Graphing Trigonometric Functions

b= π/2

2π/| π2|

|π/2|=π/2

2π/π2

DivByFracD

a/b/c= a * c/b

4π/π

a/b=.a /π./.b /π.

4/1

DivByOne

a/1=a

Therefore, the period of the function is 4. This means that every 4 seconds the curve repeats itself, which can be seen on the graph.

Discussion

The Cosine Function

Let P be the point of intersection of the unit circle and terminal side of an angle in standard position. The cosine function, denoted as cos, can be defined as the x-coordinate of the point P.

The graph of the cosine function looks as follows.

Note that for x in the interval [- 2π,0] and in the interval [0,2π] the graph looks exactly the same. This means that the cosine function is a periodic function and its period is 2π.

cos(θ+2π n)=cosθ

Here, n is any integer number. Consider the function y=acosbθ, where a and b are non-zero real numbers and θ is measured in radians. With this information, the properties of the cosine function can be defined.

Properties of y=acosbθ
Amplitude	\|a\|
Number of cycles in [0,2π]	\|b\|
Period	2π/\|b\|
Domain	All real numbers
Range	[- \|a\|,\|a\| ]

Example

Operation of Submarines' Radars

Kriz visited the port on a day when a special exhibition was taking place where scientists explained how they use a submarine in ocean exploration. They learned that radars are used to monitor objects under the sea. Even more interesting, in operating radars, sine and cosine functions are involved.

A wave signal received by a radar can be modeled by the following equation. y=3.5cos 12t Here, y is the vertical displacement from the shooting point in centimeters and t is time in seconds. What are the amplitude, period, and midline of this function? Write the period in exact form.

Hint

In the general form y=acos bθ, the amplitude is |a| and the period is 2π|b|.

Solution

First, recall the general form of a cosine function. y=acos bx This function has the following properties.

Properties of y=acosbθ
Amplitude	\|a\|
Number of cycles in [0,2π]	\|b\|
Period	2π/\|b\|

Next, examine the given function and identify the values of its coefficients a and b. y= 3.5sin 12t ⇓ a= 3.5 and b= 12 The value of a is 3.5, which means that the amplitude of the function is |3.5|=3.5 centimeters. Recall that the midline of the parent cosine function is y=0. The given function has not been translated vertically, so its midline is also y=0. These two pieces of information can be shown on a graph.

To calculate the period, substitute 12 for b into the expression 2π|b| and simplify.

2π/|b|

b= 12

2π/| 12|

|12|=12

2π/12

a/b=.a /2./.b /2.

π/6

Therefore, the period of the function is π6, which means that every π6 seconds the graph of the function repeats itself. This is illustrated in the graph.

Pop Quiz

Identifying Amplitude and Period of Sine and Cosine Functions

Consider the equation or the graph of a sine or cosine function. Identify its amplitude and period. If needed, round the answers to two decimal places.

Different graphs and equations of sine and cosine functions are shown

Discussion

Graphing the Sine and Cosine Functions

Sine and cosine functions can be graphed by closely analyzing their function rules and the graphs of their parent functions, which are y = sin x and y = cos x. As an example, consider the following function. f(x) = 0.5 cos 2x + 1 In order to graph it, there are four steps to follow.

Find the Amplitude, Period, and Translation of the Function

First, recall the general form of the cosine function and identify the value of each coefficient by comparing it with the given function. y&= acos bθ f(x)& = 0.5 cos 2x + 1 The amplitude of the cosine function is |a| and the period is 2π|b|. Therefore, the amplitude of the given function is | 0.5|=0.5. To find its period, substitute 2 for b into the corresponding expression. 2π/| 2| = π The period of the function is π. Finally, if y=0.5cos 2x is considered, 1 must be added to obtain the given function. f(x)=0.5cos 2x + 1 This means that the function is translated 1 unit upward.

Draw the Midline

The midline of the parent cosine function is y=0. However, since the considered function is translated 1 unit upward, its midline is also translated. This means that the equation of the midline is y=1.

Plot Some Key Points on the Graph

Key points ranging over at least one cycle of the function should now be plotted. These key points are the maximums, minimums, and intersections with the midline. The maximums of y = cos x occur at even multiples of π. x = 0, 2π, 4π, ... The period of f is π, which is 12 of the parent function's period 2π. Also, f has not been translated horizontally, so the maximums neither shifted to right nor to the left. Therefore, the maximums of f occur at the following x-coordinates. x = 12( 0), 12( 2π), 12( 4π), ... ⇕ x = 0, π, 2π, ... This means that the maximums of f occur at multiples of π. Since the midline is y = 1 and the amplitude is 0.5, these maximums will all have a y-value of 1 + 0.5 = 1.5. Now, plot the points on the graph with the midline.

