More Trig TransformationsObjectives: To consolidate

understanding of combinations of transformations with trig graphs.

To work out the order of combined transformations.To be confident plotting and

recognising key properties of trig graphs on a GDC.

What function is this the graph of?

Lesson Outcomes

• It is really important that as we build our knowledge of trigonometric functions (and any others) it is incorporated in our knowledge on other areas like domain, range, composite functions, transformations and modulus functions.

• In this lesson we will aim to do this by looking over different ways that these areas may be linked.

If a stretch and a translation are in the same direction we have to be very careful.

xy sine.g. A stretch s.f. parallel to the y-axis on3

followed by a translation of gives

10

With the translation first, we get 1sin xy

3sin3 xy

1sin3 xy xy sin

)1(sin3 xy xy sin

xy sin3

Where Order is Important

• Where Translations in x are combined with stretches in x or reflections in the y axis the order is important.

• So too for Translations in y with stretches in y or reflections in the x-axis.

• For each of the following questions draw shapes on your board to illustrate the combination of transformations needed to change the object into the image.

Activity

Working in circlesposter

Mini whiteboards

What combination?

• Transforms y=sin-1x into y=sin-1(3x-2)

Translate +2 in x Translate

-2 in x

Stretch sf 3 in x

Stretch sf 1/3 in x

Translate +2 in x

Stretch sf 1/3 in

x

What combination?

• Transforms y=secx into y=3sec(x)+2

Translate +2 in y Translate

-2 in y

Stretch sf 3 in y

Stretch sf 1/3 in y

Translate +2 in y

Stretch sf 3 in y

What combination?

• Transforms y=sin-1x into y=sin-1(3-x)

Translate -3 in x Translate

3 in x

Reflect in x axis

Reflect in y axis

Translate -3 in x

Reflect in y axis Translate

3 in x

ORReflect

in y axis

What combination?

• Transforms y=cotx into y=cot(2x+1)

Translate -1 in x Translate

-1/2 in x

Stretch sf 2 in x

Stretch sf 1/2 in x

Stretch sf 1/2 in x Translate

-1/2 in x

OR

Translate -1 in x

Stretch sf 1/2 in x

What combination?

• Transforms y=sin(x) into y=3sin-1(x+1)

Translate -1 in x Reflect

in y=x

Reflect in x-axis

Stretch sf 3 in

y

Translate -1 in xReflect

in y=x

Stretch sf 3 in

y

What combination?

• Transforms y=cosec(x) into y=sec(x)

Translate -π/2 in x Reflect

in y=x

Reflect in x-axis

Translate π/2 in x Translate

-π/2 in xReflect in x-

axisOR

Translate π/2 in x

Transformations and Trig

• You are expected to be familiar with how to transform trig functions (as well as any others).

• One of the tricky things with trigonometric functions is that they may be simpler to write as a single function or one with fewer transformations.

• By considering transformations of secx show that sec(π/2+2x) is the same as –cosec2x

• Hence solve sec(π/2+2x) = 2 for 0≤x≤π

Modulus and Composites

• If f(x)=secx and g(x)=|x| sketch the graph of y=gf(x)

• Solve gf(x)=2√3/3 where 0≤x≤2π

Activity

Order of TransformationsExercise E starts page 67

• Trivia: Arlie Oswald Petters is a Belizean Mathematical Physicist who is considered one of the greatest scientists of African descent. He has numerous achievements including being the first person to develop a mathematical theory of gravitational lensing.

More TransformationsGeneral Translations and

Stretches

ba

• The function is a translation of by)(xfy

baxfy )( Translation

s

Stretches

)(kxfy • The function is obtained from )(xfy by a stretch of scale factor ( s.f. ) ,parallel to the x-axis. k

1

• The function is obtained from)(xkfy )(xfy by a stretch of scale factor ( s.f. ) k,parallel to the y-axis.

More TransformationsSUMMARY

Reflections in the axes

• Reflecting in the x-axis changes the sign of y )()( xfyxfy

)()( xfyxfy

• Reflecting in the y-axis changes the sign of x

More Transformations

then (iii) a reflection in the x-axis

(i) a stretch of s.f. 2 parallel to the x-axisthen (ii) a translation of

20

e.g. Find the equation of the graph which is obtained from by the following transformations, sketching the graph at each stage. ( Start with ).

xy cos

20 x

More Transformations

xcos

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

axis x21cos

xy 21cos

xy cos2

stretch

xy cos

More Transformations

(ii) a translation of :

20 x2

1cos 2cos 21 x

2cos 21 x 2cos 2

1 x

2cos 21 xy

2

2cos 21 xy

translate reflect

x

x

(iii) a reflection in the x-axis

xy 21cos

More TransformationsSUMMARY

we can obtain stretches of scale factor k by

When we cannot easily write equations of curves in the form )(xfy

kx

• Replacing x by and by replacing y by k

y

we can obtain a translation of by

qp

• Replacing x by )( px • Replacing y by )( qx