The Central Idea
A coordinate graph connects numbers with position.
In the Cartesian plane, a point is represented by an ordered pair
\[ \boxed{(x,y)}. \]The first coordinate tells us the horizontal position. The second tells us the vertical position.
But coordinates have meaning only after the axes and their scales are known. The point
\[ (3,2) \]means three horizontal coordinate units from the origin and two vertical coordinate units from the origin. If the tick marks represent different numerical increments, the physical spacing on the screen or page may not correspond to one numerical unit.
Graphing therefore depends on three linked ideas:
- coordinates describe position;
- scale assigns numerical meaning to visual spacing;
- plotting places mathematical objects according to those coordinates and scales.
From the Number Line to the Coordinate Plane
On a number line, one coordinate is enough to locate a point.
For example, the number
\[ -3 \]identifies a position three units to the left of zero.
In a plane, one number is no longer enough. We need two independent directions.
The Cartesian coordinate plane uses:
- a horizontal number line, called the \(x\)-axis;
- a vertical number line, called the \(y\)-axis.
Their intersection provides a common reference point.
The Origin
The horizontal and vertical axes meet at the origin.
\[ \boxed{(0,0)}. \]The origin is the zero point for both coordinate directions.
From the origin:
- positive \(x\)-coordinates lie to the right;
- negative \(x\)-coordinates lie to the left;
- positive \(y\)-coordinates lie above;
- negative \(y\)-coordinates lie below.
Ordered Pairs
A point in the Cartesian plane is written as
\[ (x,y). \]The order matters.
The point
\[ (2,5) \]has horizontal coordinate \(2\) and vertical coordinate \(5\).
The point
\[ (5,2) \]is generally a different point.
Coordinates are therefore called an ordered pair, not merely a pair of numbers.
Reading the Coordinates in Order
A useful habit is:
\[ \boxed{\text{horizontal first, vertical second}}. \]For
\[ (-4,3), \]begin at the origin, move four coordinate units left, then three coordinate units up.
The first movement locates the correct vertical line. The second locates the final point on that line.
The Four Quadrants
The coordinate axes divide the plane into four regions called quadrants.
| Quadrant | Sign of \(x\) | Sign of \(y\) |
|---|---|---|
| I | positive | positive |
| II | negative | positive |
| III | negative | negative |
| IV | positive | negative |
The quadrants are numbered counterclockwise beginning in the upper-right region.
Quadrant Examples
The point
\[ (4,7) \]lies in Quadrant I.
The point
\[ (-4,7) \]lies in Quadrant II.
The point
\[ (-4,-7) \]lies in Quadrant III.
The point
\[ (4,-7) \]lies in Quadrant IV.
Points on the Axes Are Not in Quadrants
If a point has
\[ y=0, \]then it lies on the \(x\)-axis.
For example,
\[ (5,0) \]lies on the \(x\)-axis, not in Quadrant I or IV.
Similarly, if
\[ x=0, \]the point lies on the \(y\)-axis.
The origin lies on both axes and in no quadrant.
Coordinates Describe Lines of Position
The first coordinate of
\[ (3,5) \]tells us that the point lies on the vertical line
\[ x=3. \]The second coordinate tells us that it lies on the horizontal line
\[ y=5. \]The point is where those two position conditions meet.
\[ \boxed{x=3 \quad\text{and}\quad y=5}. \]A Coordinate System Is More Than Two Drawn Lines
To interpret a coordinate graph correctly, we need to know more than where the axes appear.
A usable coordinate system identifies:
- the coordinate directions;
- the origin or other reference values;
- the numerical scale on each axis;
- the quantities represented;
- the units, when the quantities are measured.
Missing any of these can make an otherwise accurate picture ambiguous.
What Is Scale?
A graph's scale tells us what numerical change is represented by a given visual distance.
Suppose consecutive labeled tick marks on the \(x\)-axis are
\[ 0,\quad 5,\quad 10,\quad 15. \]Then one tick interval represents
\[ \boxed{5\text{ coordinate units}}. \]Moving one tick to the right does not mean adding \(1\). It means adding \(5\).
