Viewing Angle to an Object Suppose that a person whose eyes are h1 feet from the ground

Concept Check Which one of the following statements is true about the graph of y = 4 sin 2x? A. It has amplitude 2 and period p 2 . B. It has amplitude 4 and period p. C. Its range is 30, 44. D. Its range is 3-4, 04.

Concept Check Which one of the following statements is false about the graph of y = -3 cos 1 2 x? A. Its range is 3-3, 34. B. Its domain is 1-, 2. C. Its amplitude is 3, and its period is 4p. D. Its amplitude is -3, and its period is p.

Concept Check Which of the basic circular functions can have y-value 12 ?

Concept Check Which of the basic circular functions can have y-value 2?

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = 2 sin x

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = tan 3x

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = - 12 cos 3x

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = 2 sin 5x

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = 1 + 2 sin14x

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = 3 - 14 cos 23x

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = 3 cos ax + p2 b

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = -sin ax - 3p4 b

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = 12 csc a2x - p4 b

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = 2 sec1px - 2p2

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = 13 tan a3x - p3 b

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. y = cot ax2+3p4 b

Identify the circular function that satisfies each description period is p; x-intercepts have x-values of the form np, where n is any integer

Identify the circular function that satisfies each description period is 2p; graph passes through the origin

Identify the circular function that satisfies each description period is 2p; graph passes through the point Ap2 , 0B

Identify the circular function that satisfies each description period is 2p; domain is 5x ' x np, where n is any integer6

Identify the circular function that satisfies each description period is p; function is decreasing on the interval 10, p2

Identify the circular function that satisfies each description period is 2p; has vertical asymptotes of the form x = 12n + 12 p2 , where n is any integer

Provide a short explanation Suppose that defines a sine function with period 10 and 152 = 3. Explain why 1252 = 3.

Provide a short explanation Suppose that defines a sine function with period p and A 6p 5 B = 1. Explain why A - 4p 5 B = 1.

Graph each function over a one-period interval. y = 3 sin x

Graph each function over a one-period interval. y = 12 sec x

Graph each function over a one-period interval. . y = -tan x

Graph each function over a one-period interval. y = -2 cos x

Graph each function over a one-period interval. y = 2 + cot x

Graph each function over a one-period interval. y = -1 + csc x

Graph each function over a one-period interval. y = sin 2x

Graph each function over a one-period interval. . y = tan 3x

Graph each function over a one-period interval. y = 3 cos 2x

Graph each function over a one-period interval. y = 12 cot 3x

Graph each function over a one-period interval. y = cos ax - p4 b

Graph each function over a one-period interval. y = tan ax - p2 b

Graph each function over a one-period interval. y = sec a2x + p3 b

Graph each function over a one-period interval. y = sin a3x + p2 b

Graph each function over a one-period interval. y = 1 + 2 cos 3x

Graph each function over a one-period interval. y = -1 - 3 sin 2x

Graph each function over a one-period interval. y = 2 sin px

Graph each function over a one-period interval. y = - 12 cos1px - p2

A set of temperature data (in F) is given for a particular location. (Source: www.weatherbase.com) (a) Plot the data over a two-year interval. (b) Use sine regression to determine a model for the two-year interval. Let x = 1 represent January of the frst year. (c) Graph the equation from part (b) together with the data on the same coordinate axes. Average Monthly Temperature, Auckland, New Zealand Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec 67.6 68.5 65.8 61.3 57.2 53.2 51.6 52.9 55.4 58.1 61.2 64.9

A set of temperature data (in F) is given for a particular location. (Source: www.weatherbase.com) (a) Plot the data over a two-year interval. (b) Use sine regression to determine a model for the two-year interval. Let x = 1 represent January of the frst year. (c) Graph the equation from part (b) together with the data on the same coordinate axes. Average Low Temperature, Auckland, New Zealand Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Dec 60.8 61.7 58.8 54.9 51.1 47.1 45.5 46.8 49.5 52.2 55.0 58.8

Determine the simplest form of an equation for each graph. Choose b 7 0, and include no phase shifts.

