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Many real-life problems can be solved by applying trigonometric functions, such as sine, cosine, and tangent. This lesson will define these functions, show how to graph them using their function rules, and explore some of their different real-life applications.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Modeling the Population of Rabbits and Foxes

On the weekend, Kriz and their family headed to the local zoo. Kriz really loves learning about animals, so they were sure to stop at all the cool information boards that teach interesting facts about them.

Zoo.jpg

Of all the animals, Kriz likes foxes and rabbits the most. They were dying to learn more about them and discovered a table showing the state population of rabbits and foxes during the previous year.
The table on the zoo website contains the following information about the population of rabbits and foxes: m=0, r=1250, f=250; m=1, r=1000, f=243; m=2, r=817, f=225; m=3, r=750, f=200; m=4, r=817, f=175; m=5, r=1000, f=157; m=6, r=1250, f=150; m=7, r=1500, f=157; m=8, r=1683, f=175; m=9, r=1750, f=200; m=10, r=1683, f=225; m=11, r=1500, f=243
Analyzing the table more closely, Kriz arrived at some interesting conclusions. By answering the following questions, try to determine what Kriz discovered.
a What type of functions can be used to model the populations of rabbits and foxes?
b Find the appropriate function that models the population of rabbits as a function of the time in months.
c Find the appropriate function that models the population of foxes as a function of the time in months.
d Graph both functions. One function seems to chase the other. What can be a possible explanation for this?
Discussion

Trigonometric Functions

Trigonometric functions are functions that relate an input, which represents an acute angle of a right triangle, to a trigonometric ratio of two of the triangle's side lengths. The angle is usually measured in radians.
Unit Circle Trig Ratios
Trigonometric functions are also defined for angles that are not acute by using the appropriate reference angle
For input values that are not between and the value of the coterminal angle that belongs to this interval is used instead. Therefore, trigonometric functions are periodic functions. Because of their close relation with the unit circle, trigonometric functions are also called circular functions. The domain of the sine and cosine functions is the set of all real numbers. Their range is the interval that goes from to
Trigonometric Function Domain Range
All real numbers
All real numbers

The tangent, cotangent, secant, and cosecant functions are defined as rational functions that involve the sine and cosine functions. The domain of each function does not include values that would make their denominator zero.

Trigonometric Function Ratio Domain Range
Real numbers except odd multiples of All real numbers
Real numbers except multiples of All real numbers
Real numbers except odd multiples of
Real numbers except multiples of

Now, one of the main trigonometric functions, the sine function, will be defined and examined more closely.

Concept

The Sine Function

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

Unit Circle
The graph of the sine function is called a sine curve.
The graph of the sine function y=sin(x) over the domain [-2pi,2pi].

Note that for in the interval and in the interval the graph looks exactly the same. This means that the sine function is a periodic function and its period is

Here, is any integer number. Consider the function where and are non-zero real numbers and is measured in radians. With this information, the properties of the sine function can be defined.

Properties of
Amplitude
Number of cycles in
Period
Domain All real numbers
Range
Example

Vertical Displacement of a Buoy

Kriz is interested in a maritime topic and wants to become a sailor one day. Kriz often goes sailing with their father and loves watching how waves crash on the shore, how buoys bob up and down as waves go past, and how the sun slowly melts into the water on the horizon.
A buoy bobs up and down on the waves in the sea
Kriz was very surprised when they learned in a math lesson that the vertical displacement of the buoys with respect to the sea level at the nearest beach can be modeled by a trigonometric function.
Here, is the vertical displacement in feet and the time in seconds. What are the amplitude, period, and midline of this function?