Tick Marks and Tick Intervals
A tick mark is a marked position on an axis. The numerical change from one equally spaced tick to the next is the tick interval.
If the labels are
\[ -20,\quad -10,\quad 0,\quad 10,\quad 20, \]then the tick interval is
\[ \boxed{10}. \]If a point lies halfway between the ticks labeled \(10\) and \(20\), its coordinate is approximately
\[ 15. \]Find the Scale Before Reading the Point
A common graph-reading mistake is to count tick marks as though every tick represents one unit.
Instead, first compute the change between labeled values.
If consecutive labels are
\[ 100,\quad 125,\quad 150, \]then one tick interval represents
\[ 125-100=25. \]The scale is \(25\) units per tick interval.
The Two Axes May Use Different Scales
The \(x\)-axis and \(y\)-axis do not have to use the same numerical scale.
A graph might use:
\[ \text{horizontal scale: }1\text{ unit per tick} \]and
\[ \text{vertical scale: }10\text{ units per tick}. \]The point
\[ (3,20) \]would then appear three horizontal tick intervals from the \(y\)-axis but only two vertical tick intervals above the \(x\)-axis.
Visual distance alone does not determine numerical distance.
Equal Visual Lengths Need Not Mean Equal Numerical Changes
Suppose one square horizontally represents \(2\) units while one square vertically represents \(50\) units.
Moving one square right changes \(x\) by
\[ 2, \]while moving one square up changes \(y\) by
\[ 50. \]The physical movements look equal on the page, but their numerical meanings are very different.
Consistent Scale Within an Axis
On an ordinary linear axis, equal visual intervals must represent equal numerical changes.
A sequence such as
\[ 0,\quad 10,\quad 20,\quad 30 \]can be placed at equally spaced ticks because each step changes by \(10\).
But the values
\[ 0,\quad 10,\quad 100,\quad 1000 \]should not be placed at equally spaced positions on an ordinary linear axis.
Specialized nonlinear scales exist, but they must be identified explicitly.
Choosing a Useful Scale
A good scale makes the important values easy to locate while using the graph space efficiently.
Suppose the \(x\)-values range from \(0\) to \(50\).
Convenient possibilities might include:
\[ 1,\quad 2,\quad 5,\quad 10 \]units per tick, depending on the size of the graph.
A scale of \(1000\) units per tick would compress all of the relevant values into a tiny region and waste most of the available space.
Convenient Scale Increments
Scales based on simple decimal multiples are often easy to read.
Common increments include values such as
\[ 1,\quad 2,\quad 5,\quad 10,\quad 20,\quad 50,\quad 100. \]Fractional or decimal increments may also be useful:
\[ \frac12,\quad 0.1,\quad 0.25,\quad 0.5. \]The goal is not to follow one mandatory pattern. The goal is to make the coordinate values easy to interpret accurately.
The Graph Window
A graph window is the portion of a coordinate system that is currently visible.
A window may be described by bounds such as
\[ -10\le x\le10 \]and
\[ -5\le y\le15. \]Mathematical objects may continue beyond the visible window. The window does not usually define the object; it defines what part of the object we can see.
Changing the Window Does Not Change the Mathematics
Consider the point
\[ (20,30). \]A window showing only
\[ -10\le x\le10 \]does not display the point.
The point still exists in the coordinate plane. It is simply outside the current view.
This distinction becomes important when graphs appear to have missing intercepts, extrema, or other features that actually lie beyond the window.
Zooming In and Zooming Out
Changing the graph window is analogous to changing magnification.
Zooming in shows a smaller coordinate range with more local detail.
Zooming out shows a larger coordinate range with less local detail.
Neither view is automatically more correct. The useful view depends on the feature we need to inspect.
Aspect Ratio
The horizontal and vertical coordinate units may also be displayed with different physical lengths.
When one numerical unit horizontally is drawn with the same physical length as one numerical unit vertically, the graph uses equal unit scaling in both directions.
If those physical lengths differ, shapes and slopes may look visually stretched or compressed even though the coordinate data are unchanged.
This is one reason numerical labels matter more than visual intuition alone.
