Grade 8 · Grade 8: Linear Equations, Systems and Functions · Linear Relationships and Functions
Functions: Rules, Graphs and Models
Define a function, compare functions across representations and build a linear model from data.
Learning objectives
- Explain why each input of a function has exactly one output.
- Compare functions given as equations, tables, graphs or descriptions.
- Interpret rate of change and initial value in context.
- Construct and sketch a linear model.
AERO Mathematics alignment
AERO.M8.AE.5
Understand that a function is a rule that assigns to each input exactly one output, and that the graph of a function is the set of ordered pairs consisting of an input and its output.
AERO.M8.AE.6
Compare properties of two functions represented algebraically, graphically, numerically in tables or by verbal descriptions, and interpret the rate of change and initial value in context.
AERO.M8.AE.7
Construct a function to model a linear relationship between two quantities, determining the rate of change and initial value from a description, a table or a graph, and sketch graphs of qualitative features.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
A machine takes a number, doubles it and adds three. Another machine sometimes returns 5 and sometimes 9 for the same input.
Think about it
Only one of these is a function. Which one, and why does the difference matter?
Hint: Count the outputs for a single input.
EXPLAIN
Functions and linear models
A function is a rule that assigns exactly one output to each input. Its graph is the set of ordered pairs (input, output). If any vertical line crosses a graph twice, the graph is not a function.
Comparing representations
| Representation | Rate of change | Initial value |
|---|---|---|
| Equation y = 40 − 3x | −3 | 40 |
| Table with x = 0, 1, 2 and y = 40, 37, 34 | −3 per step | 40 |
| Graph | Steepness of the line | Where it crosses the y-axis |
| Words: starts at 40 and falls 3 each day | −3 | 40 |
Linear or not?
A function is linear when the rate of change is constant. In a table, equal steps in x must give equal steps in y. If the differences change, the relationship is not linear.
Sketching qualitatively
Even without numbers you can sketch a graph that increases, then levels off, then decreases — for example a journey with a rest stop.
Common misconception
A function may repeat outputs. y = x² sends both 3 and −3 to 9 and is still a function; the rule is broken only when one input gives two outputs.
INVESTIGATION
Try it yourself
Record a real quantity that changes steadily, such as water draining from a bottle every 10 seconds. Tabulate it, decide whether it is linear, then write the function and use it to predict a value beyond your data.
Watch
What Are Functions?
Math Antics · 10:00
Defines a function precisely, with clear non-examples.
Before you watch: Can one input give two different outputs in a function?
- Is x² + y² = 25 a function of x? Explain.
- Find the rate of change and initial value of y = 40 − 3x.
Interactive simulation · PhET
Function Builder
Chain operations into a machine, then write the equation your machine represents.
While you explore
- Does every input give exactly one output in your machine?
- What is the initial value and the rate of change of your rule?
Key vocabulary
- Initial value
- The starting amount before any change.
- Function
- A rule giving exactly one output for each input.
Practice questions
0/1 correct
Level 1 · Criterion A
Which relation is NOT a function?
MYP criterion tasks
Level 2 · Criterion C
For y = 50 − 4x, state the rate of change and initial value, and interpret both if y is water in litres and x is minutes.
Reflect & track
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