The Sine Function vs. The graph of y = 1/x.

One may not immediately see the similarities and difference between the graphs of y=sin(x) and y=1/x. But we will examine them closely.

Let us first look at the basic graphs of y = sin(x) and y = 1/x.

The graph of y = sin (x)
The graph of y = 1/x

What are the similarities and differences of the basic graphs?

Differences

One difference that can easily be seen is the vertical asymptote that is in y = 1/x and not in the graph of y = sin(x). The graph of y = sin(x) has a minimum and maximum value for y which ranges from -1 to 1. The graph of y = 1/x just cannot have a value for y = 0 or x = 0.

Now lets look at the similarities and differences when you add a constant k to the variable x. Let k = 5 for y = 1/(x + k) and let k = pi/2 for y=sin(x +/k).

The graph of y = sin(x + pi/2)
The graph of y = 1/(x+5)

 

Similarities	The graphs both shifted to the left k units from the basic graph. The limits and horizontal asymptotes on the graphs did not change. Both graphs maintained their shapes.
Differences	The vertical asymptote on y = 1/(x+5) changed k units to the left.

Now lets look at the similarities and differences when you add a constant k to the equation. Let k = 5 for y = (1/x) + k and let k = 3 for y=sin(x)+k.

y = sin(x)+3
y = (1/x) + 5

Similarities	The graphs both shifted up k units from the basic graph. The limits and horizontal asymptotes on the graphs changed k units up. Both graphs maintained their shapes.
Differences	The vertical asymptote on y = (1/x)+5 did not change.

 

The final comparison will be on what happens when you multiply the equation by some constant. We will let k = 5 for both equations.

y = 5sin(x)
y = 5*(1/x)
Similarities	The graphs changed.
Differences	The amplitude of the sine graph changed by a factor of 5.

Homepage

Write Up #2