Hint

In the general form the amplitude is and the period is

Solution

Start by recalling the general form of the sine function.
This function has the following properties.
Properties of
Amplitude
Number of cycles in
Period
Now, analyze the given function and identify the values of the coefficients and
Since the value of is the amplitude of the function is feet. Recall that the midline of the parent sine function is the horizontal line Since the given function has not been translated up or down, its midline is also
The midline and amplitude of the function
To calculate the period, substitute for into the expression and evaluate.
Therefore, the period of the function is This means that every seconds the curve repeats itself, which can be seen on the graph.
The period of the function
Discussion

The Cosine Function

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

Unit Circle
The graph of the cosine function looks as follows.
The graph of cosine function (function equation) that has the domain:(-2pi,2pi)

Note that for in the interval and in the interval the graph looks exactly the same. This means that the cosine function is a periodic function and its period is

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

Properties of
Amplitude
Number of cycles in
Period
Domain All real numbers
Range
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 submarine and fish are under the sea
External credits: @freepik, @upklyak
A wave signal received by a radar can be modeled by the following equation.
Here, is the vertical displacement from the shooting point in centimeters and 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 the amplitude is and the period is

Solution

First, recall the general form of a cosine function.
This function has the following properties.
Properties of
Amplitude
Number of cycles in
Period
Next, examine the given function and identify the values of its coefficients and
The value of is which means that the amplitude of the function is centimeters. Recall that the midline of the parent cosine function is The given function has not been translated vertically, so its midline is also These two pieces of information can be shown on a graph.
The midline and amplitude of the function
To calculate the period, substitute for into the expression and simplify.
Therefore, the period of the function is which means that every seconds the graph of the function repeats itself. This is illustrated in the graph.
The period of the function
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 and As an example, consider the following function.
In order to graph it, there are four steps to follow.
1
Find the Amplitude, Period, and Translation of the Function
expand_more
First, recall the general form of the cosine function and identify the value of each coefficient by comparing it with the given function.
The amplitude of the cosine function is and the period is Therefore, the amplitude of the given function is To find its period, substitute for into the corresponding expression.
The period of the function is Finally, if is considered, must be added to obtain the given function.
This means that the function is translated unit upward.
2
Draw the Midline
expand_more

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

The midline of y=1 on a coordinate plane
3
Plot Some Key Points on the Graph
expand_more
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 occur at even multiples of
The period of is which is of the parent function's period Also, has not been translated horizontally, so the maximums neither shifted to right nor to the left. Therefore, the maximums of occur at the following coordinates.
This means that the maximums of occur at multiples of Since the midline is and the amplitude is these maximums will all have a value of Now, plot the points on the graph with the midline.
The maximums (0,1.5), (pi,1.5), (2pi,1.5) are plotted on a coordinate plane
The minimums of are horizontally located between the maximums.
These points have values of
The minimums (0.5pi,0.5), (1.5pi,0.5) plotted on the coordinate plane
Lastly, in-between every neighboring maximum and minimum are the intersections with the midline.
Since these points lie on the midline, their coordinate is
The intersections with the midline (0.25pi,1), (0.75pi,1), (1.25pi,1) are plotted on the coordinate plane
4
Draw the Graph
expand_more

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.

The graph of the function f

Extra

Formulas for the Key Points

There are formulas for the key points such as intercepts, maximum value, and minimum value of a sine function of the form

Formula
intercepts
Maximum


Minimum



Similarly, there are also formulas for the intercepts, maximum, and minimum of a cosine function of the form

Formula
intercepts
Maximum


Minimum


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 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 coordinate of a point on this circle. Therefore, as the point is rotated, its coordinates will be plotted on a coordinate plane.
Sine function is plotted as a point on a unit circle is rotated
The graph of the parent cosine function can be drawn in a similar manner. The values of cosine are represented by the coordinates of a point on a unit circle. Rotate the point and plot its coordinates with the respective values on a coordinate plane.
Cosine function is plotted as a point on a unit circle is rotated
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.

Light waves of different colors

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 nanometers, an amplitude of units, and whose midline is
b Graph the bright violet light wave modeled by the following equation.

Answer

a
The graph of the function
b
The graph of the function

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 and and use the fact the period is 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
The midline at y=0.5 is graphed on a coordinate plane

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.

The interceptions with the midline y=0 of the function y=sin(x)
In this case, the period is so its half-period is nanometers. Therefore, the coordinates of the intersections of the function and the midline occur at values that are multiples of
These points lie on the midline, so their coordinate is
The intersections with the midline are plotted

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.

