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15: More Transformations © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

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Trig Transformations Combined Translations x axis translation of –a and y axis translation of b x axis translation of +a and y axis translation of b Note Opposite sign Note Opposite sign

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Trig Transformations Stretches The function is obtained from by a stretch of scale factor ( s.f. ) k, parallel to the y -axis. The function is obtained from by a stretch of scale factor ( s.f. ), parallel to the x -axis.

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Trig Transformations Reflections Reflections in the axes Reflecting in the x -axis changes the sign of y Reflecting in the y -axis changes the sign of x

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Trig Transformations e.g. 1 Sketch the graph of the function is a stretch of s.f. 2, parallel to the y -axis. Solution: We can use the fact that is a stretch of. xysin2

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Trig Transformations e.g. 1 Sketch the graph of the function Solution: We can use the fact that is a stretch of. xysin2 is a stretch of s.f. 2, parallel to the y -axis. The scale factor of the stretch gives the amplitude of the function.

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Trig Transformations e.g. 2 Sketch the graph of the function Solution: is a stretch of s.f., parallel to the x -axis. So,

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Trig Transformations e.g. 2 Sketch the graph of the function Solution: is a stretch of s.f., parallel to the x -axis. So, The period of is or radians.

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Trig TransformationsExercises 1. Give the equation of the function that is shown on the sketch below. Ans:

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Trig Transformations Solution: A stretch of s.f. 2 parallel to the x -axis. Sketch both functions on the same axes for the interval 2. Describe in words the transformation Exercises

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Trig Transformations Solution: 3.Sketch the graph of for showing the scales clearly. What is the period of the function? The period is Exercises

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Trig Transformations Reflection in the x -axis Every y -value changes sign when we reflect in the x -axis e.g. So, x x In general, a reflection in the x -axis is given by

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Trig Transformations then (iii) a reflection in the x -axis (i) a stretch of s.f. 2 parallel to the x -axis then (ii) a translation of e.g.3 Find the equation of the graph which is obtained from by the following transformations, sketching the graph at each stage. ( Start with ).

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Trig Transformations Solution: (i) a stretch of s.f. 2 parallel to the x -axis stretch

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Trig Transformations Brackets aren’t essential here but they make it clearer. (ii) a translation of : translate

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Trig Transformations (ii) a translation of : translatereflect x x (iii) a reflection in the x -axis

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Trig Transformations Exercises 1. Describe the transformations that map the graphs of the 1 st of each function given below onto the 2nd. Sketch the graphs at each stage. (a) to ( Draw for ) (b) y = cosx to y = 2cos(x – 30)

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Trig Transformations Solutions: (a) to Translation Stretch s.f. parallel to the x -axis

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Trig Transformations Vertical stretch factor 2 (b) to Solutions: Translation parallel to the x -axis

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Trig Transformations (i) (ii) The diagram shows part of the curve with equation. Copy the diagram twice and on each diagram sketch one of the following: x y

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Trig Transformations Solution: (ii) x y xy (i)

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Trig Transformations In an earlier section, we met stretches. is a stretch of scale factor ( s.f. ) k, parallel to the y -axis e.g. is a stretch of s.f. 2, parallel to the y -axis Reminder: ( multiplied by k )

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Trig Transformations is a stretch of scale factor ( s.f. ), parallel to the x -axis. e.g. is a stretch of s.f. parallel to the x -axis. ( x multiplied by k )

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Trig Transformations e.g. 1 Sketch the graph of the function is a stretch of s.f. 2, parallel to the y -axis. Solution: We can use the fact that is a stretch of. xysin2

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Trig Transformations e.g. 1 Sketch the graph of the function Solution: We can use the fact that is a stretch of. xysin2 is a stretch of s.f. 2, parallel to the y -axis. The scale factor of the stretch gives the amplitude of the function.

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Trig Transformations e.g. 2 Sketch the graph of the function Solution: is a stretch of s.f., parallel to the x -axis. So,

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Trig Transformations e.g. 2 Sketch the graph of the function Solution: is a stretch of s.f., parallel to the x -axis. So, The period of is or radians.

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Trig TransformationsExercises 1. Give the equation of the function that is shown on the sketch below. Ans:

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Trig Transformations Solution: A stretch of s.f. 2 parallel to the x -axis. Sketch both functions on the same axes for the interval 2. Describe in words the transformation Exercises

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Trig Transformations Solution: 3.Sketch the graph of for showing the scales clearly. What is the period of the function? The period is Exercises

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Trig Transformations

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