The minimums of f are horizontally located between the maximums. x = 0.5π, 1.5π, 2.5π, ... These points have y-values of 1 - 0.5 = 0.5.

Lastly, in-between every neighboring maximum and minimum are the intersections with the midline. x = 0.25π, 0.75π, 1.25π, ... Since these points lie on the midline, their y-coordinate is 1.

Draw the Graph

By connecting the plotted points with a smooth curve and continuing it periodically in both directions, the graph of the function can finally be drawn.

Extra

Formulas for the Key Points

There are formulas for the key points such as x -intercepts, maximum value, and minimum value of a sine function of the form y= a sin bx.

	Formula
x-intercepts	(0,0), (1/2*2π/b,0), (2π/b,0)
Maximum	a>0 (1/4 * 2 π/b, a)	a<0 (3/4*2π/b, - a)
Minimum	a>0 (3/4*2π/b, - a)	a<0 (1/4 * 2 π/b, a)

Similarly, there are also formulas for the x-intercepts, maximum, and minimum of a cosine function of the form y= a cos bx.

	Formula
x-intercepts	(1/42π/b,0), (3/42π/b,0)
Maximum	a>0 (0, a), (2π/b, a )	a<0 (1/2*2π/b, - a)
Minimum	a>0 (1/2*2π/b, - a)	a<0 (0, a), (2π/b, a )

These formulas can be useful when graphing a sine or a cosine function. By using them, the first five points of a function can be plotted. Then, the function can be extended along the x-axis by imitating the found pattern.

Extra

Graphing Parent Sine and Cosine Functions

The graph of the parent sine function can be obtained by using a unit circle. Recall that the sine values are represented by the y-coordinate of a point on this circle. Therefore, as the point is rotated, its y-coordinates will be plotted on a coordinate plane.

The graph of the parent cosine function can be drawn in a similar manner. The values of cosine are represented by the x-coordinates of a point on a unit circle. Rotate the point and plot its x-coordinates with the respective θ-values on a coordinate plane.

Example

Graphing Light Waves of Different Colors

After learning how trigonometric functions are abundant in objects related to the ocean, Kriz was stoked to go to Physics class first thing Monday. There, they learned that light travels in waves and, therefore, can be modeled by sine and cosine functions. Different colors have different wavelengths, or periods, and the amplitude of the wave affects the brightness of the color.

For example, the light visible as red has the longest period, while the light visible as violet has the shortest period. Additionally, the greater the amplitude of the light wave, the brighter it looks.

a Use a sine function to graph the dimmed red light wave with a period of 660 nanometers, an amplitude of 0.3 units, and whose midline is y=0.5.

b Graph the bright violet light wave modeled by the following equation.

y=1.4cos π/200x

Answer

Hint

a First, plot the midline and then identify the locations of the maximums, minimums, and the intersections with the midline.

b Identify the values of a and b and use the fact the period is 2π|b|. Analyze the locations of maximums, minimums, and intersections with the midline of the parent function of cosine.

Solution

a The first step is to graph the midline of the function, which is said to be y=0.5.

Next, some key points, like maximums, minimums, and intersections with the midline should be plotted. The parent sine function intersects the midline at each half-period.

In this case, the period is 660, so its half-period is 6602=330 nanometers. Therefore, the x-coordinates of the intersections of the function and the midline occur at x-values that are multiples of 330. x=0, 330, 660, ... These points lie on the midline, so their y-coordinate is 0.5.

The maximums and minimums of a sine function occur once every period between two points of intersection with the midline. Analyzing the graph of the parent sine function starting from the origin, it can be seen that the maximum of the function occurs before the minimum.

Therefore, the maximum of the given function is in the middle between the intersections (0,0.5) and (330,0.5), while the minimum is in the middle between (330,0.5) and (660,0.5). By adding and subtracting the amplitude of 0.3 to the midline, the y-coordinates of the maximum and minimum, respectively, can be found. cc Maximum& Minimum [0.2cm] (330/2,0.5 + 0.3) & (330 + 330/2,0.5 - 0.3) ⇕ & ⇕ (165,0.8) & (495,0.2) Now, plot both points on the coordinate plane.