The maximum and minimum of the function y=sin(x)
Therefore, the maximum of the given function is in the middle between the intersections and while the minimum is in the middle between and By adding and subtracting the amplitude of to the midline, the coordinates of the maximum and minimum, respectively, can be found.
Now, plot both points on the coordinate plane.
The maximum and minimum of the function are plotted

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

The graph of the function
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.
Since the value of is the amplitude of the function is units. The value of which is can be used to find the period of the function.
Evaluate right-hand side
The midline of the parent cosine function is 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
The midline of the function located at y=0 is graphed on a coordinate plane

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

The graph of cosine function y=cos(x)
As can be seen, the maximums occur at which are the multiples of its period This means that using the period of of the considered function, its maximums can be found.
Furthermore, since the equation of the midline is and the amplitude is the coordinate of the maximums is The minimums of the parent function occur at which are the values of its half-period. In this case, the half-period of the function is
The coordinates of the minimums are Lastly, in-between every neighboring maximum and minimum are the intersections with the midline.
Finally, plot all the found points on the coordinate plane with the midline.
The maximums, minimums, and intersections with the midline are plotted

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

The graph of the function
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.

For example, if the period of a function is seconds, then the frequency is cycles per second. If the period is seconds, then the frequency is 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, Hz means 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 times lower than humans, while mice can hear astronomically high frequencies, up to - kHz.

An elephant and a mouse with sound ways of low and high frequencies
a Write a sine function in the form with and as positive real numbers, that models a sound wave with a frequency of Hz and an amplitude of unit that elephants can hear.
b Write a cosine function in the form with and positive real numbers, that models the sound wave with a frequency of kHz and an amplitude of units that mice can hear.

Hint

a For a sine function with the form the amplitude is and 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 is the same as

Solution

a Start by recalling the general form of a sine function.
Here, is the amplitude while is the coefficient used to find the period of the function. It is given that the amplitude of the function is unit. Therefore, Because is a positive number, it is known that
To find the value of use the fact that the function has a frequency of Hz. This means that the function has cycles per unit of time. Now, review the formula that relates frequency and period.
Substitute for frequency and solve the equation for the period.
Next, use the relationship between the coefficient and the period of the sine function.
Because is a positive number, it is known that
Since the value of the period was already found, substitute it into the equation and calculate the value of
Now that the value of is known, the equation of the sine function describing the sound wave heard by elephants can be completed.
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.
This time, the amplitude is units. Since is a positive number, the value of is
It is also known that the frequency of the function is kHz or Hz. Use this value to find the period of the function.
The period of a cosine function is also given by Because is positive, the period of the function can be written as Therefore, the value of the period can now be used to calculate
Finally, substitute the value of and complete the equation of the cosine function that models the sound wave heard by the mice.
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
Graphing the function sin(x)+cos(x)
Now, the graph of will be drawn.
Graphing the function sin(x)cos(x)
Finally, by applying the same method, try to draw the graph of
Graphing the function sin(x)/cos(x)
What function does the obtained graph resemble?
Discussion

The Tangent Function

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

Unit Circle
Recall that the and 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 to

The graph of the tangent function is as follows.

The graph of the tangent function y=tan(x) with asymptotes at -3pi/2, -pi/2, pi/2, 3pi/2, and 5pi/2 over the domain [-3pi/2,5pi/2]

The period of the tangent function is 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 where and are non-zero real numbers and is measured in radians. The properties of the tangent function can be identified from the function rule.

Properties of
Amplitude No amplitude
Interval of One Cycle
Asymptotes At the end of each cycle
Period
Domain All real numbers except odd multiples of
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.
In order to draw the graph, there are four steps to follow.
1
Find the Period of the Function
expand_more
Start by comparing the function with the general form of a tangent function to identify the value of its coefficients.
The period of a tangent function is given by Substitute for and calculate the period of the given function.
Therefore, the period of the function is units.
2
Draw the Asymptotes
expand_more
Recall that one cycle of a tangent function occurs in the interval from to Again, substitute the value of and find the interval corresponding to the considered function.
One cycle of the function occurs in the interval Furthermore, recall that the asymptotes occur at the end of each cycle of the function. Therefore, the function has asymptotes at and which can be drawn on a coordinate plane.
The asymptotes y=-1 and y=1 on a coordinate plane
3
Plot Some Points on the Graph
expand_more

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

The period is divided into fourths

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

Coordinate Substitute Simplify Evaluate

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

Three points (-0.5,-1.5), (0,0), and (0.5,1.5) are plotted on the graph
4
Draw the Graph
expand_more

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.

