Graphs of relations and functions

Overview

A graph of a relation or function shows input-output pairs as points on coordinate axes. The horizontal axis usually represents the input $x$, and the vertical axis usually represents the output $y$ or $f(x)$.

Graphs help learners see patterns that may be hidden in a formula or table. They also make it easier to read domain, range, intercepts, trends, and whether a relation behaves like a function.

A useful way to think about a graph is this: every point on the drawing is a statement. The point $(2,5)$ says, "when the input is $2$, the output is $5$." If a line or curve contains many points, it is making many input-output statements at once.

This chapter moves slowly from separate points, to tables, to full graphs. That order matters because most graph mistakes happen when a learner draws a line before understanding what the plotted points are saying.

Prerequisites

• Coordinate geometry: gradient and straight-line equations - Graphs are built from ordered pairs, axes, gradients, and intercepts.
• Relations and functions - A graph represents the input-output rule of a relation or function.
• Domain and range - Domain and range can be read from horizontal and vertical coverage.
• Algebraic expressions and equations - Straight-line graphs often come from simple algebraic rules.

Learning Scope

This chapter covers plotting relations and functions from ordered pairs, making a table of values, drawing simple line and curve graphs, reading values from graphs, identifying whether a graph represents a function, and connecting graph features to domain and range.

This page does not create finished media assets. It uses text-based planning notes for future graphs and learner aids only.

Subtopics

Ordered Pairs On Axes

Each ordered pair $(x,y)$ is plotted by moving first along the horizontal axis to $x$, then vertically to $y$.

For example, the point $(3,5)$ means:

• move $3$ units along the $x$-axis;
• move $5$ units up to the $y$-value.

Key insight: The order matters. The point $(3,5)$ is usually different from $(5,3)$.

Why this works: the axes give two directions of information. The $x$-coordinate answers "how far left or right?" The $y$-coordinate answers "how far up or down?" A point is fixed only after both answers are known.

Use this plotting routine:

1. Start at the origin $(0,0)$.
2. Move horizontally to the given $x$-value.
3. From there, move vertically to the given $y$-value.
4. Mark the point clearly and label it if several points are close together.
5. Check the quadrant by the signs of $x$ and $y$.

| Signs of coordinates | Position of point | | --- | --- | | $x>0,\ y>0$ | First quadrant | | $x<0,\ y>0$ | Second quadrant | | $x<0,\ y<0$ | Third quadrant | | $x>0,\ y<0$ | Fourth quadrant | | $x=0$ | On the $y$-axis | | $y=0$ | On the $x$-axis |

Graphing From A Table

To graph a function such as:

$$ y=2x+1 $$

choose input values, calculate outputs, and plot the ordered pairs.

| $x$ | $y=2x+1$ | Point | |---:|---:|---:| | $-1$ | $-1$ | $(-1,-1)$ | | $0$ | $1$ | $(0,1)$ | | $1$ | $3$ | $(1,3)$ | | $2$ | $5$ | $(2,5)$ |

Since the rule is linear, the points lie on a straight line.

The table is a bridge between algebra and the graph. The formula gives the rule, the table gives selected ordered pairs, and the graph shows the pattern made by those pairs.

Good table habits:

1. Choose simple $x$-values unless the question gives them.
2. Substitute each $x$-value carefully into the rule.
3. Write the ordered pair beside the calculation.
4. Plot the points using a consistent scale.
5. Check whether the plotted points match the expected shape.

For a linear rule such as $y=mx+c$, the graph is a straight line. For a non-linear rule such as $y=x^2$, the points should not be forced into a straight line.

Graphs Of Relations

A relation graph may contain points, a curve, or several disconnected parts. It does not have to pass the function test.

For example, a circle is a relation because it connects $x$-values and $y$-values, but many vertical lines through the circle meet it twice. That means the whole circle is not a function of $x$.

A relation only says that some inputs are connected to some outputs. It may allow one input to be connected to two or more outputs. A function is a more restricted kind of relation.

For a finite relation, repeated $x$-values are the warning sign. In:

$$ \{(2,1),(2,4),(5,7)\} $$

the input $2$ is connected to both $1$ and $4$. This relation can be graphed, but it is not a function of $x$.

Graphs Of Functions

A graph represents a function of $x$ if every allowed $x$-value has exactly one $y$-value.

One way to check is the vertical-line test: imagine a vertical line moving across the graph. If it ever cuts the graph in more than one point, the graph is not a function.

Key insight: The vertical-line test is a visual version of the rule "one input gives exactly one output."