Plotting a Point
To plot
\[ (x,y), \]use the coordinate scales rather than simply counting grid squares.
- Locate the origin or reference values.
- Read the scale on the \(x\)-axis.
- Move horizontally to the required \(x\)-coordinate.
- Read the scale on the \(y\)-axis.
- Move vertically to the required \(y\)-coordinate.
- Mark the point at the intersection.
Example: Plotting with Unit Scale
Plot
\[ (-3,4) \]on axes where each tick interval represents one coordinate unit.
Start at the origin.
Move three units left to
\[ x=-3. \]Then move four units up to
\[ y=4. \]The point lies in Quadrant II.
Example: Plotting When the Scale Is Not One
Suppose the horizontal labels are
\[ 0,\quad 10,\quad 20,\quad 30 \]at consecutive tick marks, while the vertical labels are
\[ 0,\quad 5,\quad 10,\quad 15. \]To plot
\[ (20,10), \]move two horizontal tick intervals right and two vertical tick intervals up.
The point appears two ticks away in both directions even though the numerical coordinate changes are different.
Plotting Fractional Coordinates
Coordinates do not have to be integers.
For example,
\[ \left(\frac32,-\frac12\right) \]means
\[ x=1.5,\qquad y=-0.5. \]On a unit grid, the point lies halfway between \(x=1\) and \(x=2\), and halfway between \(y=0\) and \(y=-1\).
Plotting Decimal Coordinates
Suppose we want to plot
\[ (2.4,1.7). \]If the grid marks tenths, both coordinates can be located directly.
If the grid marks only whole units, the point must be estimated between the labeled positions.
The graph can display the location approximately even though the coordinates themselves are exact as written.
Exact Coordinates Versus Approximate Placement
A coordinate may be exact while its hand-drawn position is approximate.
For example,
\[ \left(\sqrt2,\pi\right) \]specifies an exact point.
A practical graph might place it using
\[ \sqrt2\approx1.414 \]and
\[ \pi\approx3.142. \]The mathematical object and the drawn approximation should not be confused.
Plotting from a Table
A table can be interpreted as a list of ordered pairs.
| \(x\) | \(y\) |
|---|---|
| \(-2\) | 3 |
| 0 | 1 |
| 2 | 5 |
| 4 | 2 |
The rows represent the points
\[ (-2,3),\quad (0,1),\quad (2,5),\quad (4,2). \]Each row becomes one plotted location.
A Table Does Not Automatically Tell Us to Connect the Points
Plotting listed points and drawing a continuous curve through them are different operations.
If the data represent separate objects or discrete observations, the plotted points may stand alone.
If the relationship is defined continuously between the listed inputs, a curve or line may be appropriate.
We should not connect points merely because they appear on the same graph.
Discrete and Continuous Plotting
Suppose \(n\) counts the number of completed tasks. Values such as
\[ n=0,1,2,3,\ldots \]are discrete if partial tasks are not meaningful.
A plot may therefore show separate points only.
By contrast, time or distance may vary continuously, allowing every value in an interval to be meaningful.
The geometry of the graph should match the domain of the relationship.
Label the Axes with the Quantities
In an applied graph, the letters \(x\) and \(y\) may be less informative than the actual quantities represented.
For a distance-time graph, label the axes with quantities such as:
- horizontal axis: time;
- vertical axis: distance.
A graph should communicate what the coordinates mean, not merely where the points are located.
Include Units
If quantities are measured, include the units.
For example:
- time in seconds;
- distance in meters;
- temperature in degrees Celsius;
- cost in dollars.
The coordinate
\[ (5,120) \]means much more when we know it represents \(5\) seconds and \(120\) meters.
Coordinates May Carry Different Units
The two coordinates do not need to measure the same kind of quantity.
In a time-temperature graph, a point such as
\[ (10,72) \]could mean
\[ 10\text{ minutes},\qquad72^\circ\text{C}. \]The point is an ordered pair of related measurements, not a pair of distances in one physical space.
Reading a Point from a Graph
To read an already plotted point, reverse the plotting process.
- Project the point horizontally or vertically toward the axes.
