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**“Teach A Level Maths” Vol. 1: AS Core Modules**

7: More Graphs and Translations © Christine Crisp

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Module C1 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

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**x = 0 y = 0, so the graph goes through the origin.**

The function is an example of a cubic function. To sketch the graph we notice the following: x = 0 y = 0, so the graph goes through the origin. x As x increases, y increases quickly e.g. x = 1 y = 1; x = 2 y = 8 The graph has 180 rotational symmetry about the origin When sketching a graph, we try not to PLOT points. We want the general shape not an accurate drawing. e.g. x = -1 y = -1; x = -2 y = -8

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**We have seen that the quadratic function**

is a translation of by In a similar way, is a translation of

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**is another cubic function**

Suppose we translate this function by

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**is another cubic function**

The equation for the translation by is The same rule works for all functions!

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**x = 0 ( infinity ) x = 2 ; x = 3 e.g. (a) Sketch the graph of**

(b) Write down the translation of the graph by (c) Sketch the new graph. x = 0 ( infinity ) Solution: (a) This means that on the graph, x can never be 0 As x increases, y decreases x = 2 ; x = 3 e.g. x = 1 y = 1; The graph has 180 rotational symmetry about the origin e.g.

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The graph of As x increases, y decreases Rotational symmetry On this graph, the x-and y-axes form asymptotes Asymptotes are lines that a graph approaches as x or y approaches infinity.

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For the graph of As

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For the graph of As

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For the graph of We usually show the asymptotes with a broken line. The equations of the asymptotes must always be given

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**(b) Translating by gives**

The asymptotes have also been translated

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So the graph of is

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SUMMARY The function given by translating any function by the vector is given by So, to find the translated function, we replace by and add ( Notice that adding q is the same as replacing y by y – q. We’ll need this later. )

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Exercise Using the same axes for each pair, sketch the following functions: 1. and and 2. 3. and Check your answers using “Autograph” or a graphical calculator.

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The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

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**The function is an example of a cubic function.**

The graph has 180 rotational symmetry about the origin.

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**e.g. is a translation of of**

Translations

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SUMMARY The function given by translating any function by the vector is given by So, to find the translated function, we replace by and add ( Notice that adding q is the same as replacing y by y – q. We’ll need this later. )

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**The equation for the translation by is**

The same rule works for all functions! is another cubic function e.g.

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**We usually show the asymptotes with a broken line.**

The equations of the asymptotes must always be given The graph of

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The graph of is

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