Explorations of a Sinusoidal Curve
By: Sydney Roberts
LetÕs start of by looking at the graph of y = sin(x).
We can see that this function is a sinusoidal curve that alternates between the y values of -1 and 1 and passes through the point (0,0).
To interpret the parameters of a,b, and c I think itÕs best to see what each parameter does to this base function: y = sin(x) without the other parameters being added in.
Therefore, letÕs explore the graph of y = a sin(x) for different values of a.
At this point we can hypothesize that the parameter a changes the vertical height of the sinusoidal curve. In mathematical terms, the parameter a gives us the value of the vertical stretch as compared to the base function.
For example, if a=5 then our function y = sin(x) is being stretched vertically by a factor of 5 and will now alternate between -5 and 5.
Now what would happen if a is a negative number? Well letÕs compare.
The only difference here is that in the second set, the functions have been reflected across the x-axis. (Why the x-axis? Well, the negative sign is attached to the entire function, i.e. the negative sign affects the dependent variable. So if the dependent variable ÒyÓ is equal to one, using a negative sign in front of the function – which is equal to the dependent variable – now makes y = -1. Hence, y is now reflected across the x-axis. So the negative sign attached to the entire function will make the dependent variables reflect across the x-axis). We can see that the sign of this parameter, a, will represent whether or not the function has been reflected about the x-axis or not. Notice that the sign does not affect the height of the function. The height of a sine function is typically referred to as the Òamplitude.Ó So we can say that |a| will give us the amplitude of the function
Now letÕs move on the parameter, b. LetÕs examine the graph of the function y = sin(bx) for different values of b.
At first these graphs look muddled and hard to interpret. However, letÕs first focus on the purple graph of the function y = sin(x). Notice that the period of this function is 2¹. That is, the graph repeats every 2¹. Now look at the red graph of the function y = sin(2x). This function repeats twice within 2¹, so we know that the period is one half of 2¹ (or ¹). The final blue graph of the function y = sin(4x) repeats 4 times within 2¹, so the period should be one fourth of 2¹ (or π/2 ). Can you notice the relationship between b and the period, yet? Maybe not. LetÕs consider more graphs.
Note that y = sin(3x) repeats 3 times within 2¹ and y = sin(10x) repeats 10 times within 2¹. Now the relationship should be clear. Before drawing any conclusions though, we should investigate what happens if b is negative.
Again, we see that the negative sign only causes a reflection (this time about the y-axis) of the function. Why about the y-axis? Well this negative sign in explicitly attached to the ÒxÓ or the independent variable. Then every value of x that was positive, is now negative. So 2 is reflected about the y-axis to -2, 4 is reflected to -4, etc. So each x value is now reflected across the y-axis.
So now letÕs state that the relationship between b and 2¹ seems to determine the period of our sine function. More specifically, we can see that the period, p, can be found by
In the case of the sine function, the period can also be referred to as the horizontal stretch/compression along the x-axis where an increase in |b| will cause the function to compress more and more compared to the base function.
Now we can move onto the parameter c. Again, letÕs explore through the graphs of different functions. This time, however, I think it would be best to graph one function at a time.
What is happening now as c becomes more and more positive? It appears that the graph is ÒshiftingÓ to the left. In mathematical terms, we call this a translation. Why though, does an increase cause the graph to ÒdecreaseÓ down the x-axis? (i.e. Why does x+1 cause the graph to move -1 units to the left?)
Well this takes a bit more exploring. The easiest intuitive reason I can give for this involves Daylight Saving Time. In the spring, we set our clocks forward one hour. So essentially we are changing from time ÒxÓ to time Òx+1Ó. But what does this mean for the hours in the day that the clocks change? Well now we have one LESS hour during that day.
Well if this is true, does it not make sense that if c were to become more and more negative, then the graph would make a horizontal translation in the positive direction? Well letÕs see.
Now it appears that our theories were correct. A one unit decrease in c causes a horizontal translation in the negative direction. Hence, we know that c gives the horizontal translation in the function .
Putting all of this together, what do you think the graph of y = 2sin(5x+3) would look like? Well using what we learned, we know that this sinusodial curve will have an amplitude of 2, a period of 2π/5, and has a horizontal translation of 3 in the positive direction. Does the following graph match this description?
Hopefully, your answer is yes.
Now that we have covered all the different parameters a, b, and c we should ask ourselves, ÒDoes it matter that we only considered that a, b, and c were only represented as integers?Ó Well, no. The absolute value of a will always represent the amplitude of the function, the period will always be 2π/|b| , and c will always represent the horizontal translation of the function. These facts do not depend on what classification the parameters are. Whether they are integers, rational numbers, irrational numbers, etc., it doesnÕt matter.
Also, for further exploration, examine how the parameter d would affect the sine function when y = a sin(bx+c) + d.