# Quadratic Functions
*Mathematics 20-1: Relations and Functions*

## Slide 1: Quadratic Functions
![Quadratic Functions](https://goa-cc-uat-aili-app-001.azurewebsites.net/api/generate/7e039c17-a7b7-49c3-8ae4-bca490d1387f/asset/2062)
*Mathematics 20-1: Relations and Functions*
*Sources: [Topic](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17773), [Topic](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17774)*
> Notes: Welcome students to the unit on quadratic functions. Frame the lesson around Mathematics 20-1, Relations and Functions strand. Preview that students will analyze both vertex form and standard form, solve quadratic equations, systems, and inequalities. Mention this builds on their prior work with linear functions.

## Slide 2: Learning Intentions and Success Criteria
- I can analyze a quadratic function in vertex form y=a(x-p)²+q to state vertex, domain, range, direction of opening, axis of symmetry, and intercepts ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)).
- I can analyze a quadratic function in standard form y=ax²+bx+c to identify the same graph characteristics and use them to solve problems ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778)).
- I can solve quadratic equations using appropriate algebraic and graphical strategies ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)).
- I can solve linear-quadratic and quadratic-quadratic systems in two variables ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780)).
- I can solve linear and quadratic inequalities in two variables ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781)).
- I can solve quadratic inequalities in one variable ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782)).
*Sources: [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782), [Topic](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17774)*
> Notes: Read success criteria aloud. Have students self-rate 1 to 4 on each criterion as a pre-assessment; revisit at the end of the unit. Emphasize that this unit sits within the general outcome of developing algebraic and graphical reasoning through the study of relations ([Topic](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17774)).

## Slide 3: Two Forms, One Parabola
**Vertex Form: y = a(x − p)² + q**
Vertex is at (p, q). Direction of opening depends on the sign of a: opens upward if a is greater than 0, downward if a is less than 0. Axis of symmetry is x = p. Domain is all real numbers; range depends on q and the direction of opening. Find x- and y-intercepts by substitution ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)).
**Standard Form: y = ax² + bx + c**
The same characteristics, vertex, domain and range, direction of opening, axis of symmetry, and intercepts, can be identified from standard form, often by completing the square or using x = −b/(2a) to locate the axis of symmetry, then using these features to solve applied problems ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778)).
*Sources: [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778)*
> Notes: Model converting one specific function from standard form to vertex form by completing the square. Example: y = 2x² − 8x + 3. Show students how to identify vertex (2, −5), axis of symmetry x = 2, opens upward, and find intercepts algebraically. Checkpoint: cold-call students to identify direction of opening for three quick examples.

## Slide 4: Solving a Quadratic Equation: A General Approach
1. **Set equation to zero**: Write the equation in the form ax² + bx + c = 0 ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)).
2. **Choose a strategy**: Factoring, completing the square, or the quadratic formula, depending on the structure of the equation ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)).
3. **Solve for x**: Apply the chosen method to find the roots, which may be real or non-real ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)).
4. **Verify and interpret**: Check solutions in the original equation and interpret them in context of the problem ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)).
*Sources: [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)*
> Notes: Work through one full example live: 2x² + 5x − 3 = 0. Ask students to try factoring first, then confirm with the quadratic formula. Support: provide a formula reference card. Extension: ask students when the discriminant tells us the equation has no real solutions, connecting to direction of opening and the x-axis.

## Slide 5: Beyond a Single Equation: Systems and Inequalities
- **Linear-Quadratic Systems**: Solve algebraically and graphically to find intersection points of a line and a parabola ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780)).
- **Quadratic-Quadratic Systems**: Solve algebraically and graphically to find intersection points of two parabolas ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780)).
- **Two-Variable Inequalities**: Solve linear and quadratic inequalities in two variables, graphing regions on a coordinate plane ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781)).
- **One-Variable Quadratic Inequalities**: Solve quadratic inequalities in one variable, expressing solutions as intervals ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782)).
*Sources: [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782)*
> Notes: Use graphing technology (ICT C6-4.1, C6-4.4) to let students visualize intersection points before solving algebraically. For systems of equations, stress testing points in each region and using algebraic and graphical methods. Formative checkpoint: exit ticket with one linear-quadratic system and one quadratic-quadratic system problem. Support: provide graphed systems already plotted; extension: ask students to construct their own system with exactly one solution.

## Slide 6: Bringing It Together
**Summary:** This unit developed algebraic and graphical reasoning about quadratic functions in both vertex and standard form, including vertex, domain, range, direction of opening, axis of symmetry, and intercepts ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778)). Students solved quadratic equations ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)), linear-quadratic and quadratic-quadratic systems ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780)), and both two-variable and one-variable quadratic inequalities ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782)).
**Question:** Given a real-world scenario modeled by a quadratic function, such as the height of a projectile over time, what graph characteristics would you check first to answer a question about maximum height or time of impact, and why?
*Sources: [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782)*
> Notes: Use the closing question as a discussion or written reflection. Strong answers should reference vertex for maximum value questions and x-intercepts for time-of-impact questions. This can serve as a summative formative check before a unit quiz.

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*AILI presentation · language en · model claude-sonnet-5 · generated 2026-09-17 · id 7e039c17-a7b7-49c3-8ae4-bca490d1387f*

### Sources

- node:n1: Relations and Functions (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17773)
- node:n2: Develop algebraic and graphical reasoning through the study of relations. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17774)
- node:n3: *It is expected that students will:* Analyze quadratic functions of the form y=a(x-p)²+q and determine the: - vertex - d (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)
- node:n4: *It is expected that students will:* Analyze quadratic functions of the form $y=ax^{2}+bx+c$ to identify characteristics (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778)
- node:n5: *It is expected that students will:* Solve problems that involve quadratic equations. [C, CN, PS, R, T, V] [ICT: C6–4.1] (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)
- node:n6: *It is expected that students will:* Solve, algebraically and graphically, problems that involve systems of linear-quadr (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780)
- node:n7: *It is expected that students will:* Solve problems that involve linear and quadratic inequalities in two variables. [C, (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781)
- node:n8: *It is expected that students will:* Solve problems that involve quadratic inequalities in one variable. [CN, PS, V] (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782)
- node:n9: Relations and Functions (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17838)
- node:n10: Develop algebraic and graphical reasoning through the study of relations. (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17839)
- node:n11: *It is expected that students will:* Demonstrate an understanding of the characteristics of quadratic functions, includi (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17840)
- node:n12: *It is expected that students will:* Solve problems that involve quadratic equations. [C, CN, PS, R, T, V] [ICT: C6–4.1, (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17841)