The Function of a Sine Curve and the Nature of its Parameters

by: Al Byrnes

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

Provide a **mathematical **interpretation of the * Parameters *a, b, and, c.

Explore using animations to illustrate the impact of each parameter.

Perhaps initailly not as intuitive as linear functions, understanding the effects of changing parameter values a, b, and c on the sine function **y = a sin (bx + c) **can be developed using programs such as Graphing Calculator or Desmos. The graph of the function **y = sin x** is shown below for reference.

This visualization shows a sine curve with parameter values of a = 1, b = 1, and c = 0. What were to happen to the curve if we manipulated the values of each of the parameter values individually? Let's investigate for the curve **y = a sin (bx + c)**

Micro-investigation 1: Varying the value of parameter a; values of parameter b = 1 and parameter c = 0 held constant.

In this micro investigation we are interested in developing a mathematical interpretation of the effect of varying the value of parameter **a** on the curve **y = a sin (bx + c)**. To this end we will examine several different values for parameter a in the sin function y = a sin x; below are illustrations of y = 1 sin x, y = 2 sin x, and y = 3 sin x, and y = 4 sin x

The distance from the x-axis to the peak of the trough seems to correspond to the value of the parameter **a, **eg: the peak of the trough of the sine curve is |a| from the x-axis. However, it is important to note that the points of intersection with the x-axis are not changed. This makes intuitive sense when some functions are set equal to zero and evaluated over the domain x = (0, π]

0 = sin(x) , x = π

0 = 2 • sin(x), x = π

0 = 3• sin(x), x = π

The point at which the value of the function y = a•sin(x) is zero is unaffected by the value of the parameter **a**. Therefore it we should not be suprized that multplication by a factor of** a **only affects the function when sin(x) does not equal 0.

Micro-investigation 2: Varying the value of parameter b; values of parameter a = 1 and parameter c = 0 held constant. Now let us investigate the effect on the shape of the sine curve when varying the value of parameter **b**. To get a base line for comparison, the function y = sin (x) is displayed below:

First, let's double the value of the parameter **b**, resulting in a new function: y = sin (2x); see the function graphed in the plane alone and in the second illustration with the functions y = sin (x) and y = sin (2x) graphed on the same plane:

... and now y = sin (x) and y = sin (2x) on the same graph:

What's happening here as we increase the value of the parameter **b** in the function y = sin (bx)? The y = sin(2x) curve appears to be squeezed together in such a way as half the wave period (note the value at which the y = sin(x) curve crosses the x-axis for the first time right of the origin appears to be approximately at x = 3.14, where the function y = sin(2x) seems to first cross the x-axis at a value half of the first intersection with the x-axis for the function y = sinx. Algebraically, this observation makes sense set y = 0 and compare the two equasions. Domain for x = (0, π]

0 = sin(x), x = π

0 = sin(2x), x = π/2, π

Lets try y = sin (3x):

... and compared to y = sin(x):

Here, we observe that the sine curve is "compressing" even more. This leads to another interesting algebraic connection. Set both functions y = sinx and y = sin3x equal to 0. Again, domain for x for this demonstration is x = (0, π]

0 = sin (x), x = π

0 = sin (3x), x = π/3, 2π/3, π

So as values of **b **continue to get larger, we can predict a corresponding "compression" of the sine curve, shortening of the wave period, and more solutions to the function y = sin(bx) over a given domain.

What might be a prediction for the effect on the sine function curve's shape if the parameter **b **is assigned values less that 1? My guess is a "stretch" of the curve, rather than the previously observed compression of the curve. The function y = sin(.5x) is shown below:

Now, y = sin (.5x) and y = sin(x) graphed in the same plane:

Indeed, our hunch seems to have been confirmed. The function's first intersection with the x-axis to the right of the origin is additionally where we might have expected it to be, a distance twice as far from the origin point. We can confirm this with algebra, by setting both y = sinx and y = sin.5x equal to 0 and considering possible solutions for the functions in the domainx = (0, 2π]:

0 = sinx, x = π and 2π

0 = sin x/2, x = 2π

Therefore, the parameter **b **can be interpreted as a compression or stretch, or in another way, a dialation of the sine curve by a factor of **b**.

Micro-investigation 3: Varying the value of parameter c; values of parameter a = 1 and parameter b = 1 held constant. For this micro investigation, we shift our attention to the final parameter of the sin function, **c**. For a basis of comparison, we should first take a look at the graph of the function y = sinx:

... and now the graph of y = sin(x + π/2) displayed along with the graph of y = sin(x + 0):

Note that the red sine curve, representitive of the function y = sin (x + π/2) seems to have "shifted" π/2 units to the left. What about y = sin (x + π)? y = sin (x + 3π/2)? y = sin (x + 2π)? How would these graphs of these functions look in the plane?

y = sinx and y = sin (x + π)

y = sin x and y = sin (x + 3π/2)

y = sin x and y = sin (x + 2π)

Note that with the final graph, the original function has just "shifted" or translated so far to the left that the function y = sin (x + 2π) (indicated in yellow) is sitting directly a top the function y = sinx. Alegbriacally, this translation makes sense as well: consider the functions y = sin x and y = sin (x + π/2), as shown in the graph below:

If evaluated at π, we will receive the following result:

y = sin (π) = 0

y = sin (π + π/2) = sin (3π/2) = -1

The addition of a positive parameter c should then be thought of as a translation of the curve -c units of the curve along the x-axis. Algebrically, this addition of c might also be thought of as a shift further along the unit circle; sin (x + c) an just be interpreted as the sine of the value of the angle x + a.

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