More Trig Transformations Objectives: 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. e.g. A stretch s.f. parallel to the y -axis on followed by a translation of gives With the translation first, we get

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 circles poster

Mini whiteboards

What combination? Transforms y=sin -1 x 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 -1 x 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 OR Reflect 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 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 x Reflect 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 x Reflect in x- axis OR 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 Transformations Exercise 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 Transformations General Translations and Stretches The function is a translation of by  Translations  Stretches The function is obtained from by a stretch of scale factor ( s.f. ), parallel to the x -axis. The function is obtained from by a stretch of scale factor ( s.f. ) k, parallel to the y -axis.

More Transformations SUMMARY  Reflections in the axes Reflecting in the x -axis changes the sign of y 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 -axis then (ii) a translation of e.g. Find the equation of the graph which is obtained from by the following transformations, sketching the graph at each stage. ( Start with ).

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

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

More Transformations SUMMARY we can obtain stretches of scale factor k by When we cannot easily write equations of curves in the form Replacing x by and by replacing y by we can obtain a translation of by Replacing x by Replacing y by