Key ideas:
- Polynomial functions of the same degree have similar characteristics
- The degree and the leading coefficient in the equation of a polynomial function indicate the end behaviours of the graph
- The degree of a polynomial function provides information about the shape, turning points, and zeros of the graph.
Positive vs Negative Degrees
- Whether the leading coefficient of a function is positive or negative has an effect of the graph of the function
- if its NEGATIVE, then the function flips
ex: Positive Negative
The maximum amount of turning points can be easily determined by subtracting 1 from the degree of our polynomial.
Example: a polynomial of Degree 4 will have 3 turning points or less
The most is 3, but there can be less.
End behaviours
The end behaviour of a polynomial is a description of what happens as x becomes large in the positive or negative direction. To describe end behaviour, we use the following notation: X --> infinity means : "x becomes large in the positive direction"
X--> negative infinity means : "x becomes large in the negative direction"
An odd degree polynomial function has opposite end behaviours.
- If the leading coefficient is negative, then the function extends from the second quadrant to the fourth quadrant.
- If the leading coefficient is positive, then the function extends from the third quadrant to the first quadrant.
An even degree polynomial has the same end behaviours.
- If the leading coefficient is negative, then the function extends from the third quadrant to the fourth quadrant.
- If the leading coefficient is positive, then the function extends from the second quadrant to the first quadrant.
Click here for more information on end behaviours
Click here to test your knowledge on odd vs even functions
- A polynomial function of degree n may have up to n distinct zeros.
- A polynomial function of odd degree must have at least one zero.
- A polynomial function of even degree must have at least 2 zeros.