The Central Idea
A graph represents mathematical objects by position in a coordinate system.
For an equation in two variables, the graph is the set of all ordered pairs that make the equation true.
If the equation is
\[ y=2x+1, \]then a point \((x,y)\) belongs to the graph exactly when
\[ y=2x+1 \]is true for those coordinates.
This gives the fundamental interpretation:
\[ \boxed{\text{graph}=\text{geometric representation of a solution set}}. \]A Point Can Be a Solution
Consider the equation
\[ y=2x+1. \]Test the point
\[ (2,5). \]Substitute \(x=2\) and \(y=5\):
\[ 5=2(2)+1. \]Since
\[ 5=5, \]the statement is true. Therefore,
\[ \boxed{(2,5)\text{ lies on the graph of }y=2x+1}. \]A Point Can Fail to Be a Solution
Test
\[ (2,4) \]in the same equation:
\[ y=2x+1. \]Substitution gives
\[ 4=2(2)+1, \]or
\[ 4=5. \]This is false, so
\[ \boxed{(2,4)\text{ does not lie on the graph}}. \]Graphs Collect Many Solutions at Once
A single ordered pair may solve an equation, but a graph usually displays many solutions simultaneously.
For
\[ y=2x+1, \]some solutions are
\[ (-2,-3),\qquad(-1,-1),\qquad(0,1),\qquad(1,3),\qquad(2,5). \]These points are not isolated accidents. They belong to one continuous line containing infinitely many real solutions.
Building a Graph from a Table
One way to begin graphing an equation is to choose convenient values of one variable and calculate corresponding values of the other.
For
\[ y=2x+1, \]| \(x\) | \(y=2x+1\) | Point |
|---|---|---|
| \(-2\) | \(-3\) | \((-2,-3)\) |
| \(-1\) | \(-1\) | \((-1,-1)\) |
| \(0\) | \(1\) | \((0,1)\) |
| \(1\) | \(3\) | \((1,3)\) |
| \(2\) | \(5\) | \((2,5)\) |
Plotting these points reveals the linear pattern.
A Table Samples the Graph
A table usually lists only selected solutions.
The graph of
\[ y=2x+1 \]contains solutions such as
\[ \left(\frac12,2\right) \]even if that point was not listed in the table.
The table is therefore a sample of the solution set, while the complete graph represents every allowed solution.
Equations Define Geometric Sets
Different equations can create very different geometric objects.
For example,
\[ y=3 \]produces a horizontal line, while
\[ x=-2 \]produces a vertical line.
The equation
\[ y=x^2 \]produces a parabola, and
\[ x^2+y^2=25 \]produces a circle of radius \(5\) centered at the origin.
The graph is determined by the condition the coordinates must satisfy.
Graphs of Equations Need Not Be Functions
Every equation in two variables can be treated as a relation between coordinates, but not every such relation defines \(y\) as a function of \(x\).
The circle
\[ x^2+y^2=25 \]contains both
\[ (0,5) \]and
\[ (0,-5). \]The same input \(x=0\) is paired with two different \(y\)-values. Therefore, the full circle is not the graph of a function \(y=f(x)\).
What Is a Relation?
A relation is a collection of ordered pairs or, more generally, a rule or condition that associates objects with one another.
In the coordinate plane, a relation may be represented by a set of points.
For example,
\[ R=\{(-2,1),(0,3),(2,1),(2,4)\} \]is a relation.
Because \(x=2\) is paired with both \(1\) and \(4\), this relation is not a function of \(x\).
What Is a Function?
A function is a relation in which each allowed input is associated with exactly one output.
In function notation,
\[ y=f(x), \]the input \(x\) determines one output \(f(x)\).
The graph of a function is the set
\[ \boxed{\{(x,f(x)):x\text{ is in the domain of }f\}}. \]A Function Is a Special Kind of Relation
Every function is a relation, but not every relation is a function.
The relation
\[ \{(1,4),(2,4),(3,5)\} \]is a function because each input appears with only one output.
Notice that two different inputs may share the same output. The repeated output \(4\) does not violate the function rule.
