A graph is a visual display of numbers, relationships, or data. Graphs help you see patterns, compare values, and understand relationships quickly. In algebra, the graph of an equation is the picture of all its solutions on a coordinate plane; in statistics, bar graphs, line graphs and scatter plots display data.
In algebra, a graph is the picture of every solution of an equation. In statistics, the kind of graph depends on the data: categories, change over time, or two variables.
Formal definition and notation
The graph of an equation in two variables is the set of all points (x, y) in the coordinate plane whose coordinates make the equation true; it often forms a curve, which may be a line. The graph of a functionf is the set of points (x, f(x)). In statistics, a graph (or chart) is a display of data in which lengths, positions or areas stand for values, read against labeled axes with a stated scale.
Symbol
Read it as
Meaning
(x, y)
the point x comma y
an ordered pair: move x units across, then y units up or down
y = 2x − 1
y equals two x minus one
the equation whose graph is the line in the picture
the horizontal and vertical number lines that set the scale
Key facts
Every point on the graph of an equation is a solution, and every solution is a point on the graph.
The x-axis is horizontal and the y-axis is vertical; they cross at the origin, (0, 0).
Choose the graph that fits the data: a bar graph compares categories, a line graph shows change over time, a scatter plot shows how two variables relate, and a histogram shows how numbers are spread out.
Always read the axis labels and scale before interpreting a graph. One grid square might be 1, 5 or 1,000.
A graph that rises from left to right shows an increasing relationship; one that falls shows a decreasing relationship.
How to graph an equation
How to graph an equation
pick x→ compute y→ plot (x, y)→ connect
pick x
choose a few easy inputs, such as −1, 0, 1, 2
compute y
substitute each x into the equation
plot (x, y)
mark each solution as a point
connect
draw the line or curve through the points and extend it
For y = 2x − 1: x = 0 gives y = −1, x = 1 gives 1, x = 2 gives 3. The three points line up, and the line through them is the graph.
Try it: plot points
Interactive
Drag the points, or select one and use the arrow keys. The readout gives each point’s coordinates and its quadrant. Try to place A, B and C on the line y = x + 1.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Graph examples, step by step
Example 1: Read a value from a graph's equation
Warm-up
The line in the picture is the graph of y = 2x − 1. Extend it: what is y when x = 4?
Do the points (3, 8) and (−2, −5) lie on the graph of y = 3x − 1?
Test (3, 8): 3(3) − 1 = 8. The equation is true, so the point is on the graph.
Test (−2, −5): 3(−2) − 1 = −6 − 1 = −7.
−7 ≠ −5, so (−2, −5) is not a solution and not on the graph. The point (−2, −7) is.
Answer:(3, 8) is on the graph; (−2, −5) is not.
Example 3: Interpret a distance-time graph
Test-style
A straight-line graph shows a car’s distance in miles (vertical axis) against time in hours (horizontal axis). It passes through (0, 0) and (4, 240). How far has the car gone after 2.5 hours?
Read the labels: the vertical axis is miles and the horizontal axis is hours.
The line rises 240 miles in 4 hours: 240 ÷ 4 = 60 miles per hour.
A straight line means the speed is constant, so after 2.5 hours the distance is 60 × 2.5.
60 × 2.5 = 150.
Answer: After 2.5 hours the car has gone 150 miles.
Common mistakes
✗ MistakeEach grid line on the y-axis is 5, but a bar reaching the 4th line is read as 4.
✓ Instead4 grid lines × 5 per line = 20.
Always read the axis labels and scale before interpreting a graph. The scale tells you what one grid step is worth.
✗ MistakePlotting (3, 5) by going up 3 and right 5.
✓ InsteadGo right 3 (the x-coordinate), then up 5 (the y-coordinate).
Ordered pairs always list x first. (3, 5) and (5, 3) are different points.
✗ MistakeTwo bars at 50 and 55 on an axis that starts at 45: “the second is twice as big.”
✓ Instead55 is only 10% more than 50; the bar looks twice as tall because the axis does not start at 0.
A broken or shifted axis stretches small differences. Check where the vertical axis starts before comparing bar heights.
The coordinate plane is the empty grid with two axes. A graph is what you draw on it, such as the line y = 2x − 1.
Graphvs.Table
A table lists values in rows and columns; a graph shows the same values as a picture. The table x: 0, 1, 2 and y: −1, 1, 3 and the line y = 2x − 1 hold the same information.
Graphvs.Graph (in graph theory)
In discrete math, a graph is a set of dots (vertices) joined by segments (edges), like a map of cities and roads. It has no axes or scale.
Quick check: 3 questions
Question 1 of 3
On a coordinate graph, which axis runs up and down?