The graph of the function for the cycle between -1 and 1

Replicate the branch to obtain the graph for other intervals.

The graph of the function

Extra

Key Characteristics of a Tangent Function

When drawing one period of a function of the form the following characteristics of a tangent function can be used.

Formula
intercept
Asymptotes
Halfway Points

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.
First, draw the graphs of the sine and cosine functions.
Sine and cosine functions graphs

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

Asymptotes are graphed 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.

Sine and Cosine

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

The second point of the tangent function is plotted for each cycle

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 values, the value of the tangent function will be

The third point of the tangent function is plotted
Finally, the branches are drawn by starting from the bottom of the left asymptote and moving towards the top of the right asymptote.
The graph of the parent tangent function is drawn
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 feet building. As Kriz was approaching the building about feet from its base, they saw Zain going down in the elevator and waved to them.

Kriz on the way to Zain seeing Zain in the elevator; city skyline in on the background
External credits: @upklyak
a Write an equation that models Zain's distance in feet from the top of the building as a function of the angle of elevation
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
b
The graph of d
c The part in Quadrant I.
The graph of d restricted to the first quadrant

Hint

a Which trigonometric function relates the legs of a right triangle with an acute angle? Set up an equation and solve it for
b Use the fact that the period of a tangent function is given by and one cycle occurs in the interval from to
c Recall what and represent. What values cannot they have?

Solution

a Begin by analyzing the triangle formed by Kriz, Zain and the base of the building.
Kriz on the way to Zain seeing Zain in the elevator; city skyline in on the background
External credits: @upklyak
It is a right triangle in which the side that measures feet is adjacent The side representing the distance between Zain and the base of the building equals and is opposite Therefore, the tangent ratio can be used to find a relationship between the sides and the angle.
To find the equation that models Zain's distance from the top of the building, solve this equation for
b The function found in Part A can be seen as a vertical translation of the function by units up.
Therefore, to obtain the graph of first graph 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 and
Recall that the period of the parent tangent function is given by Since the period of the function can be calculated.
One cycle of a tangent function occurs in the interval from to Substitute the value of and find this interval for the given function.
Additionally, the asymptotes occur at the end of each cycle. This means that two asymptotes are and Graph them on a coordinate plane.
The asymptotes of the function y
Next, three points should be plotted between the asymptotes. By dividing the interval between and into four equal parts, the coordinates of three points are obtained.
Substitute them into the function rule for and find the corresponding coordinates.
Coordinate Substitute Evaluate

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

The three points (175,-1), (0,0), (1,-175) plotted on the coordinate plane

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

The graph of y

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

The graph of d
c To determine which part of the graph makes sense in the context of the given situation, recall that represents the distance between Zain and the top of the building, while 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
The graph of d restricted to the first quadrant

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.

Kriz with their sister and mom at the zoo entrance where they can see a giraffe and an elephant
External credits: @storyset
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.
The table on the zoo website contains the following information about the population of rabbits and foxes: m=0, r=1250, f=250; m=1, r=1000, f=243; m=2, r=817, f=225; m=3, r=750, f=200; m=4, r=817, f=175; m=5, r=1000, f=157; m=6, r=1250, f=150; m=7, r=1500, f=157; m=8, r=1683, f=175; m=9, r=1750, f=200; m=10, r=1683, f=225; m=11, r=1500, f=243
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 as a function of the time in months.
c Find the appropriate function to model the population of foxes as a function of the time 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
c
d Graph of :
The graph of the function r

Graph of :