Why a vertical line is used: a vertical line has one fixed $x$-value. If that vertical line meets the graph twice, the same input has produced two different outputs. That breaks the definition of a function.

Checking routine:

1. Choose several $x$-positions across the graph.
2. Imagine or draw a vertical line at each chosen $x$.
3. Count how many times the vertical line meets the graph.
4. If every vertical line meets the graph at most once, the graph passes the test.
5. If one vertical line meets the graph more than once, the graph fails the test.

The test must be applied to the whole graph, not only the neat part near the origin.

Reading Domain And Range From A Graph

The domain is the set of $x$-values covered by the graph. The range is the set of $y$-values covered by the graph.

For a finite plotted relation:

$$ \{(-2,1),(0,3),(2,5)\} $$

the domain is:

$$ \{-2,0,2\} $$

and the range is:

$$ \{1,3,5\} $$

For a continuous graph, domain and range are usually described using inequalities.

Think of domain as the graph's shadow on the $x$-axis. Think of range as the graph's shadow on the $y$-axis. This image helps because domain is about horizontal coverage and range is about vertical coverage.

For a line segment from $(-1,2)$ to $(3,6)$, the graph covers all $x$-values from $-1$ to $3$ and all $y$-values from $2$ to $6$. So:

$$ \text{domain: } -1 \le x \le 3 $$

and:

$$ \text{range: } 2 \le y \le 6 $$

Closed endpoints are included. If an endpoint is open on a drawn graph, that endpoint value is not included.

Intercepts

The $y$-intercept is where a graph crosses the $y$-axis. It happens when $x=0$.

The $x$-intercept is where a graph crosses the $x$-axis. It happens when $y=0$.

For:

$$ y=2x+4 $$

the $y$-intercept is:

$$ y=2(0)+4=4 $$

The $x$-intercept is found by setting $y=0$:

$$ \begin{aligned} 0 &= 2x+4 \\ 2x &= -4 \\ x &= -2 \end{aligned} $$

So the graph crosses the axes at $(0,4)$ and $(-2,0)$.

Why this works: every point on the $y$-axis has $x=0$, because it has not moved left or right from the origin. Every point on the $x$-axis has $y=0$, because it has not moved up or down from the origin.

Intercept checking routine:

1. For the $y$-intercept, substitute $x=0$.
2. Write the answer as a point $(0,y)$.
3. For the $x$-intercept, substitute $y=0$.
4. Solve for $x$.
5. Write the answer as a point $(x,0)$.
6. Check the graph sketch: the intercepts should lie on the correct axes.

Inverse Graphs

Graphs of inverse relations are reflections of each other in the line:

$$ y=x $$

This is because every point $(a,b)$ on the original relation becomes $(b,a)$ on the inverse relation.

The inverse relation reverses the input-output order. If the original graph says $2$ gives $7$, the inverse relation says $7$ gives $2$. On the coordinate plane, this swap changes $(2,7)$ into $(7,2)$.

The line $y=x$ is the mirror line because points on that line already have equal coordinates, such as $(3,3)$ and $(-2,-2)$. Swapping their coordinates does not move them.

Key Terms

• Coordinate axes: The horizontal and vertical reference lines used for plotting points.
• Ordered pair: A point written as $(x,y)$.
• Graph: A visual representation of a relation or function.
• Vertical-line test: A graph test for deciding whether a relation is a function of $x$.
• Intercept: A point where a graph crosses an axis.
• Domain on a graph: The horizontal set of input values covered by the graph.
• Range on a graph: The vertical set of output values covered by the graph.

Worked Examples

Example 1: Plot A Function From A Table

Draw the graph of:

$$ y=x+2 $$

for $x=-2,-1,0,1,2$.

Make a table:

| $x$ | $y=x+2$ | Point | |---:|---:|---:| | $-2$ | $0$ | $(-2,0)$ | | $-1$ | $1$ | $(-1,1)$ | | $0$ | $2$ | $(0,2)$ | | $1$ | $3$ | $(1,3)$ | | $2$ | $4$ | $(2,4)$ |

Plot the points and join them with a straight line because the rule is linear.

Check:

• The $y$-values increase by $1$ each time $x$ increases by $1$.
• The point $(0,2)$ is on the $y$-axis, so the $y$-intercept should be $2$.
• The points should form a straight line, not a bent path.

Example 2: Decide Whether A Graph Could Be A Function

Suppose a relation contains the points:

$$ (1,2),\ (1,5),\ (3,7) $$

The input $1$ appears with two different outputs, $2$ and $5$.

