Algebra 2 Module 2 Answers

Appoint NY Eureka Math Algebra 2 Module 2 Lesson 13 Reply Key

Eureka Math Algebra 2 Module 2 Lesson 13 Instance Answer Key

Example 1.
Write a Sinusoidal Function to Model a Set of Data
a. On the besprinkle plot y'all created in the Opening Exercise, sketch the graph of a sinusoidal function that could exist used to model the data.
Reply:

b. What are the midline, amplitude, and catamenia of the graph?
Respond:
Answers will vary depending on educatee graphs merely should exist close to those listed below.
Midline: y = 1.25
Amplitude: ane.25
Period: 12

c. Estimate the horizontal distance between a bespeak on the graph that crosses the midline and the vertical centrality.
Answer:
The distance is about 12.5 hours.

d. Write a function of the grade f(ten) = A sin(ω(x – h)) + k to model these data, where x is the hours since midnight on May 21, and f(x) is the height in feet relative to the MLLW.
Answer:
f(x) = i.25 sin (\(\frac{2 \pi}{12}\)(x – 12.v)) + 1.25.

Case 2. Digital Sampling of Sound
When audio is recorded or transmitted electronically, the continuous waveform is sampled to convert it to a discrete digital sequence. If the sampling rate (represented by the horizontal scaling) or the resolution (represented by the vertical scaling) increases, the sound quality of the recording or manual improves.

The data graphed below represent 2 dissimilar samples of a pure tone. Ane is sampled viii times per unit of time at a 2-bit resolution (4 equal intervals in the vertical direction), and the other is sampled 16 times per unit of time at a four-flake resolution (8 equal intervals in the vertical direction).
a. Which sample points would produce a better model of the actual sound moving ridge? Explain your reasoning.

Respond:
You will become a better model with the sample points in Figure 2. If the office we use to model the data passes through more data points on the original sound wave, the bend we use to fit the data will accurately represent the original audio.

b. Find an equation for a function one thousand that models the pure tone sampled in Figure 1.
Answer:
g(x) = 8.6436 sin(0. 5089x – 0. 8891) + viii.3651

c. Find an equation for a function h that models the pure tone sampled in Figure two.
Answer:
h(x) = 7.8464 sin(0. 4981x – 0.7327) + seven.8760

Eureka Math Algebra ii Module 2 Lesson 13 Opening Practice Answer Key

Anyone who works on or around the ocean needs to accept information well-nigh the changing tides to conduct their concern safely and effectively. People who go to the embankment or out onto the bounding main for recreational purposes besides need information well-nigh tides when planning their trip. The tabular array below shows tide data for Montauk, NY, for May 21-22, 2014. The heights reported in the tabular array are relative to the Mean Lower Low Water (M 11W). The MLLW is the boilerplate height of the everyman tide recorded at a tide station each day during the recording menses. This reference point is used by the National Oceanic and Atmospheric Administration (NOAA) for the purposes of reporting tidal information throughout the United States. Each different tide station throughout the United States has its ain MLLW. Loftier and depression tide levels are reported relative to this number. Since it is an boilerplate, some depression tide values tin be negative. NOAA resets the MLLW values approximately every 20 years.

a. Create a scatter plot of the data with the horizontal centrality representing time since midnight on May 21 and the vertical axis representing the tiptop in anxiety relative to the MLLW.
Reply:

b. What type of office would best model this set of data? Explain your choice.
Answer:
Fifty-fifty though the maximum points do not all have the same value, a sinusoidal function would best model this data considering the data repeat the pattern of high point, low indicate, high bespeak, depression point, over fairly regular intervals of fourth dimension-roughly every six hours.

Eureka Math Algebra two Module two Lesson thirteen Practice Answer Central

Practice i.
The graph of the tides at Montauk for the week of May 21 – 28 Is shown below. How accurately does your model predict the time and height of the high tide on May 28?

Reply:
The maximum value of f is ii.v. The high tide on May 28 was almost 3 feet in a higher place MLLW. The model is off by approximately 6 inches.

Example 2: Digital Sampling of Sound When audio Is recorded or transmitted electronically, the continuous waveform Is sampled to convert it to a discrete digital sequence. If the sampling rate (represented by the horizontal scaling) or the resolution (represented by the vertical scaling) Increases, the sound quality of the recording or transmission improves.

The data graphed below correspond two different samples of a pure tone. One is sampled 8 times per unit of measurement of time at a two-fleck resolution (iv equal intervals in the vertical management), and the other is sampled sixteen times per unit of measurement of time at a 4-bit resolution (8 equal intervals in the vertical direction).
a. Which sample points would produce a better model of the bodily sound moving ridge? Explain your reasoning.

