more trig transformations objectives: to consolidate understanding of combinations of...
TRANSCRIPT
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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.
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What function is this the graph of?
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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.
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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
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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.
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Activity
Working in circlesposter
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Mini whiteboards
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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
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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
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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
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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
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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
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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
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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.
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• 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
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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
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Activity
Order of TransformationsExercise E starts page 67
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• 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.
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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.
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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
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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
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More Transformations
xcos
Solution:(i) a stretch of s.f. 2 parallel to the x-
axis x21cos
xy 21cos
xy cos2
stretch
xy cos
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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
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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