Properties of Graphs

Test yourself

Estimating Gradient

For a line that isn't straight, we can estimate roughly what the gradient is.

Gradient at a point

To estimate the gradient at a point draw a tangent to the curve at that point.
Work out the gradient of the tangent.
Remember that gradient = change in y ÷ change in x.

Gradient = change in y ÷ change in x

To estimate the gradient use gradient = change in y ÷ change in x.
But remember it is only an estimate because the curve isn’t straight.

Estimating Area Under a Graph

Straight lines only - exact calculation

To calculate the area under a graph, split the area into trapeziums and rectangles and work out the area of each section.
Use the formula:
- Area (Trapezium) = ¹⁄₂ × base × (a + b)
- Where a and b are the sides of the trapezium.

Curved line - estimate

If the graph is not made up of straight lines you can estimate the area by splitting the graph approximately into trapeziums and doing the same calculation.

Overestimate or underestimate

An area estimate is an overestimate when the trapeziums are above the curve and an underestimate when the trapeziums are below the curve.
In the example the first and second sections are underestimates and the third is an overestimate.

Gradient and Area of a Graph With Units

A real number is a number/value that shows a quantity (or amount) on a line/scale. For a graph showing two real quantities, the gradient and area also show real quantities.

Gradient

For any graph the gradient represents: (y-axis units) per (x-axis units).
For example:
- litres per second
- £ per kg.

Area

The area under the graph represents: (y-axis units) × (x-axis units).
For a velocity-time graph the area under the graph represents distance travelled because ^km⁄_h × h = km.