Finally, connect the points with a smooth curve and continue it periodically.

b To graph the given function, start by comparing it with the general form of a cosine function to identify the values of the coefficients.

General Form:& y= acos bx Given Funtion:& y= 1.4cos π/200x Since the value of a is 1.4, the amplitude of the function is 1.4 units. The value of b, which is π200, can be used to find the period of the function.

Period=2π/|b|

b= π/200

Period=2π/| π200|

▼

Evaluate right-hand side

|π/200|=π/200

Period=2π/π200

DivByFracD

a/b/c= a * c/b

Period=400π/π

a/b=.a /π./.b /π.

Period=400/1

DivByOne

a/1=a

Period=400

The midline of the parent cosine function is y=0. In the equation of the function, there is no value added to or subtracted from the cosine term, which means that the function is neither translated up nor down. Therefore, its midline is also y=0.

Next, the key points should be identified and plotted. Consider the parent cosine function.

As can be seen, the maximums occur at x=- 2π, 0, 2π,..., which are the multiples of its period 2π. This means that using the period of 400 of the considered function, its maximums can be found. Maximums x=0, 400, 800, ... y=1.4 Furthermore, since the equation of the midline is y=0 and the amplitude is 1.4, the y-coordinate of the maximums is 1.4. The minimums of the parent function occur at x=- π, π,..., which are the values of its half-period. In this case, the half-period of the function is 200. Minimums x=200, 600, 1000, ... y=- 1.4 The y-coordinates of the minimums are - 1.4. Lastly, in-between every neighboring maximum and minimum are the intersections with the midline. Intersections x=100, 300, 500, ... y=0 Finally, plot all the found points on the coordinate plane with the midline.

By connecting the points with a smooth curve and continuing it periodically, the graph of the given function can be obtained.

Discussion

Frequency of a Periodic Function

The frequency of a periodic function is the number of cycles in a given unit of time. The frequency of a function's graph is the reciprocal of the function's period.

Frequency=1/Period

For example, if the period of a function is 2 seconds, then the frequency is 12 cycles per second. If the period is 15 seconds, then the frequency is 5 cycles per second. More examples can be seen in the following applet.

The graph of sine functions with different periods and frequencies are shown

When frequency is calculated per second, it is measured with a unit called hertz. For instance, 10 Hz means 10 times per second.

Example

Frequencies That Animals Can Hear

Later that day, Kriz was excitingly sharing their impressions with their classmate Zain about their visit to the zoo. Kriz told Zain that they were impressed to learn that elephants can hear frequencies 20 times lower than humans, while mice can hear astronomically high frequencies, up to 70 - 80 kHz.

a Write a sine function in the form y=asin bx, with a and b as positive real numbers, that models a sound wave with a frequency of 10 Hz and an amplitude of 1 unit that elephants can hear.

b Write a cosine function in the form y=acos bx, with a and b positive real numbers, that models the sound wave with a frequency of 75 kHz and an amplitude of 0.2 units that mice can hear.

Hint

a For a sine function with the form y=asin bx, the amplitude is |a| and b can be used to find the period of the function.

b Use the formula relating the frequency and the period to calculate the period of the function. Be aware that 1 kHz is the same as 1000 Hz.

Solution

a Start by recalling the general form of a sine function.

y=asin bx Here, |a| is the amplitude while b is the coefficient used to find the period of the function. It is given that the amplitude of the function is 1 unit. Therefore, |a|=1. Because a is a positive number, it is known that a=1. y= 1sin bx ⇓ y=sin bx To find the value of b, use the fact that the function has a frequency of 10 Hz. This means that the function has 10 cycles per 1 unit of time. Now, review the formula that relates frequency and period. Frequency=1/Period Substitute 10 for frequency and solve the equation for the period.

Frequency=1/Period

Frequency= 10

10=1/Period

LHS * Period=RHS* Period

10* Period=1

DivEqn

.LHS /10.=.RHS /10.

Period=1/10

Next, use the relationship between the coefficient b and the period of the sine function. Period=2π/|b| Because b is a positive number, it is known that |b|=b. Period=2π/|b| b>0 ⟶ Period=2π/b Since the value of the period was already found, substitute it into the equation and calculate the value of b.