Show answer and explanation
Answer: B. The y-axis is vertical and the x-axis is horizontal. The origin is the point (0, 0) where they cross, and the scale is the value of each grid step, not an axis.
Question 2 of 3
Which point is on the graph of y = x + 3?
Show answer and explanation
Answer: C. For (2, 5): 2 + 3 = 5, true. (5, 2) swaps the coordinates, (0, −3) has the wrong sign (it should be (0, 3)), and (1, 3) forgets to add 3 to x.
Question 3 of 3
On a bar graph, each grid line on the vertical axis adds 20. A bar ends halfway between the 3rd and 4th grid lines. What value does it show?
Show answer and explanation
Answer: A. 3.5 grid lines × 20 per line = 70. 3.5 ignores the scale, 35 uses 10 per line, and 60 drops the half line.
Graph: frequently asked questions
What are the main types of graphs in math?
In algebra you graph equations and functions on the coordinate plane: lines, parabolas and other curves. In statistics you graph data: bar graphs compare categories, line graphs show change over time, histograms show how numbers are distributed, line plots and dot plots show single values, scatter plots show two variables, and circle graphs show parts of a whole.
What does it mean to graph an equation?
Graphing an equation means drawing every point (x, y) that makes it true. Make a small table: choose x-values, compute y, plot the points, and connect them with a line or smooth curve. For y = x + 2, the points (0, 2), (1, 3) and (2, 4) line up, and the line through them is the graph.
Is a graph the same as a chart?
In everyday use, mostly yes. People say bar chart or bar graph for the same picture. In math class, graph is the usual word for the picture of an equation or function on axes, and chart often means a table or a display like a pie chart. Either way, read the title, the axis labels and the scale first.
Coordinate PlaneThe coordinate plane is a flat grid formed by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). Every point is located by an ordered pair, x first then y, such as (3, −2). The two axes divide the plane into four quadrants, numbered I to IV counterclockwise from the upper right.Geometry · Grade 5, Grade 6, Grade 7FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2DataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3Ordered PairAn ordered pair is two numbers written to locate a point on the coordinate plane. It is written as (x, y): the x-value comes first, and the y-value comes second. From the origin, move x units across, then y units up or down. The order matters: (2, 5) and (5, 2) are different points.Geometry · Grade 5, Grade 6, Grade 7X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7Y-AxisThe y-axis is the up-and-down number line on the coordinate plane, with positive numbers above the origin and negative numbers below it. It meets the x-axis at the origin, (0, 0). A point’s y-coordinate is read along the y-axis, and every point on the y-axis has x-coordinate 0, like (0, 4).Geometry · Grade 5, Grade 6, Grade 7Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6OriginThe origin is the point where the x-axis and the y-axis cross on the coordinate plane. Its coordinates are (0, 0), and it is labeled O. Every ordered pair tells how far to move from the origin: (4, 3) means 4 units right, then 3 units up.Geometry · Grade 5, Grade 6, Grade 8Bar GraphA bar graph uses bars to compare amounts in different categories. The taller or longer the bar, the larger the value it represents. Each bar stands for one category, such as a favorite sport, and its length is read against a number scale. Bars can be vertical or horizontal, with gaps between them.Statistics · Grade 3, Grade 4, Grade 5Scatter PlotA scatter plot displays paired numerical data as points on a coordinate plane, one point for each pair. It helps show whether two quantities have a pattern or relationship: points rising from left to right show a positive association, points falling show a negative association, and a shapeless cloud shows no association.Statistics · Grade 8, Algebra 1HistogramA histogram uses bars to show how numerical data are grouped into intervals. The data are split into equal-width intervals, called bins, and each bar’s height is the count of values in that interval. The bars touch, because the intervals are continuous, which is what distinguishes a histogram from a bar graph.Statistics · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8MileA mile (mi) is a US customary unit of length for long distances, equal to 5,280 feet or 1,760 yards. One mile is about 1.61 kilometers, so it is longer than a kilometer. Road distances and speed limits in the United States use miles and miles per hour.Measurement · Grade 4, Grade 5, Grade 6DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6PointA point is an exact location in space. It has no length, no width and no thickness, so it has no size at all. You draw a point as a dot and name it with a capital letter, such as point A. On a coordinate plane, an ordered pair such as (3, 2) names a point.Geometry · Grade 4, Grade 5, Grade 6SlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1Line PlotA line plot shows data values along a number line, often with marks above each value. Each X or dot stands for one data value, and repeated values stack in a column, so the tallest stack is the mode. Line plots make it easy to see clusters, gaps, outliers and the spread of a small data set.Statistics · Grade 3, Grade 4, Grade 5