Do The Set Of Points Form A Line?

📝 Practice Worksheets

Ordered Pairs and Functions Linear Concepts Linear Concepts Mixed Practice Recognize Functions from Tables

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Functions Explorer

🔑 Key Concepts

Three points lie on the same straight line when the slope between each pair of consecutive points is equal.
Slope measures the rate of change between two points: \(\displaystyle m=\frac{y_2-y_1}{x_2-x_1}\).
It is often helpful to organize the coordinates in an \(x\)- and \(y\)-value table before calculating slope.
Equal slopes indicate a constant rate of change and a linear relationship.
Unequal slopes show that the three points do not lie on one straight line.
If two points have the same \(x\)-coordinate but different \(y\)-coordinates, they form a vertical segment with an undefined slope.
A relation is a function only when each input \(x\) is paired with exactly one output \(y\).
The ideas of ordered pairs, slope, linearity, and functions are closely connected but answer different mathematical questions.

✏️ Worked Example

Problem

In which choice do all three ordered pairs lie on the same straight line?

\((0,1),\ (-1,3),\ (1,3)\)

\((4,2),\ (2,1),\ (4,-2)\)

\((0,0),\ (8,0),\ (0,8)\)

\((1,2),\ (2,4),\ (4,8)\)

Things to Think About

Organize each choice in an \(x\)- and \(y\)-value table.
Compare the change in \(y\) with the change in \(x\).
Calculate two slopes using all three points.
The two slopes must be equal for the points to be collinear.
Watch for repeated \(x\)-values, which create vertical segments.

\(\displaystyle m=\frac{\Delta y}{\Delta x}\)

Choice A

\(x\)	\(y\)
\(-1\)	\(3\)
\(0\)	\(1\)
\(1\)	\(3\)

\(\displaystyle m_1=\frac{1-3}{0-(-1)}=-2\)

\(\displaystyle m_2=\frac{3-1}{1-0}=2\)

The slopes are not equal:

\(-2\ne2\)

Not one straight line

Choice B

\(x\)	\(y\)
\(2\)	\(1\)
\(4\)	\(2\)
\(4\)	\(-2\)

\(\displaystyle m_1=\frac{2-1}{4-2}=\frac12\)

\(\displaystyle m_2=\frac{-2-2}{4-4}=\frac{-4}{0}\)

The second slope is undefined because two points share \(x=4\).

Not one straight line

Choice C

\(x\)	\(y\)
\(0\)	\(0\)
\(8\)	\(0\)
\(0\)	\(8\)

\(\displaystyle m_1=\frac{0-0}{8-0}=0\)

\(\displaystyle m_2=\frac{8-0}{0-8}=-1\)

The slopes are not equal:

\(0\ne-1\)

Not one straight line

Choice D

\(x\)	\(y\)
\(1\)	\(2\)
\(2\)	\(4\)
\(4\)	\(8\)

\(\displaystyle m_1=\frac{4-2}{2-1}=2\)

\(\displaystyle m_2=\frac{8-4}{4-2}=2\)

The slopes are equal:

\(2=2\)

✓ All three points are collinear

Final Answer

Only Choice D has the same slope between each pair of consecutive points.

\(\displaystyle \frac{4-2}{2-1}=\frac{8-4}{4-2}=2\)

\(\boxed{\text{Choice D: }(1,2),\ (2,4),\ (4,8)}\)

The Linear-Relationship Question

To decide whether three points form one straight line, compare their slopes.

\(\displaystyle \frac{y_2-y_1}{x_2-x_1} = \frac{y_3-y_2}{x_3-x_2} \)

Equal slopes mean the points have one constant rate of change and lie on the same line.

The Function Question

To decide whether a relation is a function, check the inputs.

\(\text{one input}\longrightarrow\text{one output}\)

Choice B contains:

\((4,2)\quad\text{and}\quad(4,-2)\)

The same input \(4\) has two different outputs. Therefore, Choice B is not a function.

Important Distinction

A relation can be a function without being linear. These are two separate questions:

Is it linear?

Check whether the rate of change or slope remains constant.

Is it a function?

Check whether every input is paired with exactly one output.

Choice D is both a function and a linear relationship. Choice A is a function, but its three points do not form one straight line.

🧠 Math Vocabulary

Ordered pair: A pair of numbers \((x,y)\) that identifies a point on a coordinate plane.
Input: The \(x\)-value of an ordered pair.
Output: The \(y\)-value associated with an input.
Coordinate plane: A two-dimensional plane formed by a horizontal \(x\)-axis and vertical \(y\)-axis.
Slope: A measure of vertical change compared with horizontal change.
Rate of change: The amount one quantity changes in relation to another quantity.
Constant rate of change: A rate that remains the same between points in a linear relationship.
Linear relationship: A relationship whose graph forms a straight line.
Collinear points: Points that lie on the same straight line.
Relation: A collection of ordered pairs.
Function: A relation in which every input has exactly one output.
Repeated input: An \(x\)-value appearing more than once in a relation.
Undefined slope: The slope of a vertical line, produced when the change in \(x\) is zero.
Difference: The result of subtraction, used to calculate changes in \(x\) and \(y\).
Vertical line test: A graphical test in which a relation is a function only when no vertical line intersects its graph more than once.

💡 Main Idea

To determine whether three points lie on one straight line, organize the ordered pairs and compare the slope between consecutive points. The points are collinear only when the slopes are equal. This reasoning provides a bridge from graphing and slope into the study of linear functions, where students examine both constant rates of change and the relationship between inputs and outputs.

📚 What You Should Already Know

Students should know how to read and write ordered pairs, identify \(x\)- and \(y\)-coordinates, subtract positive and negative integers, calculate slope between two points, and recognize horizontal and vertical changes on a coordinate plane.

🚀 What Comes Next

Students can extend this reasoning by determining whether tables represent linear relationships, identifying functions from ordered pairs and tables, applying the vertical line test, writing linear equations, and comparing slope and initial value across multiple representations.

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