A Friendly Teacher’s Guide with Real-Life Examples & Step-by-Step Solutions
Target Focus: Math Tutor in Richmond Hill | Grade 10 MPM2D & Grade 11 MCR3U
| 💡 A Friendly Message from Your Math Teacher: Welcome! Math can sometimes feel like a series of abstract rules, but it is actually the language we use to describe patterns in our world. In this guide, we connect functions, graphs, and algebra to simple real-life stories. Let’s make math enjoyable and easy to master together! |
1. What is a Function? (The Vending Machine Example)
Imagine you are standing in front of a snack vending machine:
• Input (x): The button you press (e.g., B4).
• Output (y): The snack that drops into the tray.
When is it a Function?
Pressing button B4 always gives you a granola bar. One button gives you exactly one predictable snack. That is a function!
When is it NOT a Function?
Pressing B4 gives you a granola bar today, but potato chips tomorrow. If one input gives two different outputs, the machine is broken—it is NOT a function.
Can two different buttons give the same snack?
Yes! Buttons A1 and A2 can both give pretzels. That is completely allowed in a function.
Worksheet Solutions: Section 1 (Tables & Sets)
| Worksheet Problem | Is it a Function? | Teacher’s Explanation |
| 1. Set A: {(-3, 4), (-1, 2), (0, 5), (2, 4), (5, 1)} | YES | Each input (-3, -1, 0, 2, 5) is different! The repeated output 4 is completely allowed. |
| 2. Set B: {(-2, 1), (1, 3), (1, -4), (3, 7), (6, 0)} | NO | Input 1 produces two different outputs (3 and -4). One input cannot have multiple outputs. |
| 3. Table of Values: Inputs: -4, -2, 0, 2, 4 Outputs: 16, 4, 0, 4, 16 | YES | Every input maps to one predictable output (following y = x²). |
2. The Vertical Line Test (The Time-Traveler Rule)
To check if a graph represents a function, use the Vertical Line Test (VLT)!
Real-Life Analogy: Think of the horizontal x-axis as Time (2:00 PM) and the vertical y-axis as your Location (School or Home). You cannot be in two different places at the exact same second! If a vertical line touches a graph at more than one point, it means one input has multiple outputs, so it is NOT a function.
Figure 1: The Vertical Line Test. Red dotted lines show where relations fail.
Worksheet Solutions: Section 2 (Vertical Line Test)
| Graph Type | Function? | Teacher’s Explanation |
| 4. Graph 1: Circle | NO | A vertical line down the middle cuts through both the top and bottom. Fails VLT. |
| 5. Graph 2: Straight Line | YES | Any vertical line drawn touches the line only once. Passes VLT. |
| 6. Graph 3: Sideways Parabola | NO | A vertical line crosses both upper and lower branches. Fails VLT. |
| 7. Graph 4: Absolute Value (V-shape) | YES | A vertical line touches the V-shaped graph at only one point anywhere. Passes VLT. |
3. Graphing Functions from Equations
Graphing is just like following a recipe: pick your ingredient (x), calculate the output (y), plot the points on your coordinate grid, and connect the dots!
Problem 8: Linear Function f(x) = 2x – 3
• Table of Values: (-2, -7), (-1, -5), (0, -3), (1, -1), (2, 1)
Problem 9: Quadratic Function g(x) = x² – 4
• Table of Values: (-2, 0), (-1, -3), (0, -4) [Vertex], (1, -3), (2, 0)
Problem 10: Absolute Value Function h(x) = |x + 1| – 2
• Table of Values: (-2, -1), (-1, -2) [Vertex], (0, -1), (1, 0), (2, 1)
4. Function Notation: The Kitchen Blender
Think of f(x) as a blender named ‘f’. The number inside the parentheses is the ingredient you drop in. The equation is the recipe!
Worksheet Solutions: Section 4 (Evaluating Functions)
Given Base Functions: f(x) = 3x – 5 | g(x) = x² + 2x | h(x) = √(x + 9)
| Problem | Step-by-Step Friendly Recipe | Final Answer |
| 11. f(4) | Drop 4 into machine f: 3(4) – 5 = 12 – 5 | 7 |
| 12. g(-3) | Drop -3 into machine g: (-3)² + 2(-3) = 9 – 6 | 3 |
| 13. h(16) | Drop 16 into machine h: √(16 + 9) = √25 | 5 |
| 14. f(-2) + g(3) | f(-2) = 3(-2) – 5 = -11 g(3) = (3)² + 2(3) = 15 Combine: -11 + 15 | 4 |
| 15. Find x if f(x) = 13 | Work backwards from output 13: 3x – 5 = 13 → 3x = 18 | x = 6 |
| 16. Evaluate g(a + 1) | (a + 1)² + 2(a + 1) = a² + 2a + 1 + 2a + 2 = a² + 4a + 3 | a² + 4a + 3 |
5. Need Extra Support? Learn with a Math Tutor in Richmond Hill
| 📝 Download the Worksheet for More Practice! Print out the worksheet, test your skills on each question, and verify your steps against the detailed solutions in this guide. |
High school mathematics across York Region moves quickly. Working with an experienced math tutor in Richmond Hill provides:
• Friendly, patient 1-on-1 explanations with real-world examples.
• Step-by-step homework help aligned with Ontario curriculum (MPM2D, MCR3U, MHF4U).
• Customized practice tests to build genuine exam confidence.
