Effects of Sin

By: Russell Lawless

Sine is a function that is the vertical distance from the horizontal diameter of a circle. It is one of the three basic trigonometric functions that high school students know. The generic equation for it is y = asin(bx+c). For this exploration we will be looking at the parameters of a, b, and c to see how it affects the sine graph.

Below is y = sin(x). This is when a = 1, b = 1, and c = 0. We see that the period is 2π and the amplitude is 1. Now let's see the effects of each parameter.

Parameter a

First let's look at when a is 1, 2, 3, and 5. We are leaving b=1 and c=0 so that we just see the effects of what the parameter a does. We see that our graph seems to be stretching in the vertical direction. What it is doing is increasing our amplitude of the sine function. This occurs for when a is greater than 1.

Now let's look at when a = 1/1, 1/2, 1/3, and 1/5.We are leaving b=1 and c=0 so that we just see the effects of what the parameter a does. We see that the graph seems to be shrinking in the vertical direction. What it is doing is decreasing our amplitude of the sine function. This occurs for when a is less than 1.

Here we see what the effects of parameter a are. The graph below shows the function y = asin(x) when -10 ≤ a ≤ 10. So from looking at this graph we see that when a is negative, it reflects the sine graph over the x-axis.

Parameter b

First let's look at when a is 1, 2, 3, and 5. We are leaving a=1 and c=0 so that we just see the effects of what the parameter b does. We see that our graph seems to be shrinking in the horizontal direction. What it is doing is increasing our period of the sine function. This occurs for when b is greater than 1.

Now let's look at when b = 1/1, 1/2, 1/3, and 1/5. We are leaving a=1 and c=0 so that we just see the effects of what the parameter a does. We see that the graph seems to be stretching in the horizontal direction. What it is doing is increasing our period of the sine function. This occurs for when b is less than 1.

Here we see what the effects of parameter b are. The graph below shows the function y =sin(bx) when -10 ≤ b ≤ 10. So from looking at this graph we see that when b is negative, it reflects the sine graph over the y-axis.

Parameter c

First let's look at when c is -1, -2, 0, 1, and 2. We are leaving a=1 and b=1 so that we just see the effects of what the parameter c does. We see that our graph seems to be shifting in the horizontal direction. What it is doing is moving our graph to the left or the right depending on the sign of the constant. If c is negative then it will be shifted to the right. It c is positive then it will be shifted to the left.

Here we see what the effects of parameter c are. The graph below shows the function y =sin(x) + c when -10 ≤ c ≤ 10.