Effects of SinBy: 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 ofa, b,andcto see how it affects the sine graph.Below is y = sin(x). This is when

a= 1,b= 1, andc= 0. We see that the period is 2π and the amplitude is 1. Now let's see the effects of each parameter.

ParameteraFirst let's look at when

ais 1, 2, 3, and 5. We are leavingb=1 andc=0 so that we just see the effects of what the parameteradoes. 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 whenais greater than 1.Now let's look at when

a= 1/1, 1/2, 1/3, and 1/5.We are leavingb=1 andc=0 so that we just see the effects of what the parameteradoes. 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 whenais less than 1.Here we see what the effects of parameter

aare. The graph below shows the function y =asin(x) when -10 ≤a≤ 10. So from looking at this graph we see that whenais negative, it reflects the sine graph over the x-axis.

ParameterbFirst let's look at when

ais 1, 2, 3, and 5. We are leavinga=1 andc=0 so that we just see the effects of what the parameterbdoes. 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 whenbis greater than 1.Now let's look at when

b= 1/1, 1/2, 1/3, and 1/5. We are leavinga=1 andc=0 so that we just see the effects of what the parameteradoes. 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 whenbis less than 1.Here we see what the effects of parameter

bare. The graph below shows the function y =sin(bx) when -10≤ b≤ 10. So from looking at this graph we see that whenbis negative, it reflects the sine graph over the y-axis.

ParametercFirst let's look at when

cis -1, -2, 0, 1, and 2. We are leavinga=1 andb=1 so that we just see the effects of what the parametercdoes. 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. Ifcis negative then it will be shifted to the right. Itcis positive then it will be shifted to the left.Here we see what the effects of parameter

care. The graph below shows the function y =sin(x) +cwhen -10 ≤c≤ 10.