Functions

Understanding Functions and Linear Relationships

Grade 8

✓By the end of this lesson students will be able to define a function as a rule that assigns to each input exactly one output.
✓By the end of this lesson students will be able to identify whether a relation is a function using tables, graphs, or sets of ordered pairs.
✓By the end of this lesson students will be able to identify linear functions and explain their characteristics.
✓By the end of this lesson students will be able to calculate the rate of change (slope) of a linear function from a table, graph, or two given points.

Key concepts

Function

A function is a special type of relation where each input value (often represented by 'x') has exactly one output value (often represented by 'y'). Think of it like a machine: you put something in, and you get one specific thing out every time. The set of all possible input values is called the domain, and the set of all possible output values is called the range.

Vertical Line Test

The Vertical Line Test is a visual way to determine if a graph represents a function. If any vertical line drawn on the graph intersects the graph at more than one point, then the relation is NOT a function. If every possible vertical line intersects the graph at most one point, then it IS a function.

Linear Function

A linear function is a function whose graph is a straight line. It has a constant rate of change, meaning the output changes by the same amount for each unit change in the input. The equation of a linear function is often written in slope-intercept form: y = mx + b, where 'm' is the slope (rate of change) and 'b' is the y-intercept (the point where the line crosses the y-axis).

y = mx + b

Rate of Change (Slope)

The rate of change, also known as slope, describes how one quantity changes in relation to another. For a linear function, the rate of change is constant and represents the steepness and direction of the line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

m = (y₂ - y₁) / (x₂ - x₁)

Key facts to remember

1A function is a relation where each input has exactly one output.
2The Vertical Line Test is used to determine if a graph represents a function.
3The domain is the set of all possible input values, and the range is the set of all possible output values.
4A linear function has a constant rate of change and its graph is a straight line.
5The rate of change (slope) of a linear function is calculated as 'rise over run' or m = (y₂ - y₁) / (x₂ - x₁).
6The equation y = mx + b represents a linear function, where 'm' is the slope and 'b' is the y-intercept.

Worked examples

Example 1

Determine if the following relation is a function: {(1, 5), (2, 7), (3, 5), (2, 9)}

IRecall the definition of a function: each input value must have exactly one output value.

IIExamine the input values (the first number in each ordered pair): 1, 2, 3, 2.

IIINotice that the input value '2' appears twice, once with an output of '7' and once with an output of '9'.

IVSince the input '2' has two different output values (7 and 9), this relation does not meet the definition of a function.

Answer

No, the relation is not a function.

A function cannot have an input with more than one output.

Example 2

Consider a graph that forms a circle centered at the origin. Is it a function? Is it a linear function?

ITo determine if it is a function, apply the Vertical Line Test. Imagine drawing vertical lines across the graph of a circle.

IIA vertical line drawn through a circle (except at the very left or right edges) will intersect the circle at two points (one above the x-axis and one below).

IIISince a vertical line intersects the graph at more than one point, it fails the Vertical Line Test, so it is not a function.

IVTo determine if it is a linear function, check if the graph is a straight line.

VA circle is a curved shape, not a straight line.

Answer

No, it is not a function. No, it is not a linear function.

All linear functions are functions, but not all functions are linear.

Example 3

Calculate the rate of change (slope) for the linear function that passes through the points (2, 3) and (5, 9).

IIdentify the coordinates of the two points. Let (x₁, y₁) = (2, 3) and (x₂, y₂) = (5, 9).

IIUse the slope formula: m = (y₂ - y₁) / (x₂ - x₁).

IIISubstitute the values into the formula: m = (9 - 3) / (5 - 2).

IVPerform the subtraction in the numerator and denominator: m = 6 / 3.

VSimplify the fraction: m = 2.

Answer

The rate of change (slope) is 2.

The order of the points does not matter as long as you are consistent (e.g., if you start with y₂ in the numerator, you must start with x₂ in the denominator).

Common mistakes

✗Confusing input (x) and output (y) values, especially when determining if a relation is a function.
✗Incorrectly applying the Vertical Line Test, or forgetting to use it for graphs.
✗Assuming that all relations are functions.
✗Calculating slope incorrectly by mixing up the order of subtraction (e.g., (y₂ - y₁) / (x₁ - x₂)).
✗Not recognizing that a constant rate of change is the defining characteristic of a linear function.

Exam tips

★When determining if a relation is a function, always check the definition: 'each input has exactly one output'.
★For graphs, use a ruler or the edge of your paper to perform the Vertical Line Test accurately.
★When calculating slope, clearly label your points (x₁, y₁) and (x₂, y₂) to avoid errors in substitution.
★Practice identifying functions and linear functions from various representations: tables, graphs, equations, and verbal descriptions.

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