Algebra 2 - Graphs of Sinusoidal Functions

A sine function has the following key features: Period = 4 Amplitude = 3 Midline: y=−1 y-intercept: (0, -1) The function is not a reflection of its parent function over the x-axis.

(0,-1) and (1,2)

Graph the function. f(x)=sin(πx2) Use the sine tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point.

(0,0) and (1,1)

A sine function has the following key features: Period = 4 Amplitude = 4 Midline: y = 1 y-intercept: (0, 1) The function is not a reflection of its parent function over the x-axis.

(0,1) and (1,5)

A researcher observes and records the height of a weight moving up and down on the end of a spring. At the beginning of the observation the weight was at its highest point. From its resting position, it takes 8 seconds for the weight to reach its highest position, fall to its lowest position, and return to its resting position. The difference between the lowest and the highest points is 20 in. Assume the resting position is at y = 0.

(0,2) and (4,-10)

At an ocean depth of 20 meters, a buoy bobs up and then down 2 meters from the ocean's depth. Four seconds pass from the time the buoy is at its highest point to when it is at its lowest point. Assume at x = 0, the buoy is at normal ocean depth.

(0,20) and (2,22)

A sine function has the following key features: Period = 4π Amplitude = 2 Midline: y = 3 y-intercept: (0, 3) The function is a reflection of its parent function over the x-axis.

(0,3) and (3.14,1)

A sine function has the following key features: Frequency = 1/6π Amplitude = 2 Midline: y = 3 y-intercept: (0, 3) The function is not a reflection of its parent function over the x-axis.

(0,3) and (4.71,5)

A researcher observes and records the height of a weight moving up and down on the end of a spring. At the beginning of the observation the weight was at its highest point. From its resting position, it takes 12 seconds for the weight to reach its highest position, fall to its lowest position, and return to its resting position. The difference between the lowest and the highest points is 10 in. Assume the resting position is at y = 0.

(0,3) and (6,-5)

A sine function has the following key features: Frequency = 1/8π Amplitude = 6 Midline: y = 3 y-intercept: (0, 3) The function is not a reflection of its parent function over the x-axis.

(0,3) and (6.28,9)

A researcher observes and records the height of a weight moving up and down on the end of a spring. At the beginning of the observation the weight was at its highest point. From its resting position, it takes 20 seconds for the weight to reach its highest position, fall to its lowest position, and return to its resting position. The difference between the lowest and the highest points is 6 in. Assume the resting position is at y = 0.

(0,5) and (10,-3)

At an ocean depth of 8 meters, a buoy bobs up and then down 5 meters from the ocean's depth. Sixteen seconds pass from the time the buoy is at its highest point to when it is at its lowest point. Assume at x = 0 the buoy is at normal ocean depth.

(0,8) and (8,13)

What is the frequency of the function f(x)? f(x)=−2sin(x/4)+3 Express the answer in fraction form.

1/8pi

The average daily maximum temperature for Laura's hometown can be modeled by the function f(x)=4.5sin(πx/6)+11.8, where f(x) is the temperature in °C and x is the month. x = 0 corresponds to January. What is the average daily maximum temperature in May? Round to the nearest tenth of a degree if needed. Use 3.14 for π.

15.7 °C

What is the amplitude of this function f(x) ? f(x)=−2sin(3x)−1

2

The table shows the height in centimeters, that a weight bouncing from a spring would achieve if there were no friction, for a given number of seconds. Time (s) Height (cm) 0 0 0.75 15 1.5 0 2.25 −15 3 0 3.75 15 4.5 0 5.25 −15 6 0 From its resting position, how long does it take the weight to bounce one direction, then the other, and then back to its resting position?

3 s

Over a 24-hour period, the temperature in a town can be modeled by one period of a sinusoidal function. The temperature measures 70°F in the morning, rises to a high of 80°F, falls to a low of 60°F, and then rises to 70°F by the next morning. What is the equation for the sine function f(x), where x represents time in hours since the beginning of the 24-hour period, that models the situation?

f(x) = 10sin(pix/12)+70

A cosine function has a period of 5, a maximum value of 20, and a minimum value of 0. The function is not a reflection of its parent function over the x-axis. Which function could be the function described?

f(x)=10cos(2π/5 x)+10

A sinusoidal function whose frequency is 1/6π, maximum value is 12, minimum value is −6 has a y-intercept of 3. Which function could be the function described?

f(x)=9sin(x/3)+3

The graph of f(x)=cos(x) is transformed to a new function, g(x), by reflecting it over the x-axis and shifting it 2 units down. What is the equation of the new function g(x)?

g(x)= -cos(x)-2

What is the equation of the midline for the function f(x)? f(x)=1/2cos(x)+5

y = 5

What is the period of the function f(x)=cos2x?

π pi

Graph the function. f(x)=−2sin(x) Use the sine tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point.

(0,0) and (1.57, -2)

A sine function has the following key features: Period = 12 Amplitude = 4 Midline: y = 1 y-intercept: (0, 1) The function is not a reflection of its parent function over the x-axis.

(0,1) and (3,5)

At an ocean depth of 10 meters, a buoy bobs up and then down 6 meters from the ocean's depth. Ten seconds pass from the time the buoy is at its highest point to when it is at its lowest point. Assume at x = 0 the buoy is at normal ocean depth.

(0,10) and (5,16)

A sine function has the following key features: Frequency = 1/4π Amplitude = 2 Midline: y = 2 y-intercept: (0, 2) The function is not a reflection of its parent function over the x-axis.

(0,2) and (3.14,4)

The temperature of a liquid during an experiment can be modeled by the function f(x)=3.8cos(πx/20)+2.2, where f(x) is the temperature in °C and x is the number of minutes into the experiment. What is the lowest temperature the liquid reached during the experiment? Round to the nearest tenth of a degree if needed. Use 3.14 for π.

-1.6 °C

The graph shows the distance y, in inches, a pendulum moves to the right (positive displacement) and to the left (negative displacement), for a given number of seconds x. How many seconds are required for the pendulum to move from its resting position and return? max -> 5 min -> -5 midline -> 0

1 s

In the function f(x), x is replaced with 3x. f(x)=1/2sin(x)−4 What effect does this have on the graph of the function?

The graph is horizontally compressed by a factor of 1/3.

A sine function has the following key features: Period = π Amplitude = 2 Midline: y=−2 y-intercept: (0, -2) The function is a reflection of its parent function over the x-axis.

(0,2) and (1,-4)

Graph a sine function whose amplitude is 3, period is 4π, midline is y = 2, and y-intercept is (0, 2). The graph is not a reflection of the parent function over the x-axis. Use the sine tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point.

(0,2) and (3.14, 5)

A sine function has the following key features: Period = 4π Amplitude = 3 Midline: y = 2 y-intercept: (0, 2) The function is a reflection of its parent function over the x-axis.

(0,2) and (3.14,-1)