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Further Graphs & Tangents (Cambridge (CIE) IGCSE Maths)

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  • FrontTypes of Graphs

    Why does the graph of y equals 1 over x not touch the y-axis?

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  • Why does the graph of y equals 1 over x not touch the y-axis?

    The graph of y equals 1 over x not touch the y-axis because on the y-axis, x equals 0. You cannot divide by zero therefore the graph does not have any values on the y-axis.

  • What is an asymptote?

    An asymptote is a line on a graph that a curve gets closer and closer to but never touches.

    These may be horizontal or vertical.

    E.g. y equals 1 over x has asymptotes at y equals 0 and x equals 0.

  • How many turning points does the graph of a cubic have?

    The graph of a cubic has 2 turning points; a maximum and a minimum.

    Note: the graph of y equals x cubed and y equals x cubed plus-or-minus a does not have any turning points.

  • True or False?

    You should always use a ruler when plotting the graph of a function.

    False.

    You should only use a ruler if a graph is linear (and for drawing the axes if they are not given).

    For curves, draw a single smooth freehand curve.

  • How would you find the y-intercept of a graph using its equation?

    To find the y-intercept of a graph, you would substitute x equals 0 into the equation.

  • True or False?

    The solutions to x cubed minus 4 x equals 0 are the value(s) where the graph of y equals x cubed minus 4 x crosses the y-axis.

    False.

    The solutions to x cubed minus 4 x equals 0 are not the value(s) where the graph of y equals x cubed minus 4 x crosses the y-axis.

    x cubed minus 4 x equals 0 when y equals 0 which is the x-axis. Therefore the solutions are the values where the graph crosses the x-axis.

  • The solutions of x cubed plus x squared minus 3 equals x minus 2 are the x values of the intersections between y equals x cubed plus x squared minus 3 and which other graph?

    The solutions of x cubed plus x squared minus 3 equals x minus 2 are the x values of the intersections between y equals x cubed plus x squared minus 3 and bold italic y bold equals bold italic x bold minus bold 2.

  • True or False?

    The x values of the intersections of the two graphs y equals x plus 1 and y equals x squared plus 5 x plus 4 are the solutions of x squared plus 4 x plus 3 equals 0.

    True.

    The x values of the intersections of the two graphs y equals x plus 1 and y equals x squared plus 5 x plus 4 are the solutions of x squared plus 4 x plus 3 equals 0.

    Set the equations equal to each other and rearrange: x squared plus 5 x plus 4 equals x plus 1.

  • What is meant by a tangent to a graph?

    A tangent is a line that touches a curve at a point (and does not cross it).

  • What is meant by the gradient of a curve at a point?

    The gradient of a curve at a point is defined to be the same as the gradient of the tangent to the curve at that point.

  • True or False?

    Drawing a tangent line to a curve at a point will always give the exact gradient of the curve at that point.

    False.

    Drawing a tangent line to a curve at a point will not always give the exact gradient of the curve at that point.

    A tangent line is drawn by eye so it will give an estimate of the gradient.

    Differentiation can be used to find the exact equation of the tangent.

  • How do you estimate the gradient of a curve at a point?

    To find an estimate for the gradient of a curve at a point:

    • Draw a tangent to the curve at the point.

    • Find the gradient of the tangent using

      • Gradient = rise ÷ run

      • or difference in y ÷ difference in x.

    • The gradient of the tangent will give the estimate for the gradient of the curve at the point.

Previous:Quadratic GraphsNext:Solving & Graphing Inequalities

    1. Topic 1

      Number Toolkit

    2. Topic 2

      Set Notation & Venn Diagrams

    3. Topic 3

      Prime Factors, HCF & LCM

    4. Topic 4

      Powers, Roots & Standard Form

    5. Topic 5

      Fractions Toolkit

    6. Topic 6

      Percentages Toolkit

    7. Topic 7

      Simple & Compound Interest, Growth & Decay

    8. Topic 8

      Surds

    9. Topic 9

      Working with FDP

    10. Topic 10

      Working with Ratios

    11. Topic 11

      Compound Measures (Speed, Density, Pressure)

    12. Topic 12

      Time, Currency & Conversions

    13. Topic 13

      Rounding, Estimation & Bounds

    14. Topic 14

      Using a Calculator

    1. Topic 1

      Algebra Toolkit

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding & Factorising Brackets

    4. Topic 4

      Linear Equations & Inequalities

    5. Topic 5

      Quadratic Equations

    6. Topic 6

      Rearranging Formulas

    7. Topic 7

      Simultaneous Equations

    8. Topic 8

      Algebraic Fractions

    9. Topic 9

      Forming & Solving Equations

    10. Topic 10

      Functions

    11. Topic 11

      Sequences

    12. Topic 12

      Proportion

    1. Topic 1

      Coordinate Geometry

    2. Topic 2

      Linear Graphs

    3. Topic 3

      Quadratic Graphs

    4. Topic 4

      Further Graphs & Tangents

    5. Topic 5

      Solving & Graphing Inequalities

    6. Topic 6

      Real-Life Graphs

    7. Topic 7

      Differentiation

    1. Topic 1

      Geometry Toolkit

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Constructions & Scale Drawings

    4. Topic 4

      Circle Theorems

    1. Topic 1

      Area & Perimeter

    2. Topic 2

      Circles, Arcs & Sectors

    3. Topic 3

      Volume & Surface Area

    4. Topic 4

      Congruence & Similarity

    1. Topic 1

      Right-Angled Triangles (Pythagoras & Trigonometry)

    2. Topic 2

      Sine, Cosine Rule & Area of Triangles

    3. Topic 3

      3D Pythagoras & Trigonometry

    4. Topic 4

      Trigonometric Graphs & Equations

    1. Topic 1

      Vectors

    2. Topic 2

      Transformations

    1. Topic 1

      Basic Probability

    2. Topic 2

      Probability Diagrams (Tree & Venn Diagrams)

    3. Topic 3

      Conditional Probability

    1. Topic 1

      Averages & Range

    2. Topic 2

      Statistical Diagrams

    3. Topic 3

      Histograms

    4. Topic 4

      Cumulative Frequency

    5. Topic 5

      Scatter Graphs & Correlation