PRE-CALCULUS GRAPHS OF PERIODIC FUNCTIONS Sinusoidal Graphs

Periodic Functions:

The Graph of Sine:

Domain: Range: Period: The Graph of the Cosine:

Domain: Range: Period: Transforming Graphs: A point rotates around a circle of radius 3. Sketch a graph of the y coordinate of the point.

Amplitude: If 𝒚 = 𝒓𝒔𝒊𝒏𝜽, then the value in front of the sine function is called the The letter A is designated as the amplitude factor: y = A circle with radius 3 feet is mounted with its center 4 feet off the ground. The point closest to the ground is labeled P. Sketch a graph of the height above ground of the point P as the circle is rotated, then find a function that gives the height in terms of the angle of rotation.

Equation: The Midline of a Periodic Function: The center value of a sinusoidal function is the value that the function oscillates above and below is called the midline. It corresponds to a ________________________ shift. 𝒇(𝜽) = 𝒄𝒐𝒔(𝜽) + 𝒌 𝒚 = ____ 𝒊𝒔 𝒕𝒉𝒆 𝒎𝒊𝒅𝒍𝒊𝒏𝒆 The Period of a Function: The value at which the sinusoidal function will _____________ its cycle.

𝒇(𝜽) = 𝒔𝒊𝒏(𝑩𝜽) 𝟐𝝅

𝑩 𝒅𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒆𝒔 𝒕𝒉𝒆 _______________

Find the Amplitudes, Midlines and Periods of the functions below:

𝒚 = −𝟑𝒔𝒊𝒏 (𝝅

𝟔𝜽) + 𝟐 𝒚 =

𝟏

𝟐𝒄𝒐𝒔 (

𝟏

𝟑𝒙) − 𝟑

Write a formula for the periodic function graphed below.

Homework for Sinusoidal Graphs Determine the Amplitude, Midline and Period for the Sinusoidal Functions. Sketch a

graph.

𝒚 = 𝟐𝒔𝒊𝒏 (𝝅

𝟐𝜽) 𝒚 = −𝒄𝒐𝒔(𝟐𝒙) − 𝟑

𝒚 = −𝟐𝒄𝒐𝒔 (𝝅

𝟒𝜽) + 𝟏 𝒚 = 𝒔𝒊𝒏(𝟑𝒙) − 𝟒

Determine the Amplitude, Midline and Period of the functions below. Write the function.

PRE-CALCULUS GRAPHS OF PERIODIC FUNCTIONS More on Sinusoidal Graphs

What we have learned so far: Given an equation in the form: f(t) = Asin(Bt) + k or f(t) = cos(Bt) + k A is the ________________ stretch and is the ___________________. B is the ________________ stretch/compression is determines the ____________ P, by the formula 𝑷 =

k is the _____________ shift and determines the _________________

A point completes 1 revolution every 2 minutes around a circle of radius 5. Graph, then state its amplitude, midline and period if it starts at (5, 0). Write an equation for the function.

Determine the midline, amplitude and period of the function: f(t) = 3 sin(2t) + 1 and then graph.

Find the formula for the sinusoidal function graphed here.

Graph the following after stating the amplitude, midline and period.

𝒇(𝒙) = −𝟐 𝒄𝒐𝒔 (𝝅

𝟑𝜽) − 𝟏 𝒚 = 𝟒 𝒔𝒊𝒏 (𝟒𝜽) + 𝟏

Homework for More on Sinusoidal Graphs If a sinusoidal function starts on the midline at point (0, 3), has an amplitude of 2, and a period of 4, write the formula for the function. Outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 50 degrees at midnight and the high and low temperature during the day are 57 and 43 degrees, respectively. Assuming t is the number of hours since midnight, find a function for the temperature, D, in terms of t. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h(t) gives your height in meters above the ground t minutes after the wheel begins to turn.

a. Find the amplitude, midline, and period of h(t).

b. Find a formula for the height function h(t)

c. How high are you off the ground after 5 minutes?

Find the amplitude, midline and period of the following functions. Then graph each.

