Periodic Functions
A periodic graph has a maximum value of 14 and a minimum value of 2.
What is the amplitude and the midline of the function?
Amplitude = 6, Midline = 8
In the equation y=asin[k(x−d)]+c, which variable controls the vertical translation of the sinusoidal graph?
the variable c
Given the function f(x) = 7sin[2(x-9)]+5, find the maximum and minimum values of the function.
12 and -2
A ferris wheel has a radius of 40m. Its center is 45m above the ground. What is its amplitude?
40m
Which part of a periodic graph shows how long it takes for the pattern to repeat?
Period
A sinusoidal graph has the same amplitude as the parent function y=sinx but a shorter period. Which parameter must have changed, and explain how this change affects the graph.
the variable k must have changed. Increasing the value of k causes a horizontal compression, which makes the graph complete its cycles more quickly and results in a shorter period.
A ferris wheel's motion is expressed as a sinusoidal function:
f(x) = 3cos[(3x-12)] -2
What is the domain and range of the ferris wheel?
D : {ХER}
R : {YER|−5≤y≤1}
A wave has a maximum height of 15 m and a minimum height of 7 m. The wave completes 5 cycles in 35 seconds.
a) What is the amplitude?
b) What is the period?
a)4 m
b) 7 s
What is the amplitude of the function - y= 5 sin (x) -3
5
A sinusoidal graph has a shorter period than the parent sine function but the same amplitude and midline. Which variable in the equation most likely caused this change?
the k value
A scientist who is tracking the temperature during the day models a function of g(x) = 3cos[2(x-5)]+1, where x is the hour during the day the temperature is measured . Find the equivalent function for sine.
f(x) = 3sin[2(x+40)]+1
A student receives between 6 and 14 Instagram notifications each day.
What is the midline (C) and amplitude (A) of their notifications?
Midline: 10, Amplitude:4
What does the midline of a periodic function represent?
The center value the graph oscillates around
Starting from the parent function y=sinx, graph the function y=2sin(x−45)+3. Describe and apply each transformation step-by-step, including amplitude, horizontal translation, and vertical translation.
The amplitude is 2, so the graph is vertically stretched by a factor of 2. The expression (x−45) causes a horizontal shift 45 degrees to the right. The +3 shifts the graph upward by 3 units, moving the midline to y=3. After applying these transformations to the parent sine function, the transformed key points are plotted and connected with a smooth sinusoidal curve.
A student’s energy throughout a 12-hour school day is modeled by the function:
E(x)=3cos(4x)+5, where x is the number of hours since the student wakes up.
How many full energy cycles will she experience during the 12-hour day?
4 full energy cycles
On a math trip to Canada’s Wonderland, students were given the sinusoidal function: y=4cos(2/5(x+35))+18, that models the height of a roller coaster car, where x represents the distance travelled along the track in meters.
What distance corresponds to one full cycle of the motion?
900 m
What does the period of a periodic function describe?
How long it takes for the graph to repeat one full cycle
Explain, using proper mathematical vocabulary, how mapping notation can be used to transform the key points of the parent sine function to graph a transformed sinusoidal function.
Mapping notation shows how each point on the parent function changes under transformations. For a function in the form y=asin[k(x−d)]+c, the mapping rule is
(x,y) → (x/k+d, ay+c). This rule is applied to each key point of the parent sine function to find the new coordinates, which are then plotted to create the transformed graph
When a student analyzes the temperature pattern, she models it in a sinusoidal function :
T(x) = 7sin[15(x-6)] + 19, where x is the hour the temperature is measured at. What will the temperature be at 8 am?
22.5°C
The school soccer team is analyzing temperatures before their big game on Friday. Over the past week, temperatures were recorded as follows:
Monday (Day 1): -7°C
Sunday (Day 7): 11°C
The temperature over the week is modeled by function:
T(d)= -9cos(30(d-1))+2
(Where d represents the day of the week)
Use the function to find the predicted temperature on Friday. Round to the nearest whole number.
7°C