Cogito
Integrated Math 3 · Chapter 1 · Lesson 1
Polynomial Behaviour
Degree and leading coefficient decide the ends.
12 problems · about 21 minutes · F-IF.C.7, A-APR.B.3
What this lesson teaches
The student describes the end behaviour and turning points of a polynomial from its equation.
- Even degree sends both ends the same way; odd degree sends them opposite ways.
- A negative leading coefficient flips both ends.
- Degree n gives at most n − 1 turning points and exactly n roots.
Warm Up
Straightforward practice. Get the method working first.
5 problemsA degree-4 polynomial. At most how many turning points?
Answer 3
Why 3.
What decides end behaviour?
Answer The degree and the sign of the leading coefficient.
Why The leading term alone.
f(x) = 3x⁴. How many ends go up?
Answer 2
Why Even degree, positive coefficient.
f(x) = −3x⁴. How many ends go up?
Answer 0
Why The negative flips both.
f(x) = 2x⁵. How many ends go up?
Answer 1
Why Odd degree sends them opposite ways.
Build It Up
The same ideas with more to keep track of.
3 problemsA degree-6 polynomial. At most how many turning points?
Answer 5
Why n − 1.
(x − 3)²(x + 1) = 0. Multiplicity of the root 3?
Answer 2
Why Read the exponent.
At an even multiplicity, does the graph cross the axis? 1 yes, 0 no.
Answer 0
Why It touches and turns.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each polynomial by its end behaviour.
Answer Both ends the same way: Degree 2, Degree 4 · Ends go opposite ways: Degree 3, Degree 5
Why Even against odd degree.
A degree-3 polynomial. How many roots in total?
Answer 3
Why The degree.
The Turns: A degree-5 polynomial. At most how many turning points?
Answer 4
Why 4.
The Roots: A degree-5 polynomial. How many roots, counting repeats and complex ones?
Answer 5
Why 5.