A vertical line at $x=1$ would pass through two plotted points. Therefore this relation is not a function of $x$.

Check:

• The repeated input is the warning sign.
• A repeated output is allowed. For example, $(1,4)$ and $(2,4)$ can still be part of a function because the inputs are different.
• The problem is not that $2$ and $5$ are different numbers; the problem is that the same input $1$ gives both of them.

Example 3: Find Intercepts

Find the intercepts of:

$$ y=3x-6 $$

For the $y$-intercept, let $x=0$:

$$ y=3(0)-6=-6 $$

So the $y$-intercept is $(0,-6)$.

For the $x$-intercept, let $y=0$:

$$ \begin{aligned} 0 &= 3x-6 \\ 3x &= 6 \\ x &= 2 \end{aligned} $$

So the $x$-intercept is $(2,0)$.

Check:

• $(0,-6)$ lies on the $y$-axis because its $x$-coordinate is $0$.
• $(2,0)$ lies on the $x$-axis because its $y$-coordinate is $0$.
• Substituting $x=2$ gives $y=3(2)-6=0$, so the $x$-intercept is consistent.

Example 4: Read Domain And Range From A Finite Relation

Find the domain and range of:

$$ \{(-3,2),(-1,2),(0,5),(4,-1)\} $$

Step 1: List the input values:

$$ -3,\ -1,\ 0,\ 4 $$

So the domain is:

$$ \{-3,-1,0,4\} $$

Step 2: List the output values:

$$ 2,\ 2,\ 5,\ -1 $$

Do not repeat $2$ in the final set. So the range is:

$$ \{-1,2,5\} $$

Check:

• Domain comes from the first coordinate of each ordered pair.
• Range comes from the second coordinate of each ordered pair.
• Sets do not need repeated entries.

Example 5: Draw And Interpret A Simple Curve

Make a table for:

$$ y=x^2 $$

using $x=-2,-1,0,1,2$.

| $x$ | $y=x^2$ | Point | |---:|---:|---:| | $-2$ | $4$ | $(-2,4)$ | | $-1$ | $1$ | $(-1,1)$ | | $0$ | $0$ | $(0,0)$ | | $1$ | $1$ | $(1,1)$ | | $2$ | $4$ | $(2,4)$ |

Plot the points. The graph is curved, not straight, because the $y$-values do not change by a constant amount as $x$ increases by $1$.

Check:

• The points for $x=-2$ and $x=2$ have the same $y$-value.
• The lowest listed point is $(0,0)$.
• Joining the points with a smooth curve is reasonable only because the rule $y=x^2$ is continuous.

Example 6: Use An Inverse Point

A function graph contains the point:

$$ (3,8) $$

State the corresponding point on the inverse relation.

The inverse relation swaps the coordinates:

$$ (3,8)\rightarrow(8,3) $$

So the inverse relation contains $(8,3)$.

Check:

• The original statement is "input $3$ gives output $8$."
• The inverse statement is "input $8$ gives output $3$."
• The point is reflected across the line $y=x$.

Common Mistakes

• Reversing coordinates when plotting. Correction: plot $x$ first, then $y$. Warning sign: the point appears in the wrong quadrant, such as plotting $(-2,5)$ to the right of the $y$-axis.
• Starting the vertical movement from the origin again. Correction: after moving to the $x$-value, move vertically from that position. Warning sign: many points line up incorrectly on an axis.
• Using unequal scales without noticing. Correction: mark the scale clearly before plotting. Warning sign: one square sometimes means $1$ unit and sometimes means $2$ units on the same axis.
• Joining points when the relation is meant to be a finite set only. Correction: only join points when the rule or context is continuous. Warning sign: the question gives only a set such as $\{(1,2),(3,4)\}$ without a formula or continuous context.
• Forcing a curve to become a straight line. Correction: use the shape suggested by the rule and plotted points. Warning sign: the changes in $y$ are not constant, but the graph has been drawn as a line.
• Calling a graph a function without checking vertical lines. Correction: test whether each $x$ gives one $y$. Warning sign: a circle, sideways parabola, or doubled curve is being described as a function of $x$.
• Reading range from the horizontal axis. Correction: domain is horizontal; range is vertical. Warning sign: the answer for range uses $x$-values instead of $y$-values.
• Writing intercepts as numbers only. Correction: write intercepts as coordinate points when the question asks for where the graph crosses the axes. Warning sign: an answer says "$4$" instead of "$(0,4)$" for a $y$-intercept.
• Including repeated values in a set. Correction: write each domain or range value once. Warning sign: the range is written as $\{2,2,5\}$.
• Ignoring open and closed endpoints on a drawn graph. Correction: include closed endpoints and exclude open endpoints. Warning sign: the inequality uses $\le$ where the graph shows an open circle.

