# Quadratic Functions: Mixed Practice

This worksheet covers analyzing quadratic functions in vertex form and standard form, including vertex, intercepts, and domain and range ([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)), along with factoring and solving quadratic equations ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17775), [Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)).

## Exercises

**1. Vertex form: identify the characteristics**

For the function y = 2(x − 3)² − 8, determine each of the following ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)).

Vertex: ______________________

Direction of opening: ______________________

Axis of symmetry: ______________________

Domain: ______________________

Range: ______________________

*y*-intercept: ______________________

**2. Matching: standard form to vertex form**

Match each quadratic function in standard form on the left to its equivalent vertex form on the right by writing the correct letter in the blank ([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/17777)).

| Standard form | Match | Vertex form |
|---|---|---|
| 1. y = x² − 4x + 1 | _____ | A) y = (x − 2)² − 3 |
| 2. y = x² + 6x + 5 | _____ | B) y = (x + 3)² − 4 |
| 3. y = 2x² − 8x + 3 | _____ | C) y = 2(x − 2)² − 5 |

**3. Factoring practice**

Factor each expression fully ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17775)).

a) x² + 7x + 12 = ______________________

b) x² − 5x − 14 = ______________________

c) 2x² + 5x − 3 = ______________________

**4. Solving quadratic equations**

Solve each equation for x, showing your method ([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/17775)).

a) x² − 9x + 20 = 0

______________________________________________

______________________________________________

b) 3x² − 12 = 0

______________________________________________

______________________________________________

**5. Multiple choice: intercepts of a standard form quadratic**

Given the function y = x² − 2x − 15, which of the following gives the *x*-intercepts of the graph ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778))?

A) x = 3 and x = −5

B) x = 5 and x = −3

C) x = 5 and x = 3

D) x = −5 and x = −3

Circle the correct answer.

**6. Applied problem: modelling with a quadratic function**

A ball is thrown into the air. Its height, h, in metres, above the ground after t seconds is modelled by the function h(t) = −5t² + 20t + 1 ([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/17778)).

a) What is the initial height of the ball, at t = 0?

______________________________________________

b) Determine the vertex of this function by completing the square or an equivalent method. State what the vertex represents in this context.

______________________________________________

______________________________________________

______________________________________________

c) After how many seconds does the ball reach its maximum height?

______________________________________________

## Answer key

1. Vertex: (3, −8); Direction of opening: upward; Axis of symmetry: x = 3; Domain: all real numbers (x ∈ ℝ); Range: y ≥ −8; y-intercept: y = 10 (at x = 0, y = 2(−3)² − 8 = 10).

2. 1, A, 2, B, 3, C.

3. a) (x + 3)(x + 4) b) (x − 7)(x + 2) c) (2x − 1)(x + 3).

4. a) x = 4 or x = 5 (factors as (x − 4)(x − 5) = 0). b) x² = 4, so x = 2 or x = −2.

5. B) x = 5 and x = −3 (factors as (x − 5)(x + 3) = 0).

6. a) h(0) = 1 m. b) Vertex is (2, 21); this represents a maximum height of 21 m reached at t = 2 seconds. c) t = 2 seconds.

---
*AILI homework sheet · language en · model claude-sonnet-5 · generated 2026-09-17 · id f6b3abcc-523f-4c96-9350-085e81a678e9*

### Sources

- node:n1: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)
- node:n2: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17776)
- node:n3: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17778)
- node:n4: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)
- node:n5: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)
- node:n6: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17775)
- node:n7: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782)
- node:n8: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780)
- node:n9: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781)
- node:n10: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17776)
- resource:r1: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/L-Uj1j9H4kypxiv587R6aA)
- resource:r2: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/9Im-lhmbb0m3jiTVElTJ2A)
- node:n11: Mathematics › Mathematics (10–12) › Mathematics 20-1 › Relations and Functions › General Outcome (https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17785)
- resource:r3: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/L2B8wqn5oUawLk4_WJDe0Q)
- resource:r4: Resources › type#studentsupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/byh431Gvkk2zgLQtZNOBlA)
- resource:r5: Resources › type#studentsupport, type#teachersupport › MAT2791 (https://goa-cc-uat-aili-app-001.azurewebsites.net/library/resource/Ev9xj275wEyw23tOXVmz4w)