Polynomial Functions and Graphs
End behaviour and turning points depend on degree.
Key idea
End behaviour and turning points depend on degree.
In depth
A degree-n polynomial has at most n real roots and at most n − 1 turning points. End behaviour is set by the leading term, and roots of multiplicity cross or touch the axis depending on whether the multiplicity is odd or even.
Worked example
y = (x − 1)²(x + 2): touches at x = 1, crosses at x = −2.
Formula
Real-world uses
- Modelling roller coaster heights, population curves and smooth paths in computer graphics
Common mistake
Forgetting that a repeated root bounces rather than crosses.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Polynomial Functions and Graphs?
End behaviour and turning points depend on degree. It is a topic at the advanced level in Algebra.
What is the formula for Polynomial Functions and Graphs?
The key formula is a*x^3-b*x.
Where is Polynomial Functions and Graphs used in real life?
Real-world examples: Modelling roller coaster heights, population curves and smooth paths in computer graphics.
What should I learn before Polynomial Functions and Graphs?
Make sure you are comfortable with Even and Odd Functions, the previous topic in Algebra.
What is a common mistake with Polynomial Functions and Graphs?
Forgetting that a repeated root bounces rather than crosses.
How can I practise Polynomial Functions and Graphs?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Polynomial Functions and Graphs?
Move on to Polynomial Long Division next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.