Use the graphing tool below to explore the basic characteristics of sine curves: period, amplitude, phase shift, and vertical shift. These characteristics are each related to a simple geometric transformation. When you are done you should be able to translate the characteristics of a situation involving periodic behavior into an equation and vice-versa.
The basic sin curve graphed above can be transformed by stretching or shrinking and by sliding it both horizontally and vertically with a few simple adjustments to the function. In general, a sin curve can be represented in the form:
y = asin(b(x - c)) + d
1) The values a, b, c, and d each relate to a separate transformation of the curve. Let's keep the experiment simple by focusing on each of the four values individually. To begin, set b = 1, c and d both equal to 0, and examine the result of varying a. The resulting equation has the form y = asin(x). Change the second equation in the grapher above and experiment with different values of a (2, .5, -1, ...). Write a simple sentence describing how the basic sin curve is transformed as you vary this value.
2) Try a similar experiment varying the value of d. Leave a and b equal to 1 and c = 0. The equation, therefore, has the form y = sin(x) + d. Again, write a simple sentence describing your results.
3) Experiment with the value of c and summarize your conclusions.
4) The role b plays is more difficult. It has to do with the period of the curve - how often the basic pattern recurs. Set a = 1 and c and b both to 0 so that the equation is in the form: y = sin(bx). What is the period of the basic curve, y = 1sin(x)? (HINT: Think π) Try b = 2, 4, 0.5. What happens to the period of the curve? Suppose you want the curve to repeat approximately every 25.13 (or 4π ) units on the x-axis. What value would you enter for b? Check.
5) Determine the equation of the sine curve with these characteristics and check by graphing:
- Stretched 4 units vertically (or has an amplitude of 4)
- Shrunk horizontally so that it has a period of π/2 units
- Slid vertically (or is shifted vertically) 4 units
- Slid horizontally (or is out of phase horizontally) 3 units
6) Now let's mix things up. Predict what the characteristics of each of the following curves will be and then check your prediction by graphing.
y = 4sin(x) + 2
y = -3sin(2x)
y - sin(x + π) - 2
Graphing tool from Desmos Calculator, (Demos, Inc., 2017), embedded December 23, 2017.