The graph of f

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?
The table on the zoo website contains the following information about the population of rabbits and foxes: m=0, r=1250, f=250; m=1, r=1000, f=243; m=2, r=817, f=225; m=3, r=750, f=200; m=4, r=817, f=175; m=5, r=1000, f=157; m=6, r=1250, f=150; m=7, r=1500, f=157; m=8, r=1683, f=175; m=9, r=1750, f=200; m=10, r=1683, f=225; m=11, r=1500, f=243
It can be seen that some values, like and repeat in the table twice. This means that the function has some kind of local symmetries. Now, plot all the points on a coordinate plane and examine the shape of the graph.
Points (m,r) where m is the month and r is the rabbits population at that month are plotted on a coordinate plane
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.
The table on the zoo website contains the following information about the population of rabbits and foxes: m=0, r=1250, f=250; m=1, r=1000, f=243; m=2, r=817, f=225; m=3, r=750, f=200; m=4, r=817, f=175; m=5, r=1000, f=157; m=6, r=1250, f=150; m=7, r=1500, f=157; m=8, r=1683, f=175; m=9, r=1750, f=200; m=10, r=1683, f=225; m=11, r=1500, f=243
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 then decreases to a minimum of 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.
Points (m,f) where m is the month and f is the foxes population at that month are plotted on a coordinate plane

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.
Points (m,r) where m is the month and r is the rabbits population at that month are plotted on a coordinate plane
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 axis. By analyzing the graph's pattern, it can be concluded that for the function will have a value of the same value as for Therefore, the function has a period of months.
Additionally, the maximum value is while the minimum is By calculating the mean of these values, the amplitude can be found.
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.
Show all of these characteristics on the graph and try to identify which function, sine or cosine, can represent it.
Midline, amplitude, and period of the function are shown on the graph
The above graph resembles the graph of a sine function. However, after intersecting the midline at 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.
In this general form, represents the amplitude and represents the coefficient related to the period by the following formula.
Substitute the known value of the period and calculate the value of
Substitute for and evaluate
Note that coefficients and can have both positive and negative values. For simplicity, only the positive values will be considered.
Now that the values of the both coefficients are known, substitute them into the equation.
Furthermore, the midline at indicates that the function has been translated units up. This fact can be shown in the function rule by adding to the right-hand side.
The function that models the population of rabbits during the previous year was obtained using the graph and the values from the table, making it possible to determine the needed characteristics.
c Similar to Part B, the function that models the population of foxes can be written. Again, begin by analyzing the shape of the graph and try to draw some conclusions about it.
Points (m,f) where m is the month and f is the foxes population at that month are plotted on a coordinate plane
Since almost one full cycle of the function is shown, it can be assumed that it will repeat periodically moving to the right along the axis. Also, it can be concluded that for the function will have the value the same value as for This means that the function has a period of months.
The maximum value is while the minimum is Calculate the mean of these values to find the amplitude of the function.
Now, 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 written.
Show all of these characteristics on the graph and try to identify which function, sine or cosine, can represent it.
Midline, amplitude, and period of the function are shown on the graph
The diagram reminds the graph of a cosine function. Recall the general form of a cosine function.
Here, represents the amplitude and represents the coefficient related to the period by the following formula.
Since for this function period is also months, the value of is Taking into account only the positive values of and substitute them into the equation.
Furthermore, the midline at indicates that the function has been translated units up. This fact can be shown in the function rule by adding to the right-hand side.
The function that models the population of foxes during the previous year was obtained using the graph and the values from the table, making it possible to determine the needed characteristics.
d To graph connect the given points plotted in Part A with a smooth curve.
The graph of the function r

Similarly, by connecting the points given for the function its graph can be drawn.

The graph of f
The fox population seems to be behind the rabbit population by three months. In other words, the fox population chases the rabbit population. This can be explained by the predator-prey relationship that exists between foxes and rabbits, as rabbits are a source of food for foxes. Where there is a big population of rabbits, the fox population grows.
At the same time, when there are a lot of foxes, more rabbits are being hunted, so the rabbit population decreases.
When the rabbit population gets quite small, foxes have a limited food supply and, therefore, their population decreases.
Once the fox population is small enough, the rabbit population can recover, and this cycle continues.


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