Repeated Inputs Are the Issue
Consider
\[ \{(1,4),(1,6),(3,5)\}. \]The input \(1\) has two outputs:
\[ 4\qquad\text{and}\qquad6. \]Therefore, this relation is not a function of its first coordinate.
A function may repeat outputs, but one input cannot be assigned two different outputs.
The Vertical Line Test
On a graph in which the horizontal coordinate is the input and the vertical coordinate is the output, a relation represents a function exactly when no vertical line intersects the graph more than once.
This is called the vertical line test.
Why does it work? A vertical line has the form
\[ x=c. \]If that line meets the graph at two different heights, then the single input \(x=c\) has two different outputs.
Example: A Parabola Passes the Vertical Line Test
Consider
\[ y=x^2. \]For each real value of \(x\), squaring produces exactly one value of \(y\).
Therefore, every vertical line intersects the parabola at most once, and
\[ \boxed{y=x^2\text{ is a function of }x}. \]Example: A Circle Fails the Vertical Line Test
Consider
\[ x^2+y^2=9. \]At \(x=0\), the equation becomes
\[ y^2=9, \]so
\[ y=3\qquad\text{or}\qquad y=-3. \]A vertical line through \(x=0\) therefore meets the circle twice. The full circle is a relation, but it is not a function \(y=f(x)\).
Explicit Function Form
A relation is in explicit function form when the dependent variable has already been isolated.
For example,
\[ y=3x-2 \]explicitly tells us how to calculate \(y\) from \(x\).
The same relationship may also be written
\[ f(x)=3x-2. \]Implicit Form
A relation is written implicitly when its variables are linked by an equation without necessarily isolating one variable.
For example,
\[ x^2+y^2=16 \]describes a circle directly.
An implicit equation may define a function, several function branches, or a relation that cannot be represented globally as one function of \(x\).
Equivalent Equations Can Have the Same Graph
The equations
\[ y=2x+4 \]and
\[ 2x-y=-4 \]describe the same set of ordered pairs.
Rearranging the second equation gives
\[ y=2x+4. \]Because the solution sets are identical, the graphs are identical.
Changing Form Does Not Necessarily Change the Graph
Algebra often changes the appearance of an equation without changing its solution set.
For example,
\[ 2y=6x-8 \]can be divided by \(2\) to obtain
\[ y=3x-4. \]Every point satisfying one equation satisfies the other, so both equations produce the same graph.
Be Careful with Non-Reversible Algebra
Some algebraic operations can introduce or remove candidate solutions if domain restrictions are ignored.
Therefore, two expressions that look algebraically related do not automatically have identical graphs over every input.
Graphical comparison can help reveal missing points, extra points, or domain restrictions, but symbolic reasoning is still needed to explain why the difference occurs.
Domain Restrictions Affect the Graph
The formula
\[ y=\frac{1}{x} \]is not defined at
\[ x=0. \]Therefore, no point on the graph can have first coordinate \(0\).
A graph represents not only a formula but also the values for which that formula or relation is actually defined.
A Graph Can Be Discrete
Not every graph is a continuous curve.
Suppose a sequence begins
\[ 3,\ 5,\ 7,\ 9,\ldots \]and is indexed by positive integers. Its graph may contain points such as
\[ (1,3),\qquad(2,5),\qquad(3,7),\qquad(4,9). \]If only integer indices are allowed, the points between these coordinates do not belong to the relation.
Do Not Connect Points Without a Mathematical Reason
Connecting plotted points visually asserts the presence of intermediate points.
For a continuous relationship such as
\[ y=2x+1, \]drawing the line between sampled points is appropriate because every intermediate real input is allowed.
For a sequence defined only at integer indices, connecting the points may falsely suggest solutions that are not part of the relation.
Equations Represent Equality
An equation specifies where two quantities are equal.
The equation
\[ y=x+2 \]asks for points at which the vertical coordinate is exactly two more than the horizontal coordinate.
Points either satisfy this equality or they do not.
Inequalities Represent Ordered Conditions
An inequality replaces exact equality with an ordering condition.
Consider
\[ y>x+2. \]We now seek every point whose \(y\)-coordinate is greater than \(x+2\).
This usually produces an entire region rather than a single curve.
The Boundary of an Inequality
To graph
\[ y>x+2, \]first identify the related equality
\[ y=x+2. \]This equality forms the boundary separating points that satisfy the inequality from points that do not.