- Read the \(x\)-coordinate from the horizontal scale.
- Read the \(y\)-coordinate from the vertical scale.
- Write the result in the order \((x,y)\).
Always check both scales independently.
Estimating Between Tick Marks
A point may fall between labeled ticks.
Suppose one tick interval represents \(20\) units and a point appears about one quarter of the way from \(40\) to \(60\).
One quarter of the interval is
\[ \frac14(20)=5. \]So the coordinate is approximately
\[ 40+5=45. \]Graphical Readings Have Limited Precision
A graph is a visual representation with finite resolution.
If the smallest readable spacing corresponds to about \(0.5\) unit, reporting a coordinate as
\[ 3.472861 \]would claim far more precision than the graph supports.
The number of digits reported should reflect the quality of the graphical information.
Plotting Accuracy and Mathematical Accuracy
A correctly calculated coordinate can be plotted poorly, and a neatly drawn point can represent an incorrect calculation.
Good graphing therefore requires two kinds of accuracy:
- mathematical accuracy: the coordinate values are correct;
- graphical accuracy: the values are placed correctly according to the scales.
Verify a Plotted Point from Its Coordinates
Suppose a point is intended to represent
\[ (-6,8). \]Check the graph in two independent directions:
- its horizontal position should correspond to \(x=-6\);
- its vertical position should correspond to \(y=8\).
This simple two-coordinate check catches many sign and scale errors.
Use Known Points as Reference Checks
Some coordinates provide immediate visual checks.
For example:
\[ (0,0) \]must be at the origin,
\[ (a,0) \]must lie on the \(x\)-axis, and
\[ (0,b) \]must lie on the \(y\)-axis.
If such a point appears elsewhere, something is wrong with the plotting or the scale interpretation.
Choosing a Window from the Data
Before plotting several points, inspect their coordinate ranges.
Suppose the \(x\)-coordinates range from
\[ -4\text{ to }8 \]and the \(y\)-coordinates range from
\[ 20\text{ to }90. \]A useful window should include those ranges with enough surrounding space to read the points clearly.
The horizontal and vertical ranges need not be numerically equal.
The Origin Does Not Always Need to Be Centered
Many introductory coordinate planes place the origin at the center because both positive and negative values are being studied.
But an applied graph with only nonnegative values may place the origin near a corner.
The location of the origin on the page is a display choice. Its coordinate meaning remains
\[ (0,0). \]A Display May Omit the Origin
A graph can show a coordinate range such as
\[ 500\le x\le600 \]without showing \(x=0\) at all.
This can be useful when the important variation occurs far from zero.
However, the scale and axis labels must make the restricted range clear so that the viewer does not mistake the visible edge for zero.
Truncated Axes Require Care
Omitting part of an axis can magnify visible differences.
For example, values
\[ 98,\quad99,\quad100 \]can appear dramatically separated on a graph whose vertical window runs only from \(97\) to \(101\).
The graph may be mathematically valid, but the display choice affects visual emphasis. Always read the numerical labels before judging the size of a difference by appearance alone.
Scale Can Change Appearance Without Changing Coordinates
Suppose two points are
\[ A=(1,10) \]and
\[ B=(2,20). \]Changing the graph window or tick spacing changes where the points appear on the page, but their coordinates remain the same.
The display changes. The mathematical positions in the coordinate system do not.
Do Not Infer Numerical Distance from Screen Distance Alone
Two points that appear close together on a heavily zoomed-out graph may have a large numerical separation.
Two points that appear far apart on a zoomed-in graph may differ only slightly.
Numerical comparison should come from coordinates and scale, not from raw screen distance.
Coordinate Differences
For two points
\[ P=(x_1,y_1) \]and
\[ Q=(x_2,y_2), \]the horizontal coordinate change is
\[ \boxed{\Delta x=x_2-x_1}, \]and the vertical coordinate change is
\[ \boxed{\Delta y=y_2-y_1}. \]These differences describe coordinate change even when the axes use different visual scales.