Practice Tasks

Foundation

1. State the $x$-coordinate and $y$-coordinate of $(5,-2)$.
2. Plot the points $(-2,3)$, $(0,1)$, $(2,-1)$, and $(4,-3)$ on coordinate axes.
3. State the quadrant of each point: $(3,4)$, $(-3,4)$, $(-3,-4)$, and $(3,-4)$.
4. Explain in one sentence why $(2,6)$ and $(6,2)$ are different points.

Skill-Building

1. Make a table of values for $y=2x-3$ using $x=-1,0,1,2,3$.
2. Draw the graph of $y=-x+4$ using at least five ordered pairs.
3. Find the domain and range of $\{(-1,4),(0,2),(3,2)\}$.
4. State whether $\{(0,2),(1,3),(1,4),(2,5)\}$ would pass the vertical-line test, and give the reason.
5. Find the $x$- and $y$-intercepts of $y=x-5$.

Exam-Style

1. Draw the graph of $y=2x+1$ for $-2 \le x \le 3$ using a table of values.
2. A relation has points $(-2,1)$, $(0,3)$, $(2,5)$, and $(2,7)$. Determine whether it is a function and justify your answer.
3. A line cuts the $y$-axis at $(0,6)$ and the $x$-axis at $(3,0)$. State the intercepts and sketch the line.
4. The graph of a relation covers $-1 \le x \le 4$ and $2 \le y \le 7$. State the domain and range.

Challenge

1. Explain why a circle is a relation but not a function of $x$.
2. If a graph contains $(2,7)$, what point appears on the inverse relation graph?
3. A graph passes the vertical-line test but fails a horizontal-line test. Explain what this means about the function and its inverse relation.
4. Create a table for $y=x^2-1$ using $x=-2,-1,0,1,2$, then describe the graph shape before drawing it.

Generated Question Layer

• Plotting questions: place ordered pairs and read coordinates from axes.
• Table-to-graph questions: calculate values and identify the graph shape.
• Function-test questions: decide whether a graph or point set represents a function.
• Intercept questions: find where a graph crosses the axes.
• Domain-range graph questions: read horizontal and vertical coverage.
• Inverse-graph questions: swap coordinates and connect to reflection in $y=x$.

Learner Aid Opportunities

• graph: Create one coordinate-plane asset showing labelled points in all four quadrants, with equal scale markers and no extra decoration.
• graph: Create paired assets comparing a finite plotted relation with a continuous line graph so learners can see when points should or should not be joined.
• interactive: Build a vertical-line-test slider that reports "one output" or "more than one output" as the learner moves across sample graphs.
• animation: Show a table row becoming an ordered pair, then becoming a plotted point, then joining into a graph only after all points are placed.
• LLM tutor: Prompt learners to explain the meaning of a selected point, identify the domain and range direction, and check intercepts using $x=0$ or $y=0$.

Exam-Derived Signals

These signals are assessment leads, not verified official past-question links. They should be checked against original papers and marking schemes before being used as final learner-facing references.

| Source | Current Signal | Review Status | Use Carefully As | | --- | --- | --- | --- | | data/exam_format_topic_crosswalk_2022.jsonl | Official 2022 format group Linear Programming/Functions/Relations maps to this topic and sibling pages; one item; weight 7.14. | Official format mapping; topic-page use still unreviewed. | Evidence that graphs of functions are adjacent to the functions/relations assessment area. | | data/topic_frequency_2021_2025.json | topic-graphs-of-relations-and-functions is listed under low-or-no coverage topics in the mapped extraction summary. | Unreviewed aggregate. | A signal that direct extracted coverage may be sparse. | | data/question_map_2021_2025.jsonl | No reviewed 2021-2025 question-map item is promoted here yet. | No reviewed links. | Future review should check whether function, domain-range, or inverse-function items also required graph interpretation. |

Source And Review Notes

• Topic registry status: official in data/curriculum_map.json.
• Learner expansion status: original prose drafted from the official syllabus topic and local assessment signals.
• Exam mapping status: unreviewed except for the official exam-format crosswalk group.
• Review risk: because this page is graph-heavy, future reviewers should verify any exam-derived link against the original diagram or graph, not only extracted text.