Strict Inequalities Exclude the Boundary
In
\[ y>x+2, \]points on the line
\[ y=x+2 \]make the two sides equal rather than strictly ordered.
Therefore, the boundary is not included. A dashed boundary is commonly used to communicate this exclusion.
Inclusive Inequalities Include the Boundary
Consider
\[ y\ge x+2. \]Points satisfying
\[ y=x+2 \]are allowed because equality is included in the symbol \(\ge\).
A solid boundary is commonly used to show that the boundary belongs to the solution set.
Which Side of the Boundary?
A boundary divides the plane into regions. We must determine which region satisfies the inequality.
For
\[ y>x+2, \]the solutions are the points whose vertical coordinates are greater than the boundary value \(x+2\).
Geometrically, this is the region above the line.
Testing a Point in an Inequality
When the correct region is not obvious, choose a convenient test point that is not on the boundary.
For
\[ y>2x-3, \]test the origin \((0,0)\):
\[ 0>2(0)-3. \]This becomes
\[ 0>-3, \]which is true. Therefore, the side containing the origin is part of the solution region.
When the Origin Is on the Boundary
The origin is often convenient, but it cannot serve as a useful side test if it lies on the boundary.
For
\[ y>x, \]the boundary
\[ y=x \]passes through \((0,0)\).
Choose another point such as \((0,1)\) or \((1,0)\) instead.
Example: Graphical Meaning of a Linear Inequality
Consider
\[ y\le -x+4. \]The boundary is
\[ y=-x+4. \]Because the inequality includes equality, the boundary is solid.
The condition \(y\le -x+4\) selects points on or below that line.
Nonlinear Inequalities Also Produce Regions
Boundaries need not be lines.
Consider
\[ y\ge x^2. \]The boundary is the parabola
\[ y=x^2. \]Since equality is included, the parabola belongs to the solution set. The remaining solutions lie on the side where \(y\) is at least as large as \(x^2\), which is the region on or above the parabola.
Relations Can Be Defined by Inequalities
A relation need not be generated by an equation alone.
The condition
\[ x^2+y^2\le 9 \]relates every ordered pair whose distance from the origin is at most \(3\).
Its graph is the filled circular region consisting of the boundary circle together with its interior.
One-Variable Equations in the Coordinate Plane
An equation does not need to contain both coordinates.
The equation
\[ x=3 \]is satisfied by every point whose horizontal coordinate is \(3\), regardless of its vertical coordinate.
Therefore, its graph is a vertical line.
Horizontal Equations
Similarly,
\[ y=-2 \]is satisfied by every point whose vertical coordinate is \(-2\).
Its graph is a horizontal line.
One-Variable Inequalities in the Plane
Consider
\[ x<3. \]The boundary is the vertical line
\[ x=3. \]The solution set contains every point whose horizontal coordinate is less than \(3\). This forms the half-plane to the left of the boundary.
Another One-Variable Inequality
For
\[ y\ge -1, \]the boundary is the horizontal line
\[ y=-1. \]The solution set consists of the boundary together with every point above it.
Compound Conditions Create Intersections of Regions
Suppose a point must satisfy both
\[ y\ge x \]and
\[ y\le 4. \]The solution set consists only of points that satisfy both conditions at the same time.
Graphically, this is the overlap of the two individual solution regions.
Graphical AND Means Overlap
When two conditions are joined by and, a solution must belong to both sets.
If \(A\) is one graphical solution set and \(B\) is another, then the points satisfying both form the intersection
\[ \boxed{A\cap B}. \]This is the graphical meaning of satisfying simultaneous restrictions.
Graphical OR Means Union
When a condition allows one possibility or another, the graphical solution set combines the allowed regions.
In set notation, the combined solution set is a union:
\[ \boxed{A\cup B}. \]A point belongs to the result if it satisfies at least one of the stated alternatives.
Graphing a Function from Its Formula
For a function
\[ f(x)=x^2-1, \]every allowed input \(x\) determines the point
\[ (x,f(x)). \]For example,
\[ f(2)=2^2-1=3, \]so
\[ (2,3) \]lies on the graph.
