Valerie Russell
Assignment 1
Exploring Functions of the form y = a sin(bx + c)
Let's explore the graph of y = sin(x) and y = 2 sin(x) for x (the domain) in the interval [π , π ] and determine the range of the functions.
Exploring a
If we multiply the ycoordinate of every point on the graph of
y = sin(x) by 2, we obtain all of the points on the graph of
y = 2 sin(x). As a increases, y = sin(x) is stretched and the range
increases. When a = 0 we get y = 0 which is a horzontal line or the
xaxis.
Now let's see what happens when a has a negative value:
Notice that the graph of y = 2sin(x) is stretched by a factor of 2. The negative sign flips the sine function upside down or refects the sine function across the xaxis. 
Exploring b
Let's explore the graph of y = sin(x) and y = sin(3x).
We can see that y = sin(x) completes its cycle in the interval [0,3.14] whereas y = sin(3x) completes its cycle in the interval [0,2.09]. As b increases the period of the sine function decreases and viceversa.

Let's see what happens if b is a multiple of π:
Notice that one cycle of y = sin(π/2)x is completed in the interval [0,4] and y = sinπx is completed in the interval [0,2]. Anytime b is a multiple of π, the zeros of the sine function will be integers.

Hence, both a and b affect the shape of the curve. We saw that a affects the vertical stretch whereas b affects the period.
Exploring c
Now let's explore the graph of y=sinx and y=sin(x+π/6).
Notice that the graph of y = sin(x+π/6) is obtained by moving y = sinx a distance of π/6 (approximately .52) units to the left. The movement can be left or right and is derived by setting the expression in parenthesis equal to zero and solving for x. Ex: x + π/6 = 0 x = π/6


y = sinx and y = 3sin(2xπ)
If we factor out b and rewrite the function as y = 3sin[2(xπ/2)], we can see that a=3, b=2 and c= π/2. Notice a: The graph stretched vertically by a factor of 3. b: The period changed causing the cycle to shrink from 6.28 to 3.14 . c: The cycle shifted a distance of π/2 or 1.57 units to the right. 
Extension
y = a sin(bx + c) + d
What happens if we add a constant to the sine function?
Notice that y=3sin(2x  π) + 1 moved up 1 unit whereas y=3sin(2x  π)  2 shifted down 2 units. Adding a constant to the sine function shifts all the points vertically up or down. 
Hence, while both a and b affect the shape of the curve, c and d affect the curves location.
Here's an exercise:
Suppose the volume of air V in cubic centimeters in the lungs of a certain distance runner is modeled by the equation V = 400sin(60πx) + 900, where x is time in minutes.
a) What are the maximum and minimum volume of air in the runners lungs?
b) How many breadths does the runner take per minute?
Solutions:
a) Remember that c and d affect the curves location whereas d moved the curve up 900 cubic cm. from a, 400 cubic cm. The maximun volume of air the lungs of a certain distance runner would be 900 + 400 or 1300cc. whereas the minimum volume would be 900  400 or 500 cc.
b) The xaxis is the time axis so the amount of time required for the volume of air in a runners lung to complete 1 cycle is 2π/b and the number of cycles per unit of time is the reciprocal b/2π. Therefore, 60π/2π = 30 breadths/min.
RETURN TO HOME PAGE