As the value of b changes, it puts an effect on the sin graph’s steepness of the waves. The value of b also indicates the amount of periods the graph is able to make between pi and 2π. The larger the value of b is, the number of waves increases. When the value of b is close to zero, the wider the curves are. The effects of *b *in y=sin(bx) makes sense mathematically because the value *b *would be altering the variable x. When a variable is multiplied to x, the characteristics of the graph would change due to x representing the placement of the graph.

These are a few examples of sin(bx) with different values for *b. *

**Table for the value of b and the resulting period of the function. **

By investigating and crunching the data for the value of b and the resulting period of the function, we can formulate a formula for the period of function y=sin(bx). Looking through the table, we can notice pi is always followed by either 2 or a factor of 2. The number dividing the xπ seems to be related to the value of b for it is either the same value or a factor of that value. In short, the formula for the period of the graph can be devised as sin(bx)= 2π/b. Take 3 for example as the value of b: if we insert the numbers into the formula [sin(bx)=2π/b] it would be 2π/3. This solution is the same value as the resulting period of the function. Another example can be 2: inserting the data into the formula we will result with 2π/2. In this case, we can cross out the 2 because they can be divided by each other, leaving the answer to be simply π. As shown in the table, even if the value of *b *is negative, it does not make the period value negative as well.

When the values of b are negative, the sin graph is upside down or flipped from its standard position. It will still show the same behavior of a positive sin graph but just in a opposite way. The starting point of 0,0 will not change but the sin graph will make its first curve from the zero point towards the negative numbers when the value of *b *is negative. But as I have brought up previously, the negative value of *b *does not make the period a negative. The comparison can be seen in this graph:

The graph of y=sin(bx-c) and y=sinb(x-c) are different because of the way the equations are set up. When the graph is y=sin(bx-c), the value of “c” varies from the value of “c” in the graph of y=sinb(x-c). This is because “c” is natural in the y=sin(bx-c) but “c” in the y=sinb(x-c) will have to mulitpy to the value of b, which causes a difference in the two graphs. We can observe how the graph of y=sin(bx-c) will be shifted in a larger way because it has the extra value that multiplies to the value of c. However, the value of *c *in both of the sin equations alters the horizontal shift but depending on the formula, the position will be different from each other. The graph of y=sin(bx-c) changes both the starting point of the sin graph and the resulting period of the function. On the other hand, the graph of y=sinb(x-c) it will only impact the starting point of the graph and will not necessarily affect the starting point.

The value of *b *in the graph of y=tan(bx) work as the position of the asymptotes are different and alters the period value. When the value of b is larger, the asymptotes appear to be closer to each other and has a shorter period throughout the graph. In contrary, the smaller the value of b, the more apart the asymptotes are and a longer period length is present. Also, when the value of *b *is negative in this tan equation, the graph flips from its original standard position, just like the negative value of *b *in y=sin(bx).

The *b *in y=tan(bx) changes the width of the graph. When the value of *b *is larger, there are more tan curves in a given period and are more steeper. The tan graph will have a longer period resulting the graph to be more wider. But when the value of *b* is smaller, the tan curves are more narrow because of the shorter length of the periods. When the values are negative, the graph is flipped, meaning the graph is going the opposite direction from the original y=tanx.

Formula for the period of y=tan(bx) can be defined as π/b. Compared to the formula for the period of y=sin(bx), it does not include the 2 in front of π. This is because in the standard position of the sin graph (or when it is y=sin(1x)), the period lasts until 2π unlike the tan graph which has a period of π. This changes the way the formula is arranged. Thus the period of y=tan(bx) is: the original period of y=tan(1x) [π] divided by the value of *b*. This table gives further justifications to the effectiveness of this tan period formula:

Not only does the value of *b *change the period length of the tan graph, but it also alters the position of the asymptotes. Asymptotes are vertical lines that is never touched with the tan graph line and are situated in between each of the periods. The formula *x=**(2n+1)*** ***π/2b *can be used to find the position of the asymptotes in the graph of y=tan(bx). The π/2b indicates how the asymptotes is placed at the value that is half of the period length. Since π/b is the period of a regular tan graph, the length half of a period would result as π/2b. The following part (2n+1) is included because the asymptotes in the tan graph are situated at odd multiples of periods instead of even multiples. Adding 1 to the solution to 2 multiplied to an integer will always end with an odd number. We can use y=tan(4x) to test this asymptote formula. Firstly, we’ll plug the value of b into the formula: x=(2n+1)π/2*4 which is x=(2n+1)π/8. Because *n *is flexible in being any integer,we could place any few numbers into the equation. In this case, I’d replace *n *with the numbers through 0~4 because they are small easy numbers. The four answers would be π/8, 3π/8, 5π/8 and 7π/8. Another example when y=tan(2x), the formula would be used as x=(2n+1)π/2*2 which becomes x=(2n+1)π/4. Now as I have inserted the same four values to n as y=tan(4x), I ended up with π/4, 3π/4, 5π/4 and 7π/4. When y=tan(2x) was graphed, same results of the asymptote was present.

In short, the formula x=(2n+1)π/2bcan enable one to find the asymptotes in y=tan(bx).

You have met the requirements of the assignment in that you have found the rules. Some rules are justified. There are a few mixed up explanations and typos which detract from the quality of the work. More detailed feedback and levels of achievement will be posted on Moodle and Powerschool shortly.