Function Notation and Graph Coordinates
The statement
\[ f(4)=7 \]has an immediate graphical meaning:
\[ \boxed{(4,7)\text{ lies on the graph of }f}. \]Function notation therefore translates naturally into coordinate language.
Reading Function Values from a Graph
If a graph represents \(y=f(x)\), then reading \(f(a)\) graphically means locating the point on the graph whose horizontal coordinate is \(a\) and reading its vertical coordinate.
Symbolically,
\[ (a,b)\text{ lies on the graph of }f \]exactly when
\[ \boxed{f(a)=b}. \]Inputs and Outputs Need Not Be Time
The horizontal coordinate often represents an independent quantity and the vertical coordinate a dependent quantity, but the quantities can represent many things:
- distance and cost;
- time and position;
- temperature and pressure;
- quantity and revenue;
- radius and area;
- an index and a sequence term.
Axis labels and units are therefore part of the mathematical meaning of an applied graph.
The Same Shape Can Represent Different Relationships
Two graphs may have the same geometric shape while representing different quantities or units.
A straight line on one graph might represent cost versus quantity. On another, it might represent distance versus time.
The curve alone is not the entire interpretation. Coordinates, labels, scales, units, and the defining condition all matter.
A Graph Is Not Merely a Drawing
A hand-drawn curve is only a representation of a mathematical set.
The actual graph of an equation consists of every point satisfying the equation, whether or not the drawing shows those points with perfect visual precision.
This distinction matters when a sketch is thick, coarse, truncated by the graph window, or rendered on a low-resolution display.
Graph Windows Show Only Part of a Relationship
A graphing window displays only selected coordinate values.
If a function continues beyond the visible window, the mathematical graph is not limited to what appears on the screen.
For example, the line
\[ y=3x+1 \]extends indefinitely in both directions even if only a short segment appears in a viewing window.
Missing from the Window Does Not Mean Missing from the Graph
Suppose a viewing window includes only
\[ -5\le y\le5. \]A point with \(y=20\) will not be visible even if it satisfies the equation.
Before concluding that a graph has no point, branch, crossing, or feature, check whether the viewing window could simply be hiding it.
Graphical Verification by Substitution
If a plotted point appears to lie on an equation, substitution provides an algebraic check.
Suppose \((3,7)\) appears on
\[ y=2x+1. \]Substitute:
\[ 7=2(3)+1=7. \]The equality confirms that the point belongs to the exact solution set.
Graphical Verification of an Inequality
Suppose \((1,5)\) appears inside the shaded region for
\[ y>2x+1. \]Substitute:
\[ 5>2(1)+1. \]This gives
\[ 5>3, \]which is true. The point belongs to the solution region.
Boundary Points Require Special Attention
A point on the boundary may or may not satisfy an inequality.
For
\[ y>2x+1, \]the point \((1,3)\) lies on the boundary because
\[ 3=2(1)+1. \]But the inequality requires \(3>3\), which is false. Therefore, the boundary point is excluded.
The Same Boundary Can Belong to Several Inequalities
These inequalities all share the boundary
\[ y=2x+1: \] \[ y>2x+1, \] \[ y<2x+1, \] \[ y\ge2x+1, \] \[ y\le2x+1. \]What changes is the selected side and whether the boundary is included.
Equality Is the Boundary Case
An equation often appears naturally as the boundary between inequality regions.
For the expression
\[ y-(2x+1), \]points can make this quantity positive, negative, or zero.
- positive corresponds to \(y>2x+1\);
- negative corresponds to \(y<2x+1\);
- zero corresponds to \(y=2x+1\).
Graphing Relations from Lists of Points
A relation may be given directly as ordered pairs instead of by a formula.
For example,
\[ S=\{(-3,2),(-1,4),(2,4),(5,-1)\}. \]To graph the relation, plot exactly those four points.
Unless additional information is given, there is no reason to connect them.
Graphing Relations from Tables
A table may represent either selected samples from a continuous relationship or the complete list of a finite or discrete relation.
The context determines which interpretation is appropriate.
Before connecting table points, ask whether intermediate inputs are actually part of the domain.
Graphing Relations from Verbal Conditions
A relation can also be described in words.
