functions & graphs

Precalculus 4.5 Graphs of Sine and Cosine 1

Functions & Graphs

· What is a function?· Describes a relationship of input and output· g(x) = x5

· Function name: g· Input: x· Output: input (x) raised to the fifth power· The function g takes an input (x) and raises it to the fifth

power.· p(q) = 2q – 5

· The function p takes an input (q) and multiples it by 2, subtracts 5.


· g(t) = sin(t)· the function g takes an input of t and outputs

the sine ratio that corresponds with t (y/r)· h(t) = cos(t)

· the function h takes an input of t and outputs the cosine ratio that corresponds with t (x/r)


· "Why didn't sin and tan go to the party?"

"... just cos!"


Think Graphically

· What is the graphical representation of a function?

· g(x) = 3x5+4· A plot of all of the points (input, output) (x, g(x))· g is the function that raises x to the 5th power and v.

stretches c=3 and v. shift up 4· p(q) = 2q – 5

· A plot of all the points (q, p(q))· p is the function that takes q and v. stretches c=2

and vertically shifts down 5.


Generalize

· f(x) = x· g(x) = a f(b(x – h)) + k· g(x) = a(b(x – h)) + k· This is similar to point-slope form:

· y – y1 = m(x - x1)


· g(x) = a(bx - h) + k· How does a affect the f(x)?

· Vertical stretch/compression. · a > 1 v. stretch or 0 < a < 1 v. compression

· How does b affect the f(x)?· Horizontal stretch/compression.· 0 < b < 1 h. stretch or b > 1 h. compression

· How does h affect the f(x)?· Horizontal shift.

· How does k affect the f(x)?· Vertical shift.


Predict Graph and Verify

· Y1 = x2

· Y2 = (1/4x)2

· Y3 = (4x)2

· Y4 = (1/4)(x)2

· Y5 = 4(x)2

· Y6 = (x - 4)2

· Y7 = (x + 4)2

· Y8 = x2 - 4· Y9 = x2 + 4


Ferris Wheel

· Which trig function could describe the height of the Ferris Wheel at time t?

· What does the height of the ferris wheel correspond to with respect to ordered pairs (x, y)?

· f(t) = sin(t)· What are some key values of sine or key points

of the graph of sine?· 0, max, 0, min (periodic)


· Given a function g(t) = sin(t)· h(t) = d + ag(b(x – c))· h(t) = d + asin(b(x – c))· How does a, b, c and d affect the graph?· Remember what a graph is:

· The plot of all of the points (input, output)· (t, g(t))

Pre-calculus

4.5 Graphs of Sine and Cosine

Objectives:Sketch the graphs of basic sine and cosine functions

Use amplitude and period to sketch the graphsSketch translations of graphs


Complete each t-chart and sketch the graphs:

You Try

y sin xx y0

/2

3 /22

22/3

2/0

sin4yxxy

y cos xx y0

/2

3 /22

y 2cos xx y0

/2

3 /22


Basic Sine and Cosine Curves

2

32

2

1

1

2

32

2

1

1

y sin x

y cos x

period = 2

period = 2

1y 1

1y 1

What symmetry do you notice with the above graphs?How does graph symmetry relate to the ideas of even and odd functions?


Example

· Sketch the graph of y = 2sin(x) on the interval [-π, 4π]

· How is this similar to the parent function sin(x)?· How is this different to the parent function sin(x)?

0

2

2

32


Amplitude and Period· The amplitude is half the distance between the

maximum and minimum values of the function· How does amplitude relate to listening to

music? · The period is the distance in x needed for the

function to complete one cycle (when the values begin to repeat).

· How would changing the period affect the music?

· Pitch, speed, tempo.


Standard Form of Sine and Cosine Functions

· · · The amplitude = |a|· The period = 2π/b· Horizontal shifts are caused by c

· Interval left endpoint: solve bx – c = 0· Interval right endpoint: solve bx – c = 2π

· Vertical shifts are caused by d· Max value = d + |a|, min value = d - |a|

€

y = d + asin(b(x − c))

€

y = d + acos(b(x − c))


Steps to Graph· Find left endpoint

· bx – c = 0· Find right endpoint

· bx – c = 2π· Divide interval into four equal parts

· Interval length is the period 2π/b· Apply basic sine shape (0, max, 0, min, 0) or basic

cosine shape (max, 0, min, 0, max) with amplitude and vertical shift to get key points


Example

· Graph

· Endpoints: · Period: · Amplitude: · Key points:

y sin x2


Example

· Graph


y 12

sin x 3


Example

· Graph


y 2 3cos x


Example

· Find the amplitude, period, and phase shift for the sine function whose graph is shown. Write an equation for this graph.

· Amplitude: · Period: · Phase shift:

2

2

3

3

3


Closure

· Describe the basic shape of the sine graph and the cosine graph

· Sine starts at zero going up· 0, max, 0, min, 0

· Cosine starts at max going down· max, 0, min, 0, max


Bellwork

?42 and between difference theis What .3

.212962 .1

22

xyxy

60° 13What are the 6 trig functions for

this angle?


You Try

· Quickly evaluate each of the following:· One half of one half =· One half of one third=· One half of two thirds=· One half of five sixths=· One half of one half of three=· One half of one half of seven=

· One half of one half = · One half of one third= · One half of two thirds= · One half of five sixths= · One half of one half of three= · One half of one half of seven=


Solutions

· Quickly evaluate each of the following:· One half of one half =· One half of one third=· One half of two thirds=· One half of five sixths=· One half of one half of three=· One half of one half of seven=

· One half of one half = 1/4· One half of one third= 1/6· One half of two thirds= 1/3· One half of five sixths= 5/12· One half of one half of three= 3/4 or .75· One half of one half of seven= 7/4 or 1.75

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