Start Up Day 261. Graph each function from -2π to 2π

2.Find a polynomial function with zeros of: 0, -4, 3i

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by C.Kennedy1

The range of y = sin x is _____________.

The range of the cosine curve is ________________.

OBJECTIVE: SWBAT use trigonometric graphs to define and interpret features such as domain, range, intercepts, periods, amplitude, phase

shifts, vertical shifts and asymptotes. SWBAT to graph Sine and Cosine functions with and without

graphing technology.

EQ: What are the key points for the basic Sine and Cosine parent graphs & where do they come from? How do the values of “a” ,”b”, “h” and “k” affect the

graphs of Sine and Cosine?

HOME LEARNING: Worksheet#1 Graphing Sine Functions

Sine Waves: The Movers and the

Shapers

Graphing Trigonometric Functions

04/21/23by C.Kennedy3

The UNIT CIRCLE—unwrapped!

• Y=sin x is the basic curve: one hill, one valley, starting at 0 and ending at 2 π

• 5 KEY Points: 0, 1, 0, -1 , 0

• Y=cos x is the basic curve: an upside down bell shape, starting at 0 and ending at 2 π

• 5 KEY Points: 1, 0, -1, 0 ,1

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“h” AND “k” OR “c” & “d”, respectively: The “Movers”

• The “h”or the “c” is the most difficult to see! (although it is always in the parentheses!)– The “h” causes a “phase shift” OR

“HORIZONTAL TRANSLATION”– You have to factor out your “b” in order to see

your “h” for the HT!

• The “k”or the “d” is much more obvious!– When looking to the “k”, you get exactly what

you see!– The “k” causes a VERTICAL TRANSLATION

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12

sin

xy

A and B: The Shapers!• The “b” value: FREQUENCY = b

– Horizontal Compression—If the “b” is greater than 1

– Horizontal Stretch –If the “b” is less than one.– Period OR Wavelength = 2pi/b

• The “a” value: AMPLITUDE = Absolute Value of a– Vertical stretch--If the absolute value of “a”

is greater than one.– Vertical Compression--If the absolute value

of “a” is less than one.– Reflection over the x-axis—If the “a” is a

negative value.

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Where to begin? y=a sin b(x-h)+k

1. First-- think of the basic wave and the 5 KEY POINTSSine: (0,1, 0,-1, 0) OR Cosine:(1, 0, -1, 0, 1)

2. Next--Identify the values of a,b,”h”(c) and “k” (d)

3. Finally--determine amp, frequency, period or wave length, horizontal and/or vertical translations 04/21/23

By C.Kennedy

9

Sketching y=a sin b(x-h)+k

1) Dash in a horizontal axis at “k” or“d”2) If c=o then start at o and end at 2π/b,

otherwise continue with begin/end– BEGIN: Set your (bx-bh) =o and solve for

x—this will be your starting point.– END: Set your (bx-bh) =2π and solve for

x again—this will be your ending spot.3) Divide your “wavelength” into FOUR equal

spaces— making room for your FIVE KEY PLACES!

4) Let your “a” wipe the “ones” away and sketch your wave!– Remember that a “-a” causes a reflection

over the x-axis 04/21/23by C.Kennedy10

04/21/23by C.Kennedy11

12sin2 xy

A=-2

B=2

C=-π/2

D=1

START:

END:

Period/Wavelength:

Amplitude:

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12sin2 xy

Let’s apply it—Now you try it! Sketch each over 2 complete

periods.

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Start Up Day 27Problems #58 & 61 from p.358

Construct a sine function (sinusoid) with the given constraints:

#58 Amplitude 2, period 3π, point (0, 0)

“a” and “b”: The Shapers!

Y = a cos b(x – c) +d

“a”Amplitude = IaI½ the distance from theMax to the min.

“b”Frequency = bPeriod (wavelength)=2π/b

-aReflection over x-axis

IaI> 1,Vertical StretchIaI < 1, Vertical Compression

I b I > 1Horizontal Compression

I b I < 1Horizontal Stretch

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y=a cos b(x-h)+kThe Movers

”h”Horizontal Translation= “h” or “c”Always opposite of what you see

in the ( ) and don’tForget to factor out your “b” in

order to see yourREAL “h”OR set your () = 0 and

Solve for “x”.Changes the starting point of the

curve!

“k” or “d”K = Vertical Translation

k lifts or lowers the base line of the curve

“h” and “k” OR “c” & “d” respectively—The Movers