For example:
The vertical coordinate is twice the horizontal coordinate.
translates to
\[ y=2x. \]The graph then represents every ordered pair satisfying that verbal condition.
Graphing from a Formula Is Translation
Moving from an equation or formula to a graph is a change of representation.
The mathematics has not changed. The same relationship is being expressed geometrically instead of symbolically.
This is why a student should be able to move among
- words;
- symbols;
- tables;
- ordered pairs;
- graphs.
Graphs Can Reveal Structure Quickly
A symbolic rule may hide patterns that become visually obvious on a graph.
A graph may make it easier to notice:
- where a relation exists;
- whether it appears continuous or discrete;
- whether one input has several outputs;
- where values are positive or negative;
- where two graphical objects meet;
- where a curve changes direction;
- how rapidly one quantity changes relative to another.
Later lessons will develop these observations systematically.
Graphs Can Also Hide Structure
A graph is powerful, but it does not automatically show everything clearly.
Important features may be hidden by
- a poor scale;
- an unsuitable graph window;
- thick or low-resolution plotting;
- overlapping curves;
- large or tiny numerical values;
- missing labels or units.
Graphical reasoning should therefore be paired with symbolic or numerical checking when precision matters.
Exact and Approximate Graphical Information
If a point is known symbolically, its coordinates may be exact.
If a coordinate is read from a drawn graph, it may only be approximate.
For example, a graph may suggest that a point is near
\[ (1.4,2.1), \]but the exact coordinates could involve fractions, radicals, logarithms, or other values not visually recoverable from the picture alone.
Open and Closed Boundary Conventions
Graphs use visual conventions to communicate whether boundary values belong to a set.
On a number line, an open endpoint commonly means exclusion and a closed endpoint inclusion.
In the coordinate plane, dashed curves commonly indicate excluded inequality boundaries, while solid curves indicate included boundaries.
Always verify the convention used by the graph or software when ambiguity is possible.
Finite Relations May Be Entirely Visible
A finite relation such as
\[ \{(0,0),(1,1),(2,4),(3,9)\} \]can be represented completely by four plotted points.
In this case, the graph is not merely a sample. The listed points are the entire relation.
Infinite Relations Cannot Be Plotted Point by Point
A line, parabola, circle, or continuous region contains infinitely many points.
We cannot physically mark every point one at a time.
Instead, we use mathematical structure to draw a curve or region that stands for the complete infinite solution set.
The Graph of \(x=3\) Is Not One Point
A common misunderstanding is to see
\[ x=3 \]and think only of the number \(3\).
In a two-dimensional coordinate plane, \(y\) is unrestricted. Therefore,
\[ (3,-10),\qquad(3,0),\qquad(3,8) \]all satisfy the equation.
The full solution set is a vertical line.
The Graph of \(y=3\) Is Also Not One Point
In
\[ y=3, \]the horizontal coordinate is unrestricted.
Thus,
\[ (-4,3),\qquad(0,3),\qquad(9,3) \]are all solutions, producing a horizontal line.
The Graph of an Identity Can Fill the Plane
Consider the two-variable identity
\[ x+y=x+y. \]Every ordered pair makes this statement true.
Therefore, the solution set is the entire coordinate plane.
This reminds us that a graph need not always be a curve.
An Impossible Equation Can Have an Empty Graph
Consider
\[ x^2+y^2=-1. \]For real coordinates,
\[ x^2\ge0\qquad\text{and}\qquad y^2\ge0, \]so
\[ x^2+y^2\ge0. \]No real ordered pair can satisfy the equation. Its real graph is therefore the empty set.
A Single Point Can Be a Complete Graph
The equation
\[ x^2+y^2=0 \]requires
\[ x=0\qquad\text{and}\qquad y=0. \]Therefore, the complete real graph is the single point
\[ \boxed{(0,0)}. \]Some Equations Produce Several Separate Pieces
A graph does not need to form one connected curve.
For example,
\[ y=\frac{1}{x} \]has no point at \(x=0\), and its graph appears in separate branches on either side of the vertical axis.
The separation is a consequence of the domain restriction, not a break made merely for visual convenience.
Some Relations Produce Closed Curves
The equation
\[ x^2+y^2=4 \]produces a closed circle.
