# Determine the equation of a sine function in graph.? Relevance
• There may be some different approaches to reach the final answer. One key is to go step by step each time you understand what was changed from a "normal" sine.

Normally, a sine has an amplitude of 2 (it goes from -1 to +1).

This one seems to have an amplitude of 4 (it goes from -3 to +1).

Therefore, this graph is 2 times the sine of something

For now, let's call it

y = 2sin(w)

where w is some function of x.

The middle value of the sine is normally 0 (halfway between -1 and +1).

The middle of this one seems to be -1 (halfway between -3 and +1).

It is as if someone took a sine value and subtracted 1 from it

So far, we have

y = 2sin(w) - 1

This means: we know it is a sine, it was multiplied by 2 and then 1 was subtracted.

Normally, the sine has a period of 360 degrees. This one also has a period of 360 degrees (as far as I can tell from the graph), therefore there is no need to multiply the value of the variable. If the period had been (let's say) 60 degrees, we would say that the angle was multiplied by 6 and we would use 6w. But here, we do not have to do anything like that.

It stays (so far) as

y = 2sin(w) - 1

A sine curve normally crosses its middle axis (normally at y=0) at an angle of 0; here it "middle" is at y = -1 and it seems to cross it at around 130 (hard to see).

The normal sine reaches its maximum positive value at 90 degrees; here, the maximum positive value is around 220 degrees. It is as if the sine curve had been shifted to the right by 130 degrees (or to the left by 360-130 = 230).

Therefore, it is as if w = x - 130

We now have

y = 2sin(x-130) - 1

( you could also try y = 2sin(x+230) - 1 )

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It seems that someone took the value of x (the angle) and subtracted 130 degrees from it. Then the sine of that angle (x-130) was found, and multiplied by 2. Then take away 1 and plot the point.

So let's look at a few points to see if this works.

x = 0

x - 130 = -130

sin(-130) = -0.766

2sin(-130) = -1.532

2sin(-130) - 1 = -2.532

Does it look OK on the graph?

Try it again with another value for x (pick easy ones)

x = 180

x-130 = +50

sin(50) = 0.766

2sin(50) = 1.532

2sin(50) - 1 = 0.532

Does it look OK on the graph?

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Looking at the graph, it is difficult to be sure that the shift of the angle is 130 degrees. It could be anything between 120 and 140.  You could try to find the "zeros" by looking at the exact values of x when the graph crosses the x-axis (these are the spots where y=0). In the old days, this would be done by changing the value 130 a bit on either side and see if it fits.

• Anonymous
2 months ago

The sine graph is shifted.