1Numbers

1.1Integers

1.1.1Integers

1.1.2Ordering Numbers

1.1.3The Four Rules of Operations

1.1.4Hierarchy of Operations

1.1.5Definitions & Terms

1.1.6HCF & LCM

1.1.7Standard Form

1.1.8Standard Form 2

1.1.9Prime Numbers

1.1.10End of Topic Test - Standard Form

1.2Fractions

1.2.1Fractions

1.2.2Using Fractions

1.2.3Conversion

1.2.4Reciprocals

1.3Decimals

1.3.1Decimals

1.3.2Decimals 2

1.3.3Exam-Style Questions - Recurring Decimals

1.3.4Comparing & Converting Decimals

1.3.5End of Topic Test - Fractions & Decimals

1.4Powers & Roots

1.4.1Square Numbers

1.4.2Index Notation

1.4.3Exam-Style Questions - Index Notation

1.4.4Advanced Indices

1.4.5Surds

1.4.6End of Topic Test - Powers & Roots

1.5Set Language & Notation

1.5.1Sets

1.6Percentages

1.6.1Percentages

1.6.2Percentage Change

1.6.3Percentages in Finance

1.6.4Compound Interest

1.7Ratio & Proportion

1.7.1Manipulating Ratios

1.7.2Proportion

1.8Degree of Accuracy

1.8.1Rounding & Significant Figures

1.8.2Estimation

1.8.3Upper & Lower Bounds

1.8.4Limits of Accuracy

1.8.5Limits of Accuracy 2

1.8.6End of Topic Test - Rounding & Estimation

2Equations, Formulae & Identities

2.1Algebraic Manipulation

2.1.1Algebra Basics

2.1.2Power Rules & Collecting Terms

2.1.3Expanding Brackets

2.1.4Expanding Brackets 2

2.1.5Factorising Quadratics

2.1.6Algebraic Fractions

2.1.7Proofs

2.1.8End of Topic Test - Algebra & Proofs

2.1.9Exam-Style Questions - Algebraic Proof

2.2Expressions & Formulae

2.2.1Rearranging Formulae

2.2.2Simple Formulae

2.2.3End of Topic Test - Algebra Manipulation & Formula

2.3Linear Equations

2.3.1Solving Linear Equations

2.3.2Simultaneous Linear Equations

2.4Quadratic Equations

2.4.1Solving Quadratics

2.4.2Solving Quadratics 2

2.4.3End of Topic Test - Solving Equations

2.4.4Simulateous Quadratic Equations

2.4.5Exam-Style Questions - Simultaneous Equations

2.4.6Graphs of Simultaneous Equations

2.4.7End of Topic Test - Simultaneous Equations

2.5Proportion

2.5.1Graphs & Proportion

2.6Inequalities

2.6.1Solving Inequalities

2.6.2Graphs of Inequalities

2.6.3Exam-Style Questions - Inequalities

2.6.4End of Topic Test - Inequalities

3Sequences, Functions & Graphs

3.1Sequences

3.1.1Sequence Rules

3.1.2Finding Terms

3.1.3Exam-Style Questions - Sequences

3.1.4End of Topic Test - Sequences

3.2Functions

3.2.1Functions

3.2.2Composite & Inverse

3.2.3End of Topic Test - Functions

3.3Graphs

3.3.1Plotting Graphs

3.3.2Straight Line Graphs

3.3.3Properties of Graphs

3.3.4Quadratic Graphs

3.3.5Practical Graphs

3.3.6Graphs of Polynomials

3.3.7End of Topic Test Graphs

3.4Common Graphs

3.4.1Graphs of Common Functions

3.4.2Transformations of Graphs

3.4.3End of Topic Test - Transformations and Graph

3.5Calculus

3.5.1Derivatives of Functions

3.5.2Maxima & Minima

4Geometry

4.1Angles, Lines & Triangles

4.1.1Angles & Lines

4.1.2Angle Rules

4.1.3Angles in Triangles

4.1.4Exam-Style Questions - Angles

4.2Polygons

4.2.1Polygons

4.2.2Quadrilaterals

4.2.3Congruence

4.2.4Scale Factor

4.2.5Exam-Style Questions - Enlargements

4.2.6End of Topic Test - Geometry

4.3Symmetry

4.3.1Symmetry

4.3.2Exam-Style Questions - Reflection

4.4Measures

4.4.1Measurements

4.4.2Time

4.4.3Bearings

4.4.4Speed

4.5Circle Properties

4.5.1Properties of a Circle

4.5.2Circle Theorems

4.5.3Circle Theorems 2

4.5.4Circle Theorems 3

4.5.5Exam-Style Questions - Circle Theorems

4.6Trigonometry & Pythagoras’ theorem

4.6.1Pythagoras' Theorem

4.6.2Trigonometry

4.6.3Sine & Cosine for Obtuse Angles

4.6.4Triangle Formulae

4.6.5Exam-Style Questions - Trigonometry & Area

4.7Mensuration of 2D Shapes

4.7.1Standard Units

4.7.2Perimeter of 2D Shapes

4.7.3Area of 2D Shapes

4.7.4Area & Circumference of Circle

4.7.5End of Topic Test - Trigonometry & Shapes

4.83D Shapes & Volume

4.8.13D Shapes

4.8.2Surface Area

4.8.3Volume

4.8.4Exam-Style Questions - Volume

5Vectors & Transformation Geometry

5.1Vectors

5.1.1Vectors

5.1.2Vectors 2

5.1.3Exam-Style Questions - Vectors

5.2Transformation Geometry

5.2.1Transformations

5.2.2Rotations

5.2.3Translations & Reflections

5.2.4Enlargements

5.2.5Effects of Transformations

6Statistics & Probability

6.1Statistical Measures

6.1.1Basic Statistics

6.1.2Frequency Table

6.2Graphical Representation of Data

6.2.1Pictograms, Bar Charts & Pie Charts

6.2.2Histograms

6.2.3Cumulative Frequency

6.2.4Cumulative Frequency 2

6.2.5Exam-Style Questions - Cumulative Frequency

6.2.6End of Topic - Datasets

6.3Probability

6.3.1Probability

6.3.2Theoretical Probability

6.3.3Experimental Probability

6.3.4Venn Diagrams

6.3.5Sample Space Diagrams

6.3.6Tree Diagrams

6.3.7Independent Events

6.3.8Conditional Probability

6.3.9End of Topic Test - Probability

6.3.10Exam-Style Questions - Probability

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