“Teach A Level Maths” Vol. 1: AS Core Modules 44: Stretches of the Trigonometric Functions © Christine Crisp

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In an earlier section, we met stretches. Reminder: ( multiplied by k ) 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

( x multiplied by k ) 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 y sin 2 = e.g. 1 Sketch the graph of the function Solution: We can use the fact that is a stretch of . x y sin 2 = is a stretch of s.f. 2, parallel to the y-axis.

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

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

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

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

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

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

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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. is a stretch of scale factor ( s.f. ) , parallel to the x-axis e.g. stretch of s.f. , parallel to the x-axis. The scale factor gives the amplitude of the function.

e.g. 1 Sketch the graph of the function Solution: We can use the fact that is a stretch of . x y sin 2 = A useful application of stretches occurs with the trigonometric functions is a stretch of s.f. 2, parallel to the y-axis.

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.