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6.4 Transformations of Trigonometric Functions: y Asin K (X D) + C y A Cosk (X D) + C

This document discusses transformations of trigonometric functions. It explains that sinusoidal functions can be written in general form using parameters a, k, c, and d that represent amplitude, period, phase shift, and vertical shift respectively. These parameters allow the graph to be stretched, compressed, reflected, translated horizontally or vertically. Key points for sine and cosine functions must be memorized to graph transformations accurately. Examples demonstrate finding amplitude, period, phase shift, and vertical shift from equations, and sketching graphs using key features.

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Ashley Elliott
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330 views4 pages

6.4 Transformations of Trigonometric Functions: y Asin K (X D) + C y A Cosk (X D) + C

This document discusses transformations of trigonometric functions. It explains that sinusoidal functions can be written in general form using parameters a, k, c, and d that represent amplitude, period, phase shift, and vertical shift respectively. These parameters allow the graph to be stretched, compressed, reflected, translated horizontally or vertically. Key points for sine and cosine functions must be memorized to graph transformations accurately. Examples demonstrate finding amplitude, period, phase shift, and vertical shift from equations, and sketching graphs using key features.

Uploaded by

Ashley Elliott
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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MHF4U1 Unit 5 Lesson 4

6.4 Transformations of Trigonometric Functions

Learning Goal: TO use transformations to sketch the graphs of the primary


trigonometric functions in radians

In General form, sinusoidal functions (sine/cosine) can be written as:

y = asin k(x − d) + c or y = a cosk(x − d) + c

Each of the letters (a, k, c, and d) tells us something different about the graph.

“a”
•   the absolute value of a ( | a | ) gives the vertical stretch/compression
•   if a is negative, it represents a reflection in the x-axis.
•   | a | is the amplitude

“k”
2p
•   the period of the function is
k
•   horizontal stretch/compression factor
•   remember to factor out “k” if necessary

“d”
•   gives the horizontal translation
•   phase shift

“c”
•   gives the vertical translation
•   states the equation of the axis (centre of the function)

As soon as we aply any transformation, we are potentially altering the amplitude,


period, max/min and intercepts of the function.
!𝝅
Note if we apply a phase shirt of 𝟐
to a sine function the graph will look identical
to the cosine function.

Ex. Given the following equation, state the amplitude, period, phase shift and vertical
shift.

a. y = 15cos(2x) −12

Amplitude – Period - Phase Shift – Vertical Shift -

1
MHF4U1 Unit 5 Lesson 4

b. y = -5 sin (2 x - 6) + 8 * You must factor the 2 out of the 2x - 6.

Amplitude – Period - Phase Shift Vertical Shift

The letters a, k, c and d also tell us how to transform the basic graphs of y = sin x and
y = cos x .

The mapping notation for trig functions is

Key Points for a Transformation Mapping

y = sin x
y = cos x

x y x y
0 0 0 1
p p
1 0
2 2
p 0 p -1
3p 3p
-1 0
2 2
2p 0 2p 1

You MUST memorize the five key points for both the sine and cosine function to put into
the mapping notation.

When sketching a sine or cosine function, you must label all 5 key points and show the
general shape of the graph.

2
MHF4U1 Unit 5 Lesson 4

Ex. Find the key points (mapping notation) and sketch the graph of the following
functions.

a. y = -2 sin 4 x

Graphing Using Key Features of the


Function
y = 3sin 2(x) + 1

3
MHF4U1 Unit 5 Lesson 4

!1 $
y = 4 cos # x & − 2
"2 %

æ p ö
y = -3 sin 4ç x + ÷ + 1
è 6 ø

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