# Quadratic Functions Practice, Mathematics 20-1

This worksheet reinforces the analysis of quadratic functions in vertex form and standard form, including vertex, axis of symmetry, direction of opening, intercepts, 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)). It also covers solving quadratic equations, linear-quadratic systems, and quadratic inequalities ([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/17782)).

## Exercises

**1.** For each quadratic function written in vertex form, state the vertex, the axis of symmetry, and the direction of opening ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17777)).

| Function | Vertex | Axis of Symmetry | Direction of Opening |
|---|---|---|---|
| y = 3(x − 2)² + 5 | ______ | ______ | ______ |
| y = −(1/2)(x + 4)² − 1 | ______ | ______ | ______ |
| y = x² + 7 | ______ | ______ | ______ |

**2.** Consider the quadratic function y = x² − 6x + 5.

a) Rewrite the function in vertex form by completing the square.
______________________________________________

b) State the vertex: ______________________

c) Determine the x-intercepts by factoring: ______________________

d) State the y-intercept: ______________________

e) State the domain: ______________________

f) State the range: ______________________

**3.** A ball is thrown upward from a height of 1 m with an initial velocity of 20 m/s. Its height in metres after t seconds is modelled by h(t) = −5t² + 20t + 1. Determine, to the nearest hundredth of a second, when the ball hits the ground, that is, when h(t) = 0 ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17779)).

Show your work:
______________________________________________
______________________________________________
______________________________________________

Answer: t ≈ ______________________ s

**4.** Solve the following system of linear-quadratic equations algebraically. Express your answers exactly, then approximate to the nearest hundredth ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17780)).

y = x² + x − 6
y = 2x − 2

Show your work:
______________________________________________
______________________________________________
______________________________________________

Points of intersection: ______________________ and ______________________

**5.** Solve the quadratic inequality x² − x − 12 ≥ 0. Show your factoring and state the solution using inequality notation ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17782)).

Show your work:
______________________________________________
______________________________________________

Solution: ______________________

**6.** Which point satisfies the inequality y ≥ x² − 4 ([Outcome](https://goa-cc-uat-aili-app-001.azurewebsites.net/explore/node/17781))?

A) (0, 0)
B) (0, −5)
C) (3, 1)
D) (−1, −6)

Circle the correct letter and show your check below.
______________________________________________

## Answer key

1. y = 3(x−2)²+5: vertex (2, 5), axis x = 2, opens up. y = −(1/2)(x+4)²−1: vertex (−4, −1), axis x = −4, opens down. y = x²+7: vertex (0, 7), axis x = 0, opens up.
2. a) y = (x − 3)² − 4; b) vertex (3, −4); c) x = 1, x = 5; d) y-intercept (0, 5); e) domain: all real numbers; f) range: y ≥ −4.
3. Solving −5t² + 20t + 1 = 0 gives t = (20 ± √420)/10; taking the positive root, t ≈ 4.05 s.
4. Setting x² + x − 6 = 2x − 2 gives x² − x − 4 = 0, so x = (1 ± √17)/2. Points: ((1+√17)/2, 2x−2) ≈ (2.56, 3.12) and ((1−√17)/2, 2x−2) ≈ (−1.56, −5.12).
5. Factoring gives (x − 4)(x + 3) ≥ 0, so x ≤ −3 or x ≥ 4.
6. A) (0, 0): 0 ≥ −4 is true; B, C, and D each fail their check. Correct answer: A.

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*AILI homework sheet · language en · model claude-sonnet-5 · generated 2026-09-17 · id 1c7c915e-71f0-47e2-b737-973678c3857a*

### 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)