Lesson 7

What is the original period of both sine and cosine?

Since 2pi radians equals 360deg each parent graph completes one cycle in 2pi units.

What five angles divide one sine or cosine period into quarters?

0, pi/2, pi. 3pi/2, 2pi

These correspond to: 0, 90, 180, 270, 360

What is the five-point pattern for y = sin x

midline → maximum → midline → minimum → midline

Parent points:

(0,0), (pi/2, 1), (pi,0), (3pi/2, -1), (2pi,0)

What is the five-point pattern for a negative sine graph?

midline → minimum → midline → maximum → midline

negative sine begins on the midline and goes down.

What is the five-point pattern for y = cos x

maximum → midline → minimum → midline → maximum

Parent points:

(0,1), (pi/2, 0), (pi, -1), (3pi/2, 0), (2pi,1)

What is the five-point pattern for a negative cosine graph?

minimum → midline → maximum → midline → minimum

negative cosine begins at a minimum.

How are the parent sine and cosine graphs related?

They have the same shape, amplitude, and period, but one is shifted sideways:

cos x = sin (x + pi/2)

Cosine can be viewed as sine shifted left by pi/2

In the point (x,y) what do x and y represent?

• x: input and horizontal position

• y: output and vertical position

y = f(x)

Therefore y and f(x) represent the same output

What does x tell us on a trigonometric graph?

The x-value tells us where we are horizontally in the cycle.

It may represent an angle, time, distance, or another input.

It does not automatically mean maximum or minimum.

How is sin(theta) different from a point on the sine graph?

On the unit circle:

(Cos theta, sin theta)

On the sine graph:

(theta, sin theta)

What form does the textbook use for transformed sine and cosine functions?

S(x) = A sin (ωx - ϕ) + B

C(x) = A cos (ωx - ϕ) + B

What does A control?

amplitude and possible reflection

use the absolute value until you know the graph pattern. Always nonnegative

What does ω control?

Period constant/control. A larger omega produces a shorter period because more cycles fit into the same horizontal.

What does ϕ control?

Constant used to calculate phase shift.

What does B control?

vertical shift and midline

Phase shift formula

ϕ/ω

What does h mean when the inside is written as ω(x - h)

h = ϕ/ω

How can I find the anchor without memorizing a sign rule?

Set the entire inside expression equal to zero and solve for x.

What kind of point occurs at the anchor?

It depends on the function and the sign of A:

• Positive sine: midline, moving up

• Negative sine: midline, moving down

• Positive cosine: maximum

• Negative cosine: minimum

The anchor is an x-value. Pair it with the appropriate y-value to make a point.

How do I calculate the midline from a graph?

B = max + min/2

How do I calculate amplitude from a graph?

Amplitude = max - min/2

Amplitude = max - mid

How do I calculate the maximum and minimum from an equation?

Max = B + |A|

Min = B - |A|

These are y-values

How do I identify the period from a graph?

Find two consecutive matching points where the cycle is behaving the same way, then subtract their x-coordinates:

P = x(second) - x(first)

Examples of matching points:

• maximum to the next maximum

• minimum to the next minimum

• upward midline crossing to the next upward midline crossing

Do not use an upward crossing and a downward crossing; they are not the same place in the pattern.

Once I know the period, how do I find ω?

Begin with:

P = 2pi/|ω|

Then solve:

|ω| = 2pi/P

The sign of the graph is normally placed in A, not in ω

How do I decide whether to use sine or cosine?

Choose a convenient anchor:

• Midline crossing → sine is usually easiest.

• Maximum or minimum → cosine is usually easiest.

More than one equation can describe the same sinusoidal graph. The goal is to choose a simple, valid equation.

What order should I use to create a sine or cosine equation from a graph?

1. Read the max and min y-values

2. Calculate the midline B

3. Calculate the amplitude |A|

4. Find the period P

5. Calculate ω = 2pi/P

6. Choose sine or cosine

7. Choose a convenient anchor on the graph

8. Use the graph’s direction to choose the sign of A

9. Find the anchor’s x-coordinate h

10. Write the inside as ω (x-h)

11. Expand only if the required for is ωx - ϕ

12. Verify a second visible point

How can I avoid confusing graph information with calculated values?

Read directly from graph:

• maximum and minimum

• anchor’s x-coordinate

• two matching x-coordinates used for period

• direction of movement

How do I find the five key x-coordinates efficiently?

1. Find the anchor h

2. Find the period

3. Find the quarter period

4. Repeatedly add Q (quarter period)

After finding the five x-coordinates, how do I find their y-values?

Calculate

Midline, max, and min

Then use the appropriate pattern:

• Positive sine: mid, max, mid, min, mid

• Negative sine: mid, min, mid, max, mid

• Positive cosine: max, mid, min, mid, max

• Negative cosine: min, mid, max, mid, min

How can I graph by transforming the five parent angles?

Let the entire inside angle equal each parent angle

1. Set the inside expression equal to µ

2. Solve for x

3. Find the parent sine or cosine value

4. Apply the outside transformation A(value) + B

5. Plot

Outside operations A (*) and B control which values

y

What is the domain of every sine and cosine function?

(-oo, oo)

What is the range

[B - |A|, B + |A|}

What happens when sine or cosine has a negative input?

Sine is odd:

sin(-x) = -sin x

Cosine is even:

cos(-x) = cos x