The curve returns to itself rather than extending indefinitely.
This illustrates again why a graph of an equation should not automatically be expected to look like the graph of a familiar function \(y=f(x)\).
Some Relations Produce Multiple Curves
Consider
\[ y^2=x^2. \]This condition is equivalent to
\[ y=x \]or
\[ y=-x. \]The graph therefore consists of two lines crossing at the origin.
Functions Can Have Restricted Domains
A function may be defined only for selected inputs.
Suppose
\[ f(x)=x^2,\qquad -2\le x\le3. \]The graph contains only the portion of the parabola corresponding to inputs from \(-2\) through \(3\).
The formula alone does not determine the complete graph unless the domain is also known or understood.
Restrictions Can Be Part of a Relation
Consider the relation
\[ y=x+1,\qquad x>0. \]The line \(y=x+1\) supplies the geometric rule, while \(x>0\) restricts the allowed points to those with positive horizontal coordinates.
Graphing requires both pieces of information.
Graphical Representation of a System
Two equations can be graphed on the same coordinate plane.
A point satisfying both equations must lie on both graphs at once.
Therefore, common points of the graphs represent simultaneous solutions.
The detailed use of intersections as solution methods is developed later, but the set idea is already the same:
\[ \boxed{\text{common graphical points}=\text{common solutions}}. \]Graphical Representation of a System of Inequalities
For several inequalities, each condition produces its own solution region.
The points satisfying the entire system are those lying in every required region simultaneously.
This overlapping region is often called a feasible region when the inequalities represent practical constraints.
Graphs and Sign
When a graph represents \(y=f(x)\), vertical position communicates the sign of the output.
- points above the \(x\)-axis have \(f(x)>0\);
- points on the \(x\)-axis have \(f(x)=0\);
- points below the \(x\)-axis have \(f(x)<0\).
This will become especially useful when we study zeros, intercepts, and sign behavior.
Graphs and Coordinate Roles
In a conventional function graph, \(x\) is treated as the input and \(y\) as the output.
That convention is common, but it is not a law that every relation must follow. A relation may treat the variables symmetrically or assign them other roles.
Read the defining statement and labels before assuming what the coordinates represent.
Graphing Is Not Guessing the Shape
Familiarity with graph families is useful, but a graph should ultimately be justified by the mathematical relationship.
A reliable graph may be supported by
- substitution and tables;
- known algebraic forms;
- intercepts and special points;
- symmetry;
- domain restrictions;
- transformations;
- technology;
- verification against the defining condition.
Later graphical-method lessons will develop these tools one by one.
From Exact Conditions to Visual Objects
Symbolic conditions are exact. Drawings are finite approximations of those conditions.
If an equation says
\[ x^2+y^2=25, \]then the exact mathematical graph contains precisely the points satisfying that equality. A hand-drawn circle may not place every pixel perfectly, but the equation remains exact.
From Visual Objects Back to Conditions
Graphical reasoning also works in reverse.
Given a graph, we may ask what conditions its points appear to satisfy.
A horizontal line suggests a constant \(y\)-value. A vertical line suggests a constant \(x\)-value. A shaded half-plane suggests an inequality. A discrete collection of points suggests a restricted or discrete domain.
This reverse translation is one of the most useful graphical skills.
Common Error: Treating Every Graph as a Function
A graph may represent a relation without representing \(y\) as a function of \(x\).
Circles, vertical lines, and many other relations fail the vertical line test.
Before using function language, check whether each horizontal input has only one vertical output.
Common Error: Plotting Only One Convenient Point
One point that satisfies an equation does not usually determine the entire graph.
The point
\[ (0,1) \]lies on infinitely many possible graphs.
To identify the relationship, use enough information to represent its full structure rather than relying on one isolated solution.
Common Error: Connecting a Discrete Relation
Suppose the complete relation is
\[ \{(1,2),(2,4),(3,6)\}. \]Drawing line segments between these points would add ordered pairs that were not given as part of the relation.
Connect points only when the mathematical model includes the intermediate values.
Common Error: Using a Solid Boundary for a Strict Inequality
If the inequality is
\[ y<3x+2, \]the boundary line
\[ y=3x+2 \]is not part of the solution set.