Period=2π/b

Period= 1/10

1/10=2π/b

CrossMult

Cross multiply

1* b = 10* 2π

Multiply

b=20π

Now that the value of b is known, the equation of the sine function describing the sound wave heard by elephants can be completed. y=sin bx ⇓ y=sin 20π x Kriz once heard in a documentary that elephants can communicate when they are miles away from other elephants by listening to vibrations that travel through the ground. Kriz realized that this function could potentially describe a low-frequency sound wave that elephants can hear when communicating miles apart!

b Similarly to Part A, first recall the general form of a cosine function.

y=acos bx This time, the amplitude is 0.2 units. Since a is a positive number, the value of a is 0.2. y= acos bx ⇓ y= 0.2cos bx It is also known that the frequency of the function is 75 kHz or 75 000 Hz. Use this value to find the period of the function.

Frequency=1/Period

Frequency= 75 000

75 000=1/Period

LHS * Period=RHS* Period

75 000* Period=1

DivEqn

.LHS /75 000.=.RHS /75 000.

Period=1/75 000

The period of a cosine function is also given by 2π|b|. Because b is positive, the period of the function can be written as 2πb. Therefore, the value of the period can now be used to calculate b.

Period=2π/b

Period= 1/75 000

1/75 000=2π/b

CrossMult

Cross multiply

1* b = 75 000* 2π

Multiply

b=150 000π

Finally, substitute the value of b and complete the equation of the cosine function that models the sound wave heard by the mice. y=0.2cos bx ⇓ y=0.2cos 150 000π x Kriz imagined a hundred different possibilities of how high-frequency sound waves help mice navigate the dark forests filled with animals out to eat them!

Explore

Investigating the Graphs of the Functions Involving Sine and Cosine

It is interesting to explore the graphs of functions that are defined by applying some basic operations, like addition, multiplication, and division, to sine and cosine functions. First, try to draw the graph of f(x)=sin x+cos x.

Now, the graph of f(x)=sin xcos x will be drawn.

Finally, by applying the same method, try to draw the graph of f(x)= sin xcos x.

What function does the obtained graph resemble?

Discussion

The Tangent Function

Let P be the point of intersection of the unit circle and terminal side of an angle in standard position. The tangent function, denoted as tan, can be defined as the ratio of the y-coordinate to the x-coordinate of the point P.

Recall that the x- and y-coordinates of the point correspond to the cosine and sine of the angle, respectively. Therefore, the tangent function can also be defined as the ratio of sin θ to cos θ.

tanθ=sinθ/cos θ

The graph of the tangent function is as follows.

The period of the tangent function is π. Since each branch comes from negative infinity towards positive infinity, the tangent function has no amplitude and its range is all real numbers. Consider the function y=atanbx, where a and b are non-zero real numbers and θ is measured in radians. The properties of the tangent function can be identified from the function rule.

Properties of y=atanbx
Amplitude	No amplitude
Interval of One Cycle	(- π/2\|b\|,π/2\|b\|)
Asymptotes	At the end of each cycle
Period	π/\|b\|
Domain	All real numbers except odd multiples of π/2\|b\|
Range	All real numbers

Method

Graphing a Tangent Function

A tangent function can be graphed by examining its function rule and determining some of its key characteristics, like period and asymptotes. Consider the following function. y=1.5tan π/2x In order to draw the graph, there are four steps to follow.

Find the Period of the Function

Start by comparing the function with the general form of a tangent function to identify the value of its coefficients. General Form:& y= atan bx Given Function:& y= 1.5tan π/2x The period of a tangent function is given by π|b|. Substitute π2 for b and calculate the period of the given function.

π/|b|

b= π/2

π/| π2|

|π/2|=π/2

π/π2

DivByFracD

a/b/c= a * c/b

2π/π

CalcQuot

Calculate quotient

Therefore, the period of the function is 2 units.

Draw the Asymptotes

Recall that one cycle of a tangent function occurs in the interval from - π2|b| to π2|b|. Again, substitute the value of b and find the interval corresponding to the considered function.

π/2|b|

b= π/2

π/2| π2|

|π/2|=π/2

π/2( π2)

DenomMultFracToNumber

2 * a/2= a

π/π

QuotOne

a/a=1

One cycle of the function occurs in the interval [- 1,1]. Furthermore, recall that the asymptotes occur at the end of each cycle of the function. Therefore, the function has asymptotes at x=- 1 and x=1, which can be drawn on a coordinate plane.

Plot Some Points on the Graph

Next, divide the period of the function into four equal parts and locate three points between the asymptotes. In this case, the period is 2, so each fourth is 0.5 units long.