Example: Coordinate Change
Let
\[ P=(-2,3) \]and
\[ Q=(5,-1). \]Then
\[ \Delta x=5-(-2)=7 \]and
\[ \Delta y=-1-3=-4. \]So the movement from \(P\) to \(Q\) is seven coordinate units right and four coordinate units down.
The Same Point Can Appear in Different-Sized Graphs
A point such as
\[ (4,6) \]may appear near the upper-right corner in a window extending only to \(5\) and \(7\), but near the origin in a window extending to \(100\) in both directions.
Coordinate location is mathematical. Apparent closeness to the edge of the screen is not an invariant property of the point.
Plotting Multiple Points Clearly
When several points are plotted, choose a scale and window that separate them enough to be distinguished.
If two points have coordinates
\[ (1000,1000) \]and
\[ (1001,1002), \]a window spanning millions of units may make them visually indistinguishable.
A closer window may be needed to inspect their difference.
Plotting Categories Is Different from Numerical Coordinates
Not every graph uses two numerical coordinate axes.
A bar chart, for example, may place categories such as
January, February, March
along one axis rather than numerical coordinates.
The Cartesian methods in this lesson apply most directly when both positions are described numerically. Other graph types use related but different conventions.
Cartesian Coordinates Are One Coordinate System
The Cartesian plane is the standard coordinate system for much of algebra, but it is not the only way to describe position.
Later work will encounter systems such as:
- polar coordinates;
- three-dimensional Cartesian coordinates;
- parametric representations.
The general principle remains the same: a coordinate system provides rules for turning numbers or parameters into positions.
Preview: Three-Dimensional Coordinates
In three-dimensional Cartesian space, a point requires three coordinates:
\[ \boxed{(x,y,z)}. \]The additional coordinate records position along a third independent axis.
This is a natural extension of the progression
\[ x \quad\longrightarrow\quad (x,y) \quad\longrightarrow\quad (x,y,z). \]Higher-dimensional coordinate systems continue the same idea even when they can no longer be drawn directly on a page.
Graphs Turn Algebra into Geometry
A coordinate pair can represent a numerical relationship geometrically.
Suppose
\[ y=2x+1. \]If \(x=3\), then
\[ y=2(3)+1=7. \]The calculation produces the point
\[ (3,7). \]Plotting that point translates an algebraic fact into a geometric location.
A Point Can Verify an Equation
Suppose we want to know whether
\[ (4,9) \]satisfies
\[ y=2x+1. \]Substitute the coordinates:
\[ 9=2(4)+1. \]Since
\[ 9=9, \]the point satisfies the equation.
The next lesson develops this idea from individual points to complete graphs of equations, inequalities, functions, and relations.
Graphing Is a Representation, Not a Replacement for Algebra
A graph can reveal patterns, locations, intersections, trends, and approximate values quickly.
But a graph may be limited by:
- window size;
- scale choice;
- screen or page resolution;
- plotting precision;
- visual distortion.
Algebra and graphing should support one another. A graph can suggest and verify; symbolic work can establish exact values that a graph only approximates.
Common Error: Reversing the Coordinates
A student is asked to plot
\[ (-2,5) \]but plots
\[ (5,-2). \]The numbers are the same, but their roles have been exchanged.
Remember:
\[ \boxed{(x,y)=\text{horizontal first, vertical second}}. \]Common Error: Ignoring a Negative Sign
The points
\[ (3,4),\quad (-3,4),\quad (3,-4),\quad (-3,-4) \]occupy four different quadrants.
A missing negative sign reflects a point across one of the axes and changes its location substantially.
Common Error: Assuming One Tick Equals One Unit
Suppose the axis labels are
\[ 0,\quad 25,\quad 50,\quad 75. \]A point one tick to the right of zero has coordinate
\[ 25, \]not \(1\).
Read the labels before counting intervals.
Common Error: Using One Axis Scale for Both Axes
A graph may use \(5\) units per horizontal tick and \(100\) units per vertical tick.
Applying the horizontal scale to the vertical axis would place points at the wrong heights.
Treat the two scales independently unless the graph explicitly shows that they are the same.
