Grade 9 · Algebra & Functions
Quadratic Functions
Learn to analyze relationships that include squared quantities through clear explanations, worked examples, an interactive model, and guided practice.
Lesson coverage
Learn the whole topic, not just one problem type.
This lesson and its practice bank deliberately rotate across these core facets so students build a connected understanding of the subject.
Connect equations, graphs, and tables of quadratic relationships.
Identify vertex, axis of symmetry, roots, and direction.
Find solutions by an appropriate method.
Interpret standard, factored, or vertex form.
Use quadratics to model changing quantities.
Build the idea
Quadratic Functions is part of the formalize algebra through equations, inequalities, functions, modeling, exponentials, quadratics, and introductory statistics. Start by identifying what each quantity, symbol, diagram, or graph represents before calculating.
Representations
Connect equations, graphs, and tables of quadratic relationships. Use a representation that fits the idea, connect it to the relevant mathematical relationship, and explain how the representation supports the result.
Key features
Identify vertex, axis of symmetry, roots, and direction. Use a representation that fits the idea, connect it to the relevant mathematical relationship, and explain how the representation supports the result.
Solve
Find solutions by an appropriate method. Use a representation that fits the idea, connect it to the relevant mathematical relationship, and explain how the representation supports the result.
Compare forms
Interpret standard, factored, or vertex form. Use a representation that fits the idea, connect it to the relevant mathematical relationship, and explain how the representation supports the result.
Applications
Use quadratics to model changing quantities. Use a representation that fits the idea, connect it to the relevant mathematical relationship, and explain how the representation supports the result.
Use a reliable strategy
Translate the information into symbols, apply one valid transformation at a time, and check the result in the original relationship.
Check and explain
A correct answer is only part of mathematical understanding. Check units, signs, scale, and whether the result makes sense. Then explain the relationship in words so you can recognize the same structure when the numbers or context change.
Key terms
Words to know
- Quadratic Functions
- The central skill or relationship explored in this lesson.
- Quadratic
- A relationship whose highest variable power is two.
- Vertex
- The turning point of a parabola.
- Root
- An input that makes a function equal zero.
Interactive math lab
Change it. See it. Understand it.
Adjust the values and watch the relationship respond.
| Input | 6 |
|---|---|
| Change | 3 |
| Output | 36 |
Worked examples
See the idea in action.
Vertex form y=(x−h)²+k has vertex (h,k).
4²=16.
Guided practice
20 questions drawn from a 150-question lesson bank.
Each practice session mixes the major skills in this lesson so you practice the whole topic, not just one repeated problem type.
What shape is the graph of a quadratic function?
Curriculum reference
Built as a continuous K–12 progression.
The sequence follows widely used K–12 mathematics progressions and references the Common Core mathematics framework. Explanations, examples, interactive models, and practice questions are original FreeLearnHub material.