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Graphs & coordinates

Read a graph as a relationship between quantities. Check axes and scale, connect points to an equation, and interpret the change between two points.

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What you’ll learn

Read coordinates in the stated horizontal/vertical order.
Connect points to values that satisfy an equation.
Calculate and interpret a slope with its units.

Read the axes before the line

Check which quantity belongs on each axis, its units, and the size of each interval. A point (x, y) describes the horizontal coordinate first and the vertical coordinate second.

Connect a rule to the plotted values

A point on the graph of an equation satisfies that equation. For a straight line, slope is vertical change divided by horizontal change; its meaning depends on the quantities and units.

A mistake to avoid

A graph can look steeper after changing the axis scales without changing the underlying rate. Calculate the numerical changes instead of comparing the drawn angle alone.

Worked example

Read y = 2x + 1

When x = 0, y = 1; when x = 2, y = 5. The points (0, 1) and (2, 5) lie on the line. Its slope is (5 − 1)/(2 − 0) = 2, and its vertical intercept is 1.

Learning paths that include this skill

Choose the route that fits your goal. The same skill can be useful in more than one subject or stage of learning.

Preparing for algebraPre-algebraConnect arithmetic to algebra through signed numbers, fractions, proportions, equations, and graphs. Review the specific foundation that blocks the next step.Signed numbers · Equivalent fractions · Decimals · Proportions · +6 more Secondary / high-school algebraHigh-school algebraWork through the relationships behind secondary algebra. Reuse number skills, connect equations to graphs, and build toward systems and quadratics.Signed numbers · Equivalent fractions · Decimals · Linear equations · +4 more Secondary geometry and trigonometryGeometry & trigonometryConnect measurement and geometric reasoning to triangle ratios and the unit circle. Keep the meaning of each angle, length, and coordinate visible.Shapes & angles · Area & volume · Linear equations · Geometric reasoning & proofs · +4 more Preparing for calculusPrecalculus foundationsReview equation, graph, and angle foundations before calculus. Identify the prerequisite that needs attention and connect formal notation to its meaning.Linear equations · Systems of equations · Quadratic equations · Graphs & coordinates · +5 more Introductory university calculusUniversity calculus foundationsConnect the first calculus ideas to functions, graphs, and justified steps. Understand local change and accumulation before extending the techniques.Graphs & coordinates · Function notation · Limits · Derivatives · +2 more Introductory linear algebraLinear algebra foundationsBuild from equations and coordinates to vectors and matrices. Explain what each component represents and how a matrix-vector calculation combines quantities.Signed numbers · Graphs & coordinates · Linear equations · Systems of equations · +2 more Goal-led refresherAdult math refresherRefresh the mathematics behind a real task, from a percentage or measurement to an equation or graph. Use familiar goals and revisit only the skills you need.Place value · Addition & subtraction · Multiplication · Long division · +12 more

Function notationRead f(x) as the output of a function at an input. Evaluate a rule carefully, check its domain, and connect notation to a table or graph.The unit circleConnect an angle to a point on the unit circle. Read sine and cosine as coordinates, identify their signs, and relate degrees to radians.LimitsDistinguish a function’s value at a point from the value it approaches nearby. Work through a removable gap and explain why direct substitution can be misleading.

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