Common Error: Changing the Scale Midway
On a standard linear axis, equally spaced ticks cannot represent
\[ 0,\quad10,\quad20,\quad50 \]without some explicit break or change of convention.
The first steps increase by \(10\), while the last increases by \(30\).
Inconsistent scale destroys the normal numerical meaning of distance along the axis.
Common Error: Connecting Every Set of Points
Points may represent isolated observations, integer inputs, or separate categories.
Drawing a continuous line between them can claim intermediate values that the data or domain do not support.
Decide whether the underlying relationship is discrete or continuous before connecting plotted points.
Common Error: Choosing a Window That Hides the Important Feature
A mathematically correct graph can still be uninformative.
If every relevant point lies between \(x=2\) and \(x=5\), a window extending from
\[ -10^6\text{ to }10^6 \]may compress the entire pattern into an unreadable cluster.
Choose the view to match the question being studied.
Common Error: Reading More Precision Than the Graph Shows
If a point appears near \(x=3.4\) on a coarse graph, reporting
\[ x=3.417826 \]is not justified by the visual evidence.
Graphical estimates should communicate that they are estimates.
A Reliable Procedure for Plotting a Single Point
- Identify the ordered pair \((x,y)\).
- Locate the origin or reference values.
- Read the numerical scale on the \(x\)-axis.
- Move horizontally to \(x\).
- Read the numerical scale on the \(y\)-axis.
- Move vertically to \(y\).
- Mark the point clearly.
- Check its signs, quadrant, and axis positions.
A Reliable Procedure for Constructing a Coordinate Plot
- Identify the quantities to be plotted.
- Determine their numerical ranges.
- Choose a useful graph window.
- Select a consistent scale for each axis.
- Label each axis and include units when applicable.
- Plot the required points according to the scales.
- Connect points only when the mathematical relationship supports it.
- Check that important features are visible and readable.
A Reliable Procedure for Reading an Existing Graph
- Identify what each axis represents.
- Read the units.
- Determine the numerical increment per tick on each axis.
- Notice the visible graph window.
- Check whether zero is shown or omitted.
- Read coordinates using the scales rather than visual distance alone.
- Estimate between ticks only to a precision the graph supports.
- Watch for unequal axis scales or a distorted aspect ratio.
Worked Example: Build a Plot from a Table
Suppose a measurement table gives:
| Time \(t\) (s) | Distance \(d\) (m) |
|---|---|
| 0 | 0 |
| 2 | 30 |
| 4 | 60 |
| 6 | 90 |
The ordered pairs are
\[ (0,0),\quad(2,30),\quad(4,60),\quad(6,90). \]A useful horizontal scale might be \(1\) or \(2\) seconds per tick. A useful vertical scale might be \(10\), \(15\), or \(30\) meters per tick.
The axes should be labeled time (s) and distance (m).
Worked Example: Read a Nonunit Scale
Suppose consecutive horizontal ticks are labeled
\[ -10,\quad0,\quad10,\quad20, \]while consecutive vertical ticks are labeled
\[ 0,\quad100,\quad200,\quad300. \]A plotted point is halfway between \(x=10\) and \(x=20\), and halfway between \(y=200\) and \(y=300\).
Its coordinates are approximately
\[ \boxed{(15,250)}. \]Worked Example: Choose a Better Window
Suppose all relevant points satisfy
\[ 40\le x\le55 \]and
\[ 900\le y\le1100. \]A window such as
\[ 0\le x\le1000, \qquad 0\le y\le100000 \]technically contains the data but makes the pattern difficult to see.
A window modestly wider than the observed ranges would usually communicate the structure more effectively.
Practice 1: Identify the Quadrant
In which quadrant does the point
\[ (-7,3) \]lie?
Practice 2: Point on an Axis
Classify the location of
\[ (0,-8). \]Is it in a quadrant or on an axis?
Practice 3: Read the Tick Interval
Consecutive labeled ticks on an axis are
\[ 40,\quad55,\quad70,\quad85. \]What numerical change does one tick interval represent?
Practice 4: Different Axis Scales
The \(x\)-axis uses \(5\) units per tick and the \(y\)-axis uses \(20\) units per tick.