A solid boundary would falsely communicate inclusion. Use the appropriate excluded-boundary convention, usually a dashed line.
Common Error: Shading the Wrong Side
A correctly drawn boundary is only half of an inequality graph.
For
\[ y>3x+2, \]the solution set must contain points with \(y\)-values greater than the boundary values.
When uncertain, test a point and verify the inequality numerically.
Common Error: Forgetting the Boundary Inclusion
Compare
\[ y<3 \]with
\[ y\le3. \]Both select points below the horizontal line \(y=3\), but only the second includes the boundary itself.
Common Error: Assuming a Graph Window Is the Domain
A screen might display only
\[ -10\le x\le10. \]That does not mean the mathematical domain of the function is necessarily limited to those inputs.
The graph window is a viewing choice. The domain is a property of the relation or function.
Common Error: Trusting the Picture More Than the Condition
A graph drawn with limited precision may make a point appear to lie on a curve when it does not, or appear slightly off a curve when it actually does.
When an exact answer is required, return to the equation, inequality, table, or definition and verify symbolically or numerically.
A Reliable Procedure for Graphing an Equation
- Identify the coordinate system and variables.
- Determine any domain or variable restrictions.
- Rewrite the equation into a useful form when appropriate.
- Find or calculate representative solution points.
- Use known structure only when justified.
- Plot the points accurately using the chosen scale.
- Connect or extend points only when the relationship supports it.
- Check representative points by substitution.
- Confirm that the viewing window shows the important features.
A Reliable Procedure for Graphing an Inequality
- Replace the inequality symbol temporarily with equality to identify the boundary.
- Graph the boundary accurately.
- Determine whether the boundary is included.
- Use a solid boundary for inclusion and the appropriate excluded-boundary convention for strict inequality.
- Determine which side or region satisfies the inequality.
- Use a test point when the correct region is not obvious.
- Shade or otherwise indicate the complete solution region.
- Verify at least one included point and, when useful, one excluded point.
A Reliable Procedure for Deciding Whether a Graph Is a Function
- Identify which coordinate is being treated as the input.
- Imagine vertical lines through the graph when \(x\) is the input.
- Check whether any such line meets the graph more than once.
- If none do, the relation represents \(y\) as a function of \(x\).
- If at least one does, one input has multiple outputs and the relation is not a function of \(x\).
Example: Equation, Function, and Relation at Once
Consider
\[ y=x^2+2. \]This is an equation because it states equality.
It defines a function because each real \(x\) produces exactly one value of \(y\).
It is also a relation because its graph is a collection of ordered pairs.
These categories overlap rather than exclude one another.
Example: Equation and Relation, but Not a Function of \(x\)
Consider
\[ x^2+y^2=4. \]It is an equation and it defines a relation.
But it is not a function \(y=f(x)\) because some allowed \(x\)-values correspond to two different \(y\)-values.
Example: Inequality and Relation
Consider
\[ y\ge x^2. \]This inequality defines a relation consisting of every ordered pair on or above the parabola \(y=x^2\).
It is not the graph of one function \(y=f(x)\), because each input is paired with infinitely many allowed outputs in the shaded region.
Example: A Finite Function
Consider
\[ F=\{(-1,2),(0,4),(1,2),(2,5)\}. \]Each input occurs once, so \(F\) is a function.
The repeated output \(2\) is allowed because it belongs to two different inputs.
The complete graph consists of four isolated points.
Example: A Finite Relation That Is Not a Function
Consider
\[ R=\{(-1,2),(0,4),(0,5),(2,5)\}. \]The input \(0\) is paired with both \(4\) and \(5\).
Therefore,
\[ \boxed{R\text{ is not a function of its first coordinate}}. \]Example: Determine Whether a Point Belongs to a Relation
Does \((3,4)\) lie on
\[ x^2+y^2=25? \]Substitute:
\[ 3^2+4^2=9+16=25. \]Yes. Therefore,
\[ \boxed{(3,4)\text{ belongs to the relation}}. \]Example: Determine Whether a Point Belongs to an Inequality
Does \((2,1)\) satisfy
\[ y\le -x+4? \]Substitute:
\[ 1\le -2+4. \]Thus,
\[ 1\le2, \]which is true. The point belongs to the shaded solution region.