𝒚 = 𝟐𝒔𝒊𝒏 (𝝅

𝟐𝜽) 𝒚 = −𝒄𝒐𝒔(𝟐𝒙) − 𝟑

𝒚 = −𝟐𝒄𝒐𝒔 (𝝅

𝟒𝜽) + 𝟏 𝒚 = 𝒔𝒊𝒏(𝟑𝒙) − 𝟒

PRE-CALCULUS GRAPHS OF PERIODIC FUNCTIONS Even More on Sinusoidal Graphs

Horizontal Shifts: Given an equation in the form:

𝒇(𝒕) = 𝑨𝒔𝒊𝒏(𝑩(𝒕 − 𝒉)) + 𝒌 𝒐𝒓 𝒇(𝒕) = 𝑨𝒄𝒐𝒔(𝑩(𝒕 − 𝒉)) + 𝒌

h is the horizontal shift of the function. Give the period, shift and direction of the following:

𝒚 = 𝟑𝒄𝒐𝒔((𝟔𝜽 − 𝟑)) 𝒚 = −𝟐𝒔𝒊𝒏 ((𝝅

𝟑𝒙 + 𝝅))

Sketch a graph of 𝒇(𝒕) = 𝟑𝒔𝒊𝒏 (𝝅

𝟒𝒕 −

𝝅

𝟐)

Find a formula for the function graphed here.

When it comes to finding the shift, there are 4 different combinations using sine and cosine and shifts. Cosine shifted ___ to the ________ or a negative cosine shifted ___ to the ________. Sine shifted ___ to the _________ or a negative sine shifted _____ to the _________.

Write an equation of a sine function with a midline of -6, that is reflected about the x-axis,

with an amplitude of 3, a period of 6 and a shift of 𝝅

𝟑 to the left.

𝒚 = 𝒔𝒊𝒏( ( 𝜽 ))

Write an equation of a cosine function with a midline of 2, with an amplitude of 5, a period

of 𝟐𝝅

𝟑 and a shift of 4 to the right.

𝒚 = 𝒄𝒐𝒔( ( 𝜽 ))

Homework for Still More on Sinusoidal Graph

Write a formula for the graphs below.

For each of the following equations, find the amplitude, period, horizontal shift and midline.

𝒚 = 𝟑𝒔𝒊𝒏(𝟖(𝒙 + 𝟒)) + 𝟓 𝒚 = 𝟐𝒄𝒐𝒔(𝟑𝒙 − 𝟐𝟏) + 𝟒 𝒚 = 𝒔𝒊𝒏 (𝝅

𝟔𝒙 + 𝝅) − 𝟑

Find the formula for the graphs below.

Graph the following functions:

𝒚 = 𝟐𝒔𝒊𝒏(𝟐𝒙 − 𝟒) + 𝟏 𝒚 = −𝒄𝒐𝒔 (𝝅

𝟐𝜽 +

𝟑𝝅

𝟐) − 𝟏

PRE-CALCULUS GRAPHS OF PERIODIC FUNCTIONS Graphs of the Other Trig Functions

The Tangent Graph: 𝒚 = 𝒕𝒂𝒏 𝜽

The Period of the tangent graph is _____. Domain: _____________ Range: __________ Find the formula for the graph below.

Sketch the graph of: 𝒇(𝜽) = 𝟑𝒕𝒂𝒏 (

𝝅

𝟔𝜽)

The Graph of Cotangent: 𝒚 = 𝒄𝒕𝒏 𝜽

The Period of the Cotangent graph is _____. Domain: _____________ Range: __________

The Graph of the Secant: 𝒚 = 𝒔𝒆𝒄 𝜽

The Period of the Secant graph is _____. Domain: _____________ Range: ____________ The Graph of the Cosecant: 𝒚 = 𝒄𝒔𝒄 𝜽

The Period of the Cosecant graph is _____. Domain: _____________ Range: ____________

Sketch the graph of 𝒚 = 𝟐𝒄𝒔𝒄 (𝝅

𝟐𝜽) + 𝟏 (Hint: First sketch 𝒚 = 𝟐𝒔𝒊𝒏 (

𝝅

𝟐𝜽) + 𝟏)

Homework for Graphs of the Other Trig Functions

Given the graph of 𝒚 = 𝟐𝒄𝒐𝒔 (𝝅

𝟐𝜽) + 𝟏, sketch the graph of 𝒚 = 𝟐𝒔𝒆𝒄 (

𝝅

𝟐𝜽) + 𝟏

Label each graph as tangent, secant, cotangent or cosecant.

Find the period and horizontal shift of each and then sketch:

𝒚 = 𝟐𝒕𝒂𝒏(𝟐𝒙 − 𝟖) 𝒚 = 𝟐𝒔𝒆𝒄 (𝝅

𝟒(𝒙 + 𝟏))

Write the formula for each graph: