fact: End behavior idea
For large $|x|$, a polynomial behaves like its leading term.
Example: Classify polynomial degree and leading coefficient
For
$$P(x)=-3x^4+2x^2-7x+9,$$
find degree and leading coefficient.
Show Solution
The highest power is $4$, so degree is $4$.
Leading coefficient is coefficient of $x^4$, which is $-3$.
MyOpenMath: Practice: degree and leading coefficient (placeholder)
End Behavior from Leading Term
A quick sign check gives the graph direction on both ends.
Example: Predict end behavior
Describe end behavior of
$$Q(x)=2x^5-x^2+1.$$
Show Solution
Leading term is $2x^5$ (odd degree, positive coefficient).
So:
- As $x\to\infty$, $Q(x)\to\infty$.
- As $x\to-\infty$, $Q(x)\to-\infty$.