How many tick intervals from the origin are needed to plot
\[ (15,40)? \]Practice 5: Estimate Between Ticks
One vertical tick interval runs from \(200\) to \(240\). A point lies halfway between those ticks.
Estimate its vertical coordinate.
Practice 6: Plot from a Table
Write the ordered pairs represented by the table.
| \(x\) | \(y\) |
|---|---|
| \(-3\) | 2 |
| 1 | 0 |
| 4 | \(-5\) |
Practice 7: Exact Versus Approximate
A point has exact coordinates
\[ (\sqrt5,\pi). \]Explain why its position on a hand-drawn graph will normally be approximate.
Practice 8: Choose a Scale
A set of vertical values ranges from \(0\) to \(500\), and the graph has ten major vertical intervals.
Give one convenient scale that uses the available space well.
Practice 9: Identify the Error
A student draws equally spaced ticks and labels them
\[ 0,\quad10,\quad20,\quad50,\quad60 \]on an ordinary linear axis.
Explain the problem.
Practice 10: Window Interpretation
A graphing window displays
\[ -5\le x\le5. \]Does the fact that the point \((12,3)\) is not visible mean that the point does not exist in the Cartesian plane? Explain.
Practice Answers
Practice 1
Since
\[ x<0 \]and
\[ y>0, \]the point lies in
\[ \boxed{\text{Quadrant II}}. \]Practice 2
The first coordinate is zero, so the point lies on the \(y\)-axis:
\[ \boxed{(0,-8)\text{ is on the negative }y\text{-axis}}. \]It is not in any quadrant.
Practice 3
Subtract consecutive labels:
\[ 55-40=15. \]Therefore, one tick interval represents
\[ \boxed{15\text{ units}}. \]Practice 4
Horizontally,
\[ \frac{15}{5}=3 \]tick intervals are needed.
Vertically,
\[ \frac{40}{20}=2 \]tick intervals are needed.
So move
\[ \boxed{3\text{ ticks right and }2\text{ ticks up}}. \]Practice 5
The interval length is
\[ 240-200=40. \]Half of that is \(20\), so the midpoint value is
\[ \boxed{220}. \]Practice 6
Read each row as \((x,y)\):
\[ \boxed{(-3,2),\quad(1,0),\quad(4,-5)}. \]Practice 7
The coordinates are exact, but a finite grid cannot usually mark irrational values such as \(\sqrt5\) and \(\pi\) with unlimited precision. Their drawn locations are therefore approximations to the exact coordinates.
Practice 8
Ten equal intervals covering \(0\) to \(500\) can use
\[ \frac{500}{10}=50 \]units per interval.
One convenient scale is therefore
\[ \boxed{50\text{ units per major tick interval}}. \]Practice 9
The numerical differences are
\[ 10,\quad10,\quad30,\quad10. \]Equal visual spacing on an ordinary linear axis must represent equal numerical changes. The jump from \(20\) to \(50\) breaks that scale.
Practice 10
No. The point
\[ (12,3) \]is simply outside the visible horizontal range. Changing the window can make it visible. The display window does not determine which points exist in the Cartesian plane.
What to Remember
The Cartesian plane locates points with ordered pairs:
\[ \boxed{(x,y)}. \]- read the horizontal coordinate first and the vertical coordinate second;
- use signs to determine direction and quadrant;
- read the scale before counting tick intervals;
- allow the \(x\)- and \(y\)-axes to use different scales when useful;
- keep the numerical scale consistent within an ordinary linear axis;
- choose a graph window that makes the relevant information visible;
- label quantities and units;
- connect plotted points only when the underlying relationship supports continuity;
- treat graphical readings as approximate when the display limits precision.
The central habit is simple: never interpret position without first interpreting the coordinate system and its scale.
Where We Go Next
A coordinate system gives us the framework for locating individual points. The next step is to use many points together to represent complete mathematical relationships.
The next lesson, Graphs of Equations, Inequalities, Functions, and Relations, develops the graph as a geometric representation of solution sets and shows how different kinds of symbolic statements produce points, curves, regions, and other graphical objects.