Practice 1: Point on an Equation
Determine whether \((4,9)\) lies on the graph of
\[ y=2x+1. \]Practice 2: Point Not on an Equation
Determine whether \((3,8)\) lies on the graph of
\[ y=2x+1. \]Practice 3: Function or Not?
Determine whether the relation
\[ \{(-2,1),(0,3),(2,5),(2,7)\} \]is a function of its first coordinate.
Practice 4: Repeated Output
Determine whether
\[ \{(1,4),(2,4),(3,4),(4,5)\} \]is a function.
Practice 5: Boundary Type
For
\[ y<-2x+6, \]identify the boundary equation and state whether the boundary is included.
Practice 6: Test a Point
Determine whether \((0,0)\) satisfies
\[ y\ge x-2. \]Practice 7: Vertical or Horizontal?
Describe the graph of
\[ x=-5. \]Practice 8: Discrete or Continuous?
A relation is defined only by the ordered pairs
\[ \{(1,2),(2,4),(3,6),(4,8)\}. \]Should the points automatically be connected by line segments? Explain.
Practice 9: Circle as a Function
Is the relation
\[ x^2+y^2=16 \]a function \(y=f(x)\)? Explain using the input-output rule or vertical line test.
Practice 10: Classify the Graphical Object
Describe the real graph of
\[ x^2+y^2=0. \]Practice Answers
Practice 1
Substitute \(x=4\) and \(y=9\):
\[ 9=2(4)+1=9. \]The statement is true, so
\[ \boxed{(4,9)\text{ lies on the graph}}. \]Practice 2
Substitute \(x=3\) and \(y=8\):
\[ 8=2(3)+1=7. \]This is false, so
\[ \boxed{(3,8)\text{ does not lie on the graph}}. \]Practice 3
The input \(2\) is paired with both \(5\) and \(7\). Therefore,
\[ \boxed{\text{the relation is not a function}}. \]Practice 4
Each input appears with exactly one output. Repeating the output \(4\) is allowed. Therefore,
\[ \boxed{\text{the relation is a function}}. \]Practice 5
The boundary equation is
\[ y=-2x+6. \]Because the inequality is strict, the boundary is excluded and is commonly drawn dashed.
Practice 6
Substitute \((0,0)\):
\[ 0\ge0-2, \]so
\[ 0\ge-2. \]This is true. The origin belongs to the solution region.
Practice 7
Every point has horizontal coordinate \(-5\), while \(y\) is unrestricted. The graph is therefore
\[ \boxed{\text{the vertical line }x=-5}. \]Practice 8
No. The relation is given only as four ordered pairs. Connecting them would add intermediate points that were not stated to belong to the relation.
Practice 9
No. For example, when \(x=0\),
\[ y^2=16, \]so
\[ y=4\qquad\text{or}\qquad y=-4. \]One input has two outputs, so the circle fails the vertical line test.
Practice 10
Since squares of real numbers are nonnegative, their sum can equal zero only when both are zero:
\[ x=0,\qquad y=0. \]Therefore, the complete graph is
\[ \boxed{\{(0,0)\}}. \]What to Remember
The graph of a mathematical condition represents the ordered pairs that satisfy that condition.
- an equation graph represents points where an equality is true;
- an inequality graph usually contains a boundary and a solution region;
- strict inequalities exclude their boundaries;
- inclusive inequalities include their boundaries;
- a relation is a set or rule associating ordered pairs;
- a function is a relation assigning exactly one output to each allowed input;
- the vertical line test checks whether a graph represents \(y\) as a function of \(x\);
- discrete points should not be connected unless intermediate values belong to the relation;
- domain restrictions are part of the graph;
- a graph window shows only a selected view of the mathematical object;
- substitution can verify whether a point truly belongs to a graph.
The central habit is: read every graph as a set of coordinates satisfying a condition, not merely as a picture.
Where We Go Next
Once a graph is understood as a solution set, we can identify special points where that set meets important reference lines or another graphical object.
The next lesson, Intercepts, Zeros, and Intersections, develops the graphical meaning of axis crossings, function zeros, shared solutions, and points where two relations meet.