Therefore, the x-coordinates of the points that will be plotted are - 12, 0, and 12. Substitute these values for x into the function rule and evaluate the corresponding y-coordinates.

y=1.5tan π2x
x-Coordinate	Substitute	Simplify	Evaluate
x= - 1/2	y=1.5tan π/2( - 1/2)	y=1.5tan (- π/4)	y=- 1.5
x= 0	y=1.5tan π/2( 0)	y=1.5tan 0	y=0
x= - 1/2	y=1.5tan π/2( 1/2)	y=1.5tan π/4	y=1.5

Now that both coordinates of the three points are known, plot them on the coordinate plane with the asymptotes.

Draw the Graph

Finally, connect the three points with a smooth curve and continue the function to the left and right keeping in mind that it should get closer to the asymptotes but will never intersect them.

Replicate the branch to obtain the graph for other intervals.

Extra

Key Characteristics of a Tangent Function

When drawing one period of a function of the form y= a tan bθ, the following characteristics of a tangent function can be used.

	Formula
x-intercept	(0,0)
Asymptotes	θ = - π/2\| b\|, θ = π/2\| b\|
Halfway Points	(- π/4 b,- a), (π/4 b, a)

Extra

Graphing the Parent Tangent Function

Recall that a tangent function can be defined as the quotient of a sine and a cosine function. Therefore, the graph of the parent tangent function can be drawn as a rational function. tan x = sin x/cos x First, draw the graphs of the sine and cosine functions.

Because the tangent function has cosine in its denominator, the asymptotes of the tangent function are located at the zeros of the cosine function.

Since the sine function is the numerator of the tangent function, the zeros of the sine function will be the zeros of the tangent function as well.

The y-values of the sine and cosine functions are equal at the intersection of both graphs. This means that the y-value of the tangent function will be 1 at the x-coordinates of these points.

Likewise, three more points can be plotted between the left asymptotes of each period and the zeros. This time, however, since the sine and cosine functions have opposite x-values, the y-value of the tangent function will be - 1.

Finally, the branches are drawn by starting from the bottom of the left asymptote and moving towards the top of the right asymptote.

Example

Modeling a Distance with a Tangent Function

After school, Kriz and Zain will meet some friends to play volleyball near Zain's home. Zain lives in a modern 300-feet building. As Kriz was approaching the building about 175 feet from its base, they saw Zain going down in the elevator and waved to them.

a Write an equation that models Zain's distance d in feet from the top of the building as a function of the angle of elevation x.

b Graph one cycle of the function from Part A ignoring the limitations of the context.

c What part of the graph makes sense considering the given context?

Answer

a d=300-175tan x

c The part in Quadrant I.

Hint

a Which trigonometric function relates the legs of a right triangle with an acute angle? Set up an equation and solve it for d.

b Use the fact that the period of a tangent function is given by π|b| and one cycle occurs in the interval from - π2|b| to π2|b|.

c Recall what x and d represent. What values cannot they have?

Solution

a Begin by analyzing the triangle formed by Kriz, Zain and the base of the building.

It is a right triangle in which the side that measures 175 feet is adjacent ∠ x. The side representing the distance between Zain and the base of the building equals 300-d and is opposite ∠ x. Therefore, the tangent ratio can be used to find a relationship between the sides and the angle. tan x=opp/adj ⇒ tan x = 300-d/175 To find the equation that models Zain's distance d from the top of the building, solve this equation for d.

tan x=300-d/175

LHS * 175=RHS* 175

175tan x=300-d

AddEqn

LHS+d=RHS+d

d+175tan x=300

SubEqn

LHS-175tan x=RHS-175tan x

d=300-175tan x

b The function found in Part A can be seen as a vertical translation of the function y=- 175 tan x by 300 units up.

d= 300-175tan x Therefore, to obtain the graph of d, first graph y=- 175 tan x and then translate it vertically. Start by comparing the function to the general form of a tangent function to identify the values of the coefficients a and b. General Form:& y= atan bx Given Function:& y= - 175 tan 1x Recall that the period of the parent tangent function is given by π|b|. Since b= 1, the period of the function can be calculated. Period π/| b| ⇒ π/| 1|=π One cycle of a tangent function occurs in the interval from - π2|b| to π2|b|. Substitute the value of b and find this interval for the given function. (- π/2| 1|,π/2| 1|) ⇔ (- π/2,π/2) Additionally, the asymptotes occur at the end of each cycle. This means that two asymptotes are x=- π2 and x= π2. Graph them on a coordinate plane.

