Tiffany N. KeysŐ Assignment 1:

The graphs of all sine functions are related to the graph of y = sinx

This graph has the following characteristics:

¤     The domain of the function is all real numbers.

¤     The range of the function is -1 < y < 1 or [-1, 1]

¤     The function is periodic, meaning that its graph has a repeating pattern that goes on indefinitely.

¤     The shortest repeating portion of the pattern is called a cycle and the horizontal length of each cycle is the period.

¤     The amplitude of the function is half the vertical distance between its minimum and its maximum value.

INVESTIGATION: Examine graphs of y = a sin (bx +c) for different values of a, b, and c.

First, letŐs begin with substituting in values for a:

Y = sin(x)                     y = 2 sin(x)                      y = 3 sin(x)

Y = -sin(x)                    y = -2 sin(x)                     y = -3 sin(x)

OBSERVATIONS:

á     The graph of each function passes through the origin, (0,0).

á     Each graph is continuous along the x-axis and appears to repeat itself once it reaches a certain point on the x-axis.

á     The domain of all the graphs is all real numbers, however they reach have different ranges.

They are as follows:

Y = sin(x), [-1, 1]                        Y = -sin(x), [-1,1]

y = 2 sin(x) , [-2, 2]                    y = -2 sin(x), [-2,2]

y = 3 sin(x), [-3, 3]                     y = -3 sin(x), [-3,3]

á     The amplitude of the graph is affected when the value of a is changed. When a > 1, the slope of the graph to appears to be more steep.  When a < 1, the magnitude of the amplitude is the absolute value of a and the graph is reflected across the x-axis.

Next, letŐs observe what happens when different values for b are substituted in:

Y = sin(x)                           y = sin(2x)                             y = sin(3x)

Y = sin(-x)                    y = sin(-2x)                y = sin(-3x)

OBSERVATIONS:

á     Like the graphs of each function above, these also pass through the origin, (0,0).

á     They are also continuous along the x-axis and appear to repeat themselves once they reach a certain point on the x-axis.

á     The domain of all the graphs is all real numbers and they all have the same range of [-1, 1].

á     When b is changed in the function, the period is effected. When b > 1, the graph squeezes and the period becomes shorter.  When b < 1, the periods stayed the same and, like when a was assigned a negative value, the graph is reflected across the x-axis.

Lastly, letŐs notice the difference in the graph when values for c are substituted in:

Y = sin(x + 1)                      y = sin(x + 2)                   y = sin(x + 3)

Y = sin(x - 1)                y = sin(x - 2)                    y = sin(x - 3)

OBSERVATIONS:

á     Unlike the graphs above, these graphs do not pass through the origin.

á     They are, however, continuous along the x-axis and repeat themselves once they reach a certain point on the x-axis.

á     The domain of all the graphs is all real numbers and they all have the same range of [-1, 1].

á     The part of the graphs that are effected when the c value is changed in the function is the phase shift, which is the graphŐs movement along the horizontal axis.  When c > 1, the graph moves to the right along the x-axis and when c < 1, the phase shift is toward the left along the x-axis.