Determine the simplest form of an equation for each graph. Choose b 7 0, and include no phase shifts.

Determine the simplest form of an equation for each graph. Choose b 7 0, and include no phase shifts.

Determine the simplest form of an equation for each graph. Choose b 7 0, and include no phase shifts.

Viewing Angle to an Object Suppose that a person whose eyes are h1 feet from the ground is standing d feet from an object h2 feet tall, where h2 7 h1. Let u be the angle of elevation to the top of the object. See the figure. (a) Show that d = 1h2 - h12 cot u. (b) Let h2 = 55 and h1 = 5. Graph d for the interval 0 6 u p 2 .

(Modeling) Tides The figure shows a function that models the tides in feet at Clearwater Beach, Florida, x hours after midnight. (Source: Pentcheff, D., WWW Tide and Current Predictor.) (a) Find the time between high tides. (b) What is the difference in water levels between high tide and low tide? (c) The tides can be modeled by 1x2 = 0.6 cos30.5111x - 2.424 + 2. Estimate the tides, to the nearest hundredth, when x = 10.

(Modeling) Maximum Temperatures The maximum afternoon temperature (in F) in a given city can be modeled by t = 60 - 30 cos xp 6 , where t represents the maximum afternoon temperature in month x, with x = 0 representing January, x = 1 representing February, and so on. Find the maximum afternoon temperature, to the nearest degree, for each month. (a) January (b) April (c) May (d) June (e) August (f) October

(Modeling) Average Monthly Temperature The average monthly temperature (in F) in Chicago, Illinois, is shown in the table. (a) Plot the average monthly temperature over a two-year period. Let x = 1 correspond to January of the frst year. (b) To model the data, determine a function of the form 1x2 = a sin3b1x - d24 + c, where a, b, c, and d are constants. (c) Graph together with the data on the same coordinate axes. How well does model the data? (d) Use the sine regression capability of a graphing calculator to fnd the equation of a sine curve of the form y = a sin1bx + c) + d that fts these data.

(Modeling) Pollution Trends The amount of pollution in the air is lower after heavy spring rains and higher after periods of little rain. In addition to this seasonal fluctuation, the long-term trend is upward. An idealized graph of this situation is shown in the figure. 0 0 600 6 Circular functions can be used to model the fluctuating part of the pollution levels. Powers of the number e (e is the base of the natural logarithm; e 2.718282) can be used to model long-term growth. The pollution level in a certain area might be given by y = 711 - cos 2px21x + 102 + 100e0.2x , where x is the time in years, with x = 0 representing January 1 of the base year. July 1 of the same year would be represented by x = 0.5, October 1 of the following year would be represented by x = 1.75, and so on. Find the pollution levels on each date. (a) January 1, base year (See the screen.) (b) July 1, base year (c) January 1, following year (d) July 1, following year

(Modeling) Lynx and Hare Populations The figure shows the populations of lynx and hares in Canada for the years 18471903. The hares are food for the lynx. An increase in hare population causes an increase in lynx population some time later. The increasing lynx population then causes a decline in hare population. The two graphs have the same period. Year Canadian Lynx and Hare Populations Number 150,000 100,000 50,000 1850 1860 1870 1880 1890 1900 Hare Lynx (a) Estimate the length of one period. (b) Estimate the maximum and minimum hare populations.

An object in simple harmonic motion has position function s1t2 inches from an equilibrium point, where t is the time in seconds. Find the amplitude, period, and frequency s1t2 = 4 sin pt

An object in simple harmonic motion has position function s1t2 inches from an equilibrium point, where t is the time in seconds. Find the amplitude, period, and frequency s1t2 = 3 cos 2t

In Exercise 55, what does the frequency represent? Find the position of the object relative to the equilibrium point at 1.5 sec, 2 sec, and 3.25 sec.

In Exercise 56, what does the period represent? What does the amplitude represent?