Next, three points should be plotted between the asymptotes. By dividing the interval between - π2 and π2 into four equal parts, the x-coordinates of three points are obtained. x=- π/4, x=0, x=π/4 Substitute them into the function rule for x and find the corresponding y-coordinates.

y=- 175tan x
x-Coordinate	Substitute	Evaluate
x= - π/4	y=- 175tan ( - π/4)	y=175
x= 0	y=- 175tan 0	y=0
x= π/4	y=- 175tan ( π/4)	y=- 175

Now, plot the points on the coordinate plane with the asymptotes.

Finally, the graph of y can be drawn by connecting the points with a smooth curve such that, as x tends to - π2 and π2, the graph gets closer and closer to the asymptotes but never crosses them.

The last step is to translate the graph 300 units up vertically to obtain the graph of d.

c To determine which part of the graph makes sense in the context of the given situation, recall that d represents the distance between Zain and the top of the building, while x is the measure of the angle of elevation. In real life, the values of these two variables can only be greater than or equal to 0.

Therefore, only the part of the graph in the first quadrant makes sense considering the context.

Closure

Drawing Conclusions About the Population of Rabbits and Foxes

Earlier, it was mentioned that on the weekend Kriz went with their family to the local zoo.

Of all the great animals at the zoo, Kriz likes foxes and rabbits the most. Eager to learn more information about them, Kriz found the following table that shows the state population of rabbits and foxes from the previous year.

After analyzing the table closely, Kriz arrived at some interesting conclusions. By answering the following questions, try to determine what Kriz found.

a What type of functions can be used to model the populations of rabbits and foxes?

b Find the appropriate function to model the population of rabbits r(m) as a function of the time m in months.

c Find the appropriate function to model the population of foxes f(m) as a function of the time m in months.

d Graph both functions. One function seems to chase the other. What can be a possible explanation for this?

Answer

a Sine or cosine functions.

b r(m)=- 500sin π/6m+1250

c f(m)=50cos π/6m+200

d Graph of r(m):

Graph of f(m):

Explanation: A predator-prey relationship between foxes and rabbits.

Hint

a First, analyze the values in the table and try to find patterns. Then, plot the points and examine the shape of the curve that connects them.

b Use the graph and the values from the table to determine the amplitude, midline, and period of the function.

c The equation of the midline indicates the number of units by which the function has to be translated vertically.

d Connect the given and plotted points with a smooth curve. Analyze what happens to the population of foxes when the population of rabbits increases and decreases, and vice versa.

Solution

a Begin by analyzing the values of the rabbit population given in the table. Are there any patterns?

It can be seen that some values, like 1250, 1000, 817, 1500, and 1683, repeat in the table twice. This means that the function r(m) has some kind of local symmetries. Now, plot all the points on a coordinate plane and examine the shape of the graph.

The points remind the shape of a sine or a cosine graph. Similarly, analyze and try to find any patterns in the values of the fox population.

Again, some values in the table repeat, which implies that the function has some kind of symmetry. The number of foxes starts at a maximum value of 250, then decreases to a minimum of 150, followed by almost returning to the maximum by the end of the year. By plotting the points, the shape of the function can be seen.

Both functions have a general shape of a sine or a cosine function.

b Start by examining the shape of the graph in order to determine some of its characteristics, like midline, period, and amplitude.

On the graph almost one full cycle of the function is shown and it can be assumed that it will repeat periodically moving to the right along the x-axis. By analyzing the graph's pattern, it can be concluded that for x=12, the function will have a value of 1250, the same value as for x=0. Therefore, the function has a period of 12 months. Period 12 months Additionally, the maximum value is 1750, while the minimum is 750. By calculating the mean of these values, the amplitude can be found. Amplitude [0.2cm] 1750- 750/2=500 Next, by subtracting the value of the amplitude from the maximum, or by adding its value to the minimum, the equation of the midline can be determined. Midline y=1250 Show all of these characteristics on the graph and try to identify which function, sine or cosine, can represent it.

The above graph resembles the graph of a sine function. However, after intersecting the midline at m=0, the graph goes down to its minimum instead of going up to its maximum. This leads to the conclusion that the graph is a reflection of a sine function over the midline. General Form:& y= asin bm Reflection:& y= - asin bm In this general form, a represents the amplitude and b represents the coefficient related to the period by the following formula. Period=2π/|b| Substitute the known value of the period and calculate the value of b.

Period=2π/|b|

▼

Substitute 12 for Period and evaluate

Period= 12

12=2π/|b|

LHS * |b|=RHS* |b|

12|b|=2π

DivEqn

.LHS /12.=.RHS /12.

|b|=2π/12