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

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

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.

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

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 

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.

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

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.

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

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

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

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

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 ).

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

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

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

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)

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

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

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

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

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 )

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 )

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 

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.

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

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.

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

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

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

Trig Transformations