Reply:
Yous will go a amend model with the sample points in Effigy 2. If the function we use to model the data passes through more data points on the original audio wave, the curve we apply to fit the data will accurately represent the original sound.

b. Find an equation for a function thousand that models the pure tone sampled in Figure 1.
Answer:
yard(x) = 8.6436 sin(0. 5089x – 0.8891) + 8.3651

c. Find an equation for a function h that models the pure tone sampled in Figure two.
Answer:
h(10) = vii.8464 sin(0. 4981x – 0.7327) + vii.8760

Exercise 2 – 6
Stock prices have historically increased over time, but they also vary in a cyclical manner. Create a scatter plot of the data for the monthly stock cost for a fifteen-month fourth dimension menstruum since January one, 2003.

Months Since Jan. 1, 2003	Cost at close in dollars
0	xx.24
1	nineteen.42
2	18.25
3	nineteen.13
4	19.twenty
v	20.91
6	xx.86
vii	twenty.04
8	20.xxx
9	xx.54
10	21.94
11	21.87
12	21.51
xiii	xx.65
14	19.84

Practise 2.
Would a sinusoidal function be an advisable model for these data? Explicate your reasoning.
Reply:
A sinusoidal role would not be advisable because the information are trending upward as time passes.

We tin can model the slight upward tendency In these data with the linear role f(10) = 19. five + 0.13x.
If we add a sinusoidal function to this linear function, we can model these information with an algebraic part that displays an upward trend merely also varies in a cyclical style.

Exercise iii.
Find a sinusoidal function, yard, that when added to the linear part, f, models these information.
Answer:
g(10) = i. 1 sin (\(\frac{ii \pi}{5.5}\)(x – four)) (Student answers may vary.)

Exercise 4.
Let S exist the stock price function that Is the sum of the linear function listed above and the sinusoidal function in Exercise 3.
Answer:
S(x) = 19.5 + 0.13x + 1.i sin (\(\frac{2 \pi}{5.5}\)(x – 4))

Practice v.
Add together the graph of this function to your besprinkle plot. How well does it appear to fit the data?
Reply:
The scatter plot with the function S is shown below for the first 15 months. In general, it appears to model these data, but there are a few outliers such as the stock value ii months after data were reported.

Exercise 6.
Here is a graph of the same stock through December 2009.

a. Will your model exercise a good job of predicting stock values by the year 2005?
Answer:
Information technology will non practice a very good task of predicting the value of the stock because it rises sharply and and so plummets before starting to rise in toll afterwards a low in 2009.

b. What event occurred in 2008 to account for the sharp decline in the value of stocks?
Answer:
There was a fiscal crisis In the stock markets that marked the start of a recession.

c. What are the limitations of using any function to make predictions regarding the value of a stock at some bespeak in the future?
Answer:
There are many variables that can affect the toll of a stock. A office that is based just on time volition exist express in its power to predict futurity events during unforeseen circumstances.

Eureka Math Algebra 2 Module ii Lesson 13 Trouble Set Answer Key

Question 1.
Find equations of both a sine function and a cosine office that could each represent the graph given below.

Answer:
y = 2 sin(2x)
y = 2 cos(2 (ten + π))

Question 2.
Detect equations of both a sine function and a cosine function that could each represent the graph given beneath.

Reply:
y = 0.5 cos(3x) + 2
y = 0.five sin(3 (10 – \(\frac{\pi}{2}\))) + 2

Question 3.
Rapidly vibrating objects ship pressure waves through the air that are detected by our ears and then interpreted by our brains equally sound. Our brains analyze the amplitude and frequency of these force per unit area waves.

A speaker usually consists of a paper cone attached to an electromagnet. By sending an oscillating electric electric current through the electromagnet, the paper cone can be made to vibrate. Past adjusting the current, the amplitude and frequency of vibrations tin exist controlled.

The following graph shows the pressure intensity (I) as a office of time (x), In seconds, of the pressure waves emitted by a speaker gear up to produce a unmarried pure tone.

a. Does Information technology seem more than natural to use a sine or a cosine role to fit to this graph?
Reply:
Either a sine or a cosine function could be used, but since the graph passes through the origin, It is natural to use a sine function.

b. Find the equation of a trigonometric function that fits this graph.
Respond:
Reading from the graph, we have A = i, h = 0, k = 0, and 2p 0.0045 (because the graph has an 10-Intercept at approximately 0. 0045 after two full periods). Then, p ≈ 0.00225, and so ω = \(\frac{2 \pi}{p}\) gives west ≈ 800π. Thus, an equation that models the graph presented here is 1(ten) = sin(800πx).

