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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 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 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 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.

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

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. 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

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

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.

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

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 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. 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.

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

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

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. 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 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.

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

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. 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 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. 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. 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.

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

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.