Question iv.
Suppose that the following tabular array represents the average monthly ambient air temperature, in degrees Fahrenheit, in some subterranean caverns in southeast Australia for each of the twelve months in a year. We wish to model these data with a trigonometric function. (Observe that the seasons are reversed in the Southern Hemisphere, and so January is in summer, and July is in winter.)
a. Does information technology seem reasonable to presume that these data, if extended beyond one twelvemonth, should be roughly periodic?
Reply:
Yes

b. What seems to exist the amplitude of the data?
Answer:
There are a number of ways to gauge the amplitude. We can simply accept half of the difference of the highest and everyman temperatures, or we tin can take half of the difference of the highest temperature and the one six months later on (or half of the difference of the lowest temperature and the one that is half-dozen months earlier). The highest temperature is 64. 22°F; the lowest temperature is 49.82°F. Using these temperatures, nosotros approximate the amplitude to be \(\frac{64.22^{\circ} \mathrm{F}-49.10^{\circ} \mathrm{F}}{2}\) = vii. 56°F.

c. What seems to be the midline of the data (equation of the line through the middle of the graph)?
Answer:
The average of 64. 22°F and 49.x°F 6 months later is 56.66. Thus, the midline would bey = 56.66; so,
we have 1000 = 56.66.

d. Would it be easier to utilise sine or cosine to model these data?
Answer:
Cosine would exist easier because we can pinpoint a peak of the function more hands than where it should cross the midline.

e. What is a reasonable approximation for the horizontal shift?
Reply:
It appears that the meridian of the function would be between January and February, so let's utilise h = 1.5 for the horizontal shift. (This assumes that nosotros accept labeled January as calendar month 1 and February as month two.)

f. Write an equation for a office that could fit these data.
Respond:
F(x) = 7.56 cos(\(\frac{2 \pi}{12}\)(10 – 1.v)) + 56.66

Question v.
The table below provides data for the number of daylight hours every bit a function of day of the year, where twenty-four hour period 1 represents January 1. Graph the data and determine if they could exist represented by a sinusoidal role. If they can, determine the menses, amplitude, and midline of the function, and detect an equation for a function that models the data.

Reply:
The data appear sinusoidal, and the easiest function to model them with ¡south a cosine office. The period would be 365 days, so the frequency would be ω = \(\frac{2 \pi}{365}\) ≈ 0.017. The amplitude is \(\frac{1}{two}\)(xx. five – three. vi) = 8.45, and the midline is y = m, where yard = \(\frac{1}{2}\)(20.five + 3.6) = 12.05. We want the highest value at the peak, so the horizontal shift is 175.
H = 8.5 cos(0. 017t – 175) + 12.05.

Question vi.
The function graphed below is y = ten^sin(x). Blake says, "The office repeats on a fixed interval, and so it must be a sinusoidal function." Answer to his argument.

Answer:
While the equation for the function includes a sine function inside it, the part itself is not a sinusoidal function. It does non have a constant amplitude or midline, though it does appear to get zippo at fixed increments; information technology does non have a catamenia because the function values practice non repeat. Thus, it is not a periodic office and is non sinusoidal.

Eureka Math Algebra ii Module 2 Lesson 13 Exit Ticket Answer Key

Tidal data for New Canal Station, located on the shore of Lake Pontchartrain, LA, and Lake Charles, LA, are shown below.

Question one.
Would a sinusoidal part of the form f(x) = A sin(ω(x – h)) + chiliad be appropriate to model the given data for each location? Explain your reasoning.
Answer:
The data for New Culvert Station could be modeled with a sinusoidal role, but the other data would non work very well since in that location appear to be ii different depression tide values that vary by approximately afoot.

Question ii.
Write a sinusoidal function to model the information for New Canal Station.
Reply:
The amplitude is \(\frac{(0.53-0.115)}{2}\) = 0. 2075. The flow appears to be approximately 24. five hours. The value of k that determines the midline is \(\frac{0.53+0.115}{ii}\) = 0.3225. Since the graph starts at its lowest value at 7:22 a.m., utilise a negative cosine function with a horizontal shift of approximately seven.4.
f(10) = – 0.2075 cos(\(\frac{2 \pi}{24.5}\)(ten – seven.four)) + 0. 3225

